NUMERICAL FINITE HORIZON RUIN PROBABILITIES IN THE CLASSICAL RISK MODEL WITH STOCHASTIC RETURN ON INVESTMENTS

NUMERICAL FINITE HORIZON RUIN PROBABILITIES IN THE CLASSICAL RISK MODEL WITH STOCHASTIC RETURN ON INVESTMENTS Lubasi Simataa M.Sc. (Mathematical Modelling) Dissertation University Of Dar es Salaam June, 2011

NUMERICAL FINITE HORIZON RUIN PROBABILITIES IN THE CLASSICAL RISK MODEL WITH STOCHASTIC RETURN ON INVESTMENTS By Lubasi Simataa A Dissertation Submitted in (Partial) Fulfillment of the Requirements for the Degree of Master of Science (Mathematical Modelling) of the University of Dar es Salaam University Of Dar es Salaam June, 2011

i CERTIFICATION The undersigned certify that they have read and hereby recommend for acceptance by the University of Dar es Salaam the dissertation entitled: Numerical Finite Horizon Ruin Prbabilities in the Classical Risk Model with Stochastic Return on Investments, in partial fulfillment of the requirements for the degree of Master of Science (Mathematical Modelling) of the University of Dar es Salaam. Dr. MAHERA WILSON CHARLES ( Supervisor) Date:... Dr. JUMA KASOZI (Supervisor) Date:...

ii DECLARATION AND COPYRIGHT I, Lubasi Simataa, declare that this dissertation is my own original work and that it has not been presented and will not be presented to any other University for a similar or any other degree award. Signature: - This dissertation is copyright material protected under the Berne Convention, the Copyright Act 1999 and other international and national enactments, in that behalf, on intellectual property. It may not be reproduced by any means, in full or in part, except for short extracts in fair dealings, for research or private study, critical scholarly review or discourse with an acknowledgement, without the written permission of the Director of Postgraduate Studies, on behalf of both the author and the University of Dar es Salaam.

iii I am so much honoured to thank Almight God for keeping me healthy and taking me through my course successfully. My special thanks go to Dr. Wilson Mahera Charles and Dr. Juma Kasozi my supervisors for always being there and providing many suggestions and comments at all stages of this study.

iv DEDICATION To God The Almight and my family.

v ABSTRACT This study deals with numerically computing finite horizon ruin probabilities in insurance business with the financial world in mind. The risk process in this study has its origin in the classical risk process(a constant income stream from which is subtracted a claim process that is assumed to be a compound Poisson process). This classical risk process is then compounded by an independent return on investments process of the Black and Scholes type. The uncertainity in both processes is provided by a standard Brownian motion. We derive partial differential equations and use an implicit finite difference scheme to numerically solve the equations for finite time ruin probabilities in the absence of jumps and in the presence of jumps.

vi TABLE OF CONTENTS Certification..................................... i Declaration and Copyright............................. ii Aknowledgement iii Acknowledgements................................. iii Dedication...................................... iv Abstract....................................... v Table of Contents.................................. vi List of Figures................................... ix List of Abbreviations................................ x CHAPTER ONE: INTRODUCTION 1 1.1 General Introduction............................. 1 1.2 Examples of stochastic processes...................... 5 1.3 Investment.................................. 18 1.4 Black-Scholes investment model....................... 19 1.5 Statement of the problem.......................... 20 1.6 Research objectives............................. 20 1.7 General objective.............................. 20 1.8 Specific objectives.............................. 20

vii 1.9 Significance of the study.......................... 21 CHAPTER TWO: LITERATURE REVIEW 22 CHAPTER THREE:THE MODEL AND THEORETICAL RESULTS 28 3.1 Introduction................................. 28 4 Model formulation 28 5 The case λ P = 0 (i.e. in the absence of jumps or claims) 31 6 Limits and Continuity of Functions 34 6.1 Limits of Functions............................. 34 6.2 Continuous Functions............................ 36 7 Singularities 39 8 The case λ P 0 (i.e. in the presence of jumps or claims) 41 CHAPTER FOUR: MODEL SIMULATIONS 42 5 Introduction 42 6 Numerical Methods 42 7 Implicit- finite difference scheme 43 8 Boundary conditions 48

viii 9 Numerical integration 51 9.1 The case λ P 0 (i.e. in the presence of jumps or claims)........ 52 9.2 Simpson s Rule............................... 52 CHAPTER FIVE: NUMERICAL RESULTS 57 6 Introduction 58 7 The case λ P = 0 (i.e. in the absence of jumps or claims) 58 8 The case λ P 0 (i.e. in the presence of jumps or claims) 62

ix List of Figures 1.1 Sample path of geometric Brownian motion.............. 9 1.2 Sample path of wiener process or Brownian motion........... 13 5.1 Domain 1.................................. 32 5.2 Domain 2.................................. 33 7.1 PDE solution grid............................. 43 7.2 Pictorial representation of the scheme.................. 47 8.1 Pictorial representation of the scheme in our model........... 50

x LIST OF ABBREVIATIONS PDE Partial Differential Equation

1 CHAPTER ONE INTRODUCTION 1.1 General Introduction. Risk theory is an important issue in financial markets. Insurance companies are important financial intermediaries which have to use this theory to hedge risk against a contingent loss. An actuarial risk model is a mathematical description of the behavior of a collection of risks generated by an insurance portfolio. Determining the probability of ruin and time to ruin are important problems in classical risk theory. Lundberg s (1903) pioneer work in risk theory received rigorous mathematical treatment first by Cramer in (1930) and (1955) and later, by many authors. Lundberg s contributions were presented in his monograph Collective Risk Theory. Lundberg s model, expounded by Cramer, is termed the Cramer-Lundberg model or the classical risk model. More generalisations of this model exist and most of the research has been inclined to computing ultimate (infinite) ruin probabilities in preference to finite ruin probabilities. This study is on finite ruin probabilities which we will define in a sequel. Amongst other things, insurance companies exist to pool together risks faced by individuals such that those who may experience a loss at any one time receives compensation that would help to alleviate the financial consequences of the loss. Because of this it is vital for insurance companies to set aside an amount of money, as reserves or surplus so that it would be able to meet its commitments to pay claims whenever they occur. This protection is accomplished through a pooling mechanism whereby many individuals who are vulnerable to a particular risk are joined together into a risk pool. Each person pays a small amount of money, known as premium into the pool, which in turn is used to compensate the unfortunate individuals who do actually suffer a loss. Insurance reduces vulnerability by replacing the uncertain prospect of large losses with the certainty of making small, regular premiums. At

2 some point one is very likely to hold an insurance policy. Be it car, holiday and house. Despite this we do not often think about the mathematics behind our policies. The insurer runs the risk that the premiums they charge will not be enough to pay for the claims they receive, leaving them incapacitated to meet this obligation. The problem of finding the probability of ruin was first considered by Lundberg (1903). Since then the problem has received much attention up to present day. In his study, Lundberg considered a surplus model of the type: Surplus = Initial reserve + P remium income claims paid. (1) One question that has received much attention is what is the probability that the surplus process ever becomes negative? The first time when this happens is termed the time of ruin and the associated probability is the probability of ruin. Ruin is considered as a technical term. It does not mean that the company is bankrupt. However, if ruin occurs it is interpreted as meaning that the company has to take action in order to make the business profitable. The Cramer-Lundberg model serves as a skeleton for more realistic models that have been studied in the insurance literature. This standard model for non-life insurance is simple enough to calculate probabilities of interest, but too simple to be realistic. For example, it does not include interest earned on the invested surplus. There are several papers treating this model in many directions and forms, all with a view of finding the probability of ruin. Several authors have developed schemes for calculating finite time ruin probability. For the classical surplus process, recursive algorithms for the calculation of this probability have been developed by, for example, De Vylder and Goovaerts (1988) and Dickson and Waters (1991). In their algorithms, the time plane is replaced by a rectangular grid and the continuous surplus process is approximated by a discrete process. Recursive algorithms for a model that incorporates a constant force of interest were derived by Dickson and Waters (1999) using discretisation ideas developed by De Vylder and Goovaerts (1988) and Dickson and Waters (1991). An improvement to the algorithms above was done by Cardoso and Waters (2003). Cardoso and Waters approximate or bound the continuous surplus process by discrete

3 time Markov chains with countable state spaces. For a general background to ruin theory, we refer to Buhlmann (1970). The following terminologies will be defined for us to understand the business of insurance in general. Definition 1.1. Insurance is the mechanism whereby risk of financial loss is transferred from individual, company, organization or other entity to an insurance company. Definition 1.2. Risk is the quantifiable likelihood of loss or less than expected returns. In other words, risk is the danger or probability of loss to an investor. Definition 1.3. Risk Theory is a synonym for non- life insurance mathematics, which deals with the modeling of claims that arrive in an insurance business and which gives advice on how much premium has to be charged in order to avoid bankruptcy (ruin) of the insurance company. (Mikosch,2003). Definition 1.4. Portfolio is a collection of investment held by an institution or a private individual. Definition 1.5. A Claim is a demand made by the insured, or the insured s beneficiary for payment of the benefits as provided by the insurance policy. Definition 1.6. Insurance policy is a legal document outlining a particular insurance cover for an insured entity for a given risk. Definition 1.7. Claim number process, N t is the number of claims which occurred by time t. (Mikosch,2003). Definition 1.8. Surplus is the amount by which assets exceed liabilities. Definition 1.9. Ruin Theory is a tool to assess the probability that the company can deal with risks and avoid ruin through time. (Mikosch,2003) Definition 1.10. Time horizon is the length of time for which an investment is made, or held before it is liquidated.

4 A stopping rule problem has a finite horizon if there is a known upper bound on the number of stages at which one may stop. If stopping is required after observing X 1, X 2,...X T, we say the problem has horizon T. Claims arrive at an insurance business randomly or in unforeseen circumstances. Because of this, financial losses happen abruptly. It is for this reason that we bring the concepts or definitions from probability theory for mathematical analysis of an insurance business. Definition 1.11. Sample space, Ω is the collection or totality of all possible outcomes of a conceptual experiment. (Mood et al.,1963) Definition 1.12. A σ-algebra, is a collection F of subsets of Ω with the following properties: 1.,Ω F 2. If A F, then A c F. 3. A 1, A 2,... F, then k=1 A k, k=1 A k F. Here A c :=Ω A is the complement of A. (Evans, 2004) NOTE: Consider an event A Ω, F is a collection of all events. Events are subsets of the sample space. A collection of events A 1, A 2,... is called σ-algebra. Definition 1.13. A random variable. A random variable is a real function X(ω), (ω Ω is measurable with respect to a probability measure P) That is X : Ω R. (Chuasan and Somesh, 1997) Definition 1.14. Definition 1.15. A probability measure. Let F be a σ-algebra of subsets of Ω. We call P [0, 1] a probability measure provided: 1. P( )=0, P(Ω)=1

2. If A 1, A 2,... is a sequence of disjoint sets in F, then 5 P( A k ) k=1 P(A k ) k=1 3. If A 1, A 2,... are disjoint sets in F, then P( A k ) = k=1 P(A k ) k=1 It follows that if A,B F then A B implies P(A) P(B). (Evans, 2001) Definition 1.16. A Probability space. The triple (Ω, F, P) is called a probability space. (Oksendal, 2000). Definition 1.17. A filtration, {F} t 0. Let Ω be a non-empty finite set. A filtration is a sequence of σ-algebras such that each σ-algebra in the sequence contains all the sets contained by the previous σ-algebra. (Chuasan and Somesh, 1997) Definition 1.18. Stochastic basis is a quadruple (Ω, F, P, {F} t 0 ) An essential branch of mathematics that deals with modeling uncertainities, in this case modeling the insurance process will be introduced at this stage. This branch is called Stochastic processes. Definition 1.19. A stochastic process is a parameterised collection of random variables {N t } t T defined on a probability space (Ω, F, P) and assuming values in IR n We shall let T = [0, ). (Oksendal,2000) 1.2 Examples of stochastic processes There are many examples of stochastic processes which are used in real life to model uncertainness depending on the situation at hand. These include Poisson process, Wiener process or Brownian motion, Geometric Brownian motion, to mention but

6 a few. We shall restrict our selves to the ones listed and define them since they are important in our study. 1. Poisson process Definition 1.20. A Poisson process is a stochastic process N = {N t } t 0 which satisfies the following conditions; (a) The process starts at zero;n 0 = 0. (b) The process has independent independent increments. That is, for any t i, i=0,...,n, and n 1 such that 0 = t 0 < t 1 <... < t n, the increments N (ti 1,t i ), i = 1,..., n, are mutually independent. (c) There exists a non-decreasing right continuous function µ : [0, ) [0, ) with µ(0) = 0 such that the increments N (s,t] have a Poisson distribution P ois(µ(s, t]). We call µ the mean value function of N. (d) With probability 1, the sample paths {N (t,ω) } t 0 of the process N are right continuous for t 0 and have limits from the left for t > 0. We say that N has càdlàg sample paths. (Mikosch,2003). Definition 1.21. A Poisson process with a mean value function µ defined as µ(t) = λt, t 0 for some λ > 0 is said to be homogeneous, inhomogeneous otherwise. The quantity λ is the intensity or rate of the homogeneous Poisson process. If λ = 1, then N is called a standard homogeneous Poisson process. A homogeneous Poisson process with intensity λ has (a) càdlàg sample paths,

7 (b) starts at zero, (c) has independent and stationary increments. By stationary increments, we mean that for any 0 s t and h > 0, N (s,t] has the same distribution as N (s+h,t+h] (d) N t is P ois(λt) distributed for every t > 0. Geometric Brownian motion Definition 1.22. A Geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion or a Wiener process. A stochastic process S t is said to follow a Geometric Brownian Motion if it satisfies the following stochastic differential equation. ds t = µs t dt + σs t db t (2) where B t is a Wiener process or Brownian motion, µ is the percentage drift and σ is the percentage volatility and they are constants. (Geometric Brownian Motion, Wikipedia,2011). Geometric Brownian motion has gained a lot popularity in the finance sector to model prices of risky assets due to its very important property of not taking on negative values. Brownian motion was highly criticized for taking on negative values yet prices are non-negative. If the price of an asset follows equation (1.2), then the price at any time t is given by S t = S 0 exp{(µ 1 2 σ2 )t + σb t } (3) where µ is the rate of appreciation of S t. Figure 1.3 is a sample path of Geometric Brownian motion with σ = 0.1 and µ =

8 0.2. In insurance and finance language, µ is the potential profit where as σ is the risk exposure. Geometric Brownian motion has applications in finance, insurance, mathematical epidemiology, to mention but a few.

9 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Figure 1.1: Sample path of geometric Brownian motion Lévy process In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits cádlág modification and has stationary independent increments this phrase will be explained below. They are a stochastic analog of independent and identically-distributed random variables, and the most well-known examples are the Wiener process and the Poisson process. Definition 1.23. A stochastic process is said to be a Lévy process if, (i) N 0 = 0 almost surely (ii) Independent increments: For any 0 t 1 < t 2 <...t n <, N t2 N t1, N t3 N t2,...n tn N tn 1 are independent. Note. A continuous-time stochastic process assigns a random variable N t to each point t = 0 in time. In effect it is a random function of t. The increments of such a process are the differences

10 N s N t between its values at different times t < s. To call the increments of a process independent means that increments N s N t and N u N v are independent random variables whenever the two time intervals do not overlap and, more generally, any finite number of increments assigned to pairwise non-overlapping time intervals are mutually (not just pairwise) (iii) Stationary increments: For any s < t, N t N s is equal in distribution to N t s. Note.To call the increments stationary means that the probability distribution of any increment N s N t depends only on the length s t of the time interval; increments with equally long time intervals are identically distributed. In the Wiener process, the probability distribution of N s N t is normal with expected value 0 and variance s t. In the (homogeneous) Poisson process, the probability distribution of N s N t is a Poisson distribution with expected value λ(s t), where λ > 0 is the intensity or rate of the process. (iv) t N t is almost surely right continuous with left limits. (Lévy process-wikipedia, 2011) The homogeneous Poisson process is one of the prime examples of Lévy processes with applications in various areas such as insurance, finance, queuing theory, stochastic networks, to mention but a few. Brownian motion or Wiener process Definition 1.24. Wiener process or Brownian motion. A stochastic process B = {B t } t 0 defined on some filtered probability space (Ω, F, P, {F} t 0 ) is a Wiener process or Brownian motion if:

11 1. B 0 = 0 2. B has independent increments, that is, if 0 s t, then B t B s is independent of F s. 3. The increments B t B s are Gaussian with mean zero and variance t s. 4. The paths t [0, ) B (t,ω) of B are continuous functions of time. 5. A Wiener process is a martingale with respect to the filtration {F t } t 0 : that is E(B t /F s ) = B s for 0 s t. Bachelier (1900) proposed Brownian motion as a model of stock prices. He properly used it to analyze the motion of the Paris stock exchange and his results faired well with reality. If the price of a stock is S 0, it will be S t at time t where S t = S 0 + σb t (4) In equation (4), σ is the volatility. Figure 1.1 shows a sample path of Brownian motion with σ = 0.1. Brownian motion has mean zero where as the wealth of an insurance company or the price of a stock grows at some rate. To cater for this growth, a drift term can be added to get Brownian motion with drift S t = S 0 + µt + σb t (5) The path of Brownian motion with drift in Figure 1.2 is not so bad since the process has a long term upwards growth as we would expect. However, the process still goes negative. If it is a risk process, one would say that ruin has occured. If it is price process, then it has no meaning. One way to modify the Brownian motion with drift to eliminate negative prices is to introduce geometric Brownian motion. Brownian

12 motion has applications in physics, finance and insurance mathematics. For quite some time, Brownian motion has been used to model prices of risky assets such as share prices, foreign exchange rates, and many others.

13 1.5 1 Wiener process 0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 time Figure 1.2: Sample path of wiener process or Brownian motion. Definition 1.25. Standard Brownian motion; The standard Brownian motion B : [0, ) Ω R is a Gaussian stochastic process with independent increments for which; B 0 = 0, with probability 1 E[B t B s ] = 0 E[(B t B s ) 2 ] = t s, s < t (Onskog, 2008). In 1903, the Swedish actuary Lundberg laid the foundations of modern risk theory. One of Lundberg s main contributions is the introduction of a simple model which is capable of describing the dynamics of a homogeneous insurance portfolio that follows equation (1). There are 3 assumptions in the model.

14 Claims happen at times T i, i = 1, 2,.. satisfying 0 T 1 T 2... These are called claim times or claim arrival times in insurance literature. The i th claim arriving at time T i causes the claim size or claim severity S i. The sequence S 1, S 2,... constitute an independent and identically distributed (iid) sequence of non-negative random variables. The claim sizes process {S i } and the claim arrival process {T i } are mutually independent. (Mikosch,2003). In modern risk theory, the following terminologies appear so often and it is worthy defining them for clarity. Definition 1.26. Total claim amount Process, C is defined as where N t is defined in Definition (1.4) N t C t = S i, t 0 i=1 Definition 1.27. Collective risk theory also called Ruin theory, is a branch of actuarial science that studies an insurer s vulnerability to insolvency based on mathematical modeling of the insurer s surplus. (Ruin theory, Wikipedia, 2009). In the 1930s, Cramer, the famous Swedish statistician and probabilist, extensively developed collective risk theory by using the total claim amount process C with arrival T i which are generated by a Poisson process. In collective risk theory approach, the total gain or loss of the company is the main concern and individual policies are disregarded. (Houston, 1960). Therefore, collective risk theory addresses the following issues: 1. Models for the claim Number Process, N The most common claim number Process is the Poisson process because of its

15 good theoretical properties, as outlined in Definition 1.15. In his 1903 thesis, Filip Lundberg already exploited it as a model for the claim number process N. (Mikosch, 2003). The (homogeneous) Poisson process does not always describe claim arrivals in an adequate way, other processes to model claim number processes were developed, for example: Renewal process. Definition 1.28. Let W i be an iid sequence of almost surely positive random variables. Then the random walk T 0 = 0, T n = W 1 +W 2 +...+W n, n 1, is said to be a renewal sequence and the counting process N(t) = #{i 1 : T i t} t 0, is the corresponding renewal (counting) process. See (Andersen et al, 1993) for details. Mixed Poisson process Definition 1.29. Let Ñ be a standard homogeneous Poisson process and µ be the mean value function of a Poisson process on [0, ). Let θ > 0 almost surely be a (non-degenerate) random variable independent of Ñ. Then the process N t = {Ñθµ(t)}, t 0, is said to be a mixed Poisson process with mixing variable θ (Mikosch,2003). 2. Claim size {S i } distributions

16 A claim size distribution is a distribution that represents the sizes of claims that the insurance company has to pay. The question that arises here is What are realistic claim size distributions?. In literature, many distributions have been suggested. These are grouped into two: (a) Light tailed distributions. These are distributions of small claims. Examples include exponential distributions (which is common because of its desirable properties), Gamma, Weibul, Truncated normal and many others (See Table 3.2.17 of Mikosch, 2003). (b) Heavy tailed distributions These are distributions of large claims. Such claim size distributions typically occur in a reinsurance portfolio, where the largest claims are insured. Examples include Pareto, Log-normal, Burr and many others. (See Table 3.2.19 of Mikosch,2003). In order to determine a heavy tailed or light tailed claim size distribution, the commonest method is to use the exponential distribution as a benchmark. That is, if lim x sup F (x) e λx < for some λ > 0, where F (x) = 1 F (x), x > 0, then F is light-tailed. If lim x inf F (x) e λx > 0 for all λ > 0, then F is called heavy-tailed. Definition 1.30. A Compound Poisson process {C t } is the cumulative value process defined as N t C t = Z i, t > 0 i=1. The process {N t } t 0 (a Poisson process) and the sequence of random variables (S i ) are assumed to be independent. Depending on the choice of the counting Process N, there are different models for the total claim amount process C. For example: In the Cramer-Lundberg model, where N is a homogeneous Poisson Process,

17 C is modeled as a compound Poisson process. Another prominent model for C is called the renewal or Sparre-Andersen model; where N is a renewal process. See (Andersen et al, 1993) Note: The information about the asymptotic growth of the total claim amount enables one to give advice as to how much premium should be charged in a given time period in order to avoid bankruptcy (ruin) in the portfolio. Common classical premium calculation principles include; the net or equivalence principle, the expected value principle, the variance principle, the standard deviation principle. (See Mikosch, 2003, section 3.1.3 for details). In risk theory or mathematical term, the Cramer-Lundberg model or the classical risk model (1) is where; N t P t = y + pt S i, t 0 i=1 P t is the surplus of the insurance company at time t. y = Y (0) is the initial surplus (capital) p is the rate of premium income pt is the amount of premium received up to time t N t is a Poisson process with rate λ (which is the counting process for the claims)

18 S i are independent and identically distributed (iid) random variables representing the claim sizes, with distribution function F. N t and S i are independent. N t i=1 S i are the aggregate losses to the insurance company which are modelled by a compound Poisson process with an average number of claims per time period of λ. 1.3 Investment. The classical model does not account for interest on the reserve: in modern terms this may be expressed by saying that the insurance company may only invest in the money market. Now we deviate from the classical setting and assume that the company may also invest in a stock or market index, described by geometric Brownian motion. (Gaier et al, 2003). We shall also assume that the company can also invest some of the surplus into a risk-free asset (for example a Bond). The first attempt to incorporate investment incomes was undertaken by, e.g. Segerdahl, 1942. Segerdahl s assumption was that capital earns interest at a fixed rate r. Other researchers have expanded on this area since then and it has gained a lot of popularity (for example Scholes and Black, 1973). Definition 1.31. Investment income; The return received by insurers from their investment portfolios including interest, dividends and realised capital gains on stocks. It does not include the value of any stocks or bonds that the company currently owns.

19 1.4 Black-Scholes investment model. The Black-Scholes model was published in 1973, in two celebrated articles, one by Fischer Black and Myron Scholes, the other by Robert C. Merton. This formula which prices a wide class of derivatives has facilited an explosive growth in the market for options and more complex derivatives. In addition to filling the need for objective pricing (a starting point for valuation by creditors and auditors),it gives those who write (originate) options guidance on how to hedge the risks they are taking, and it permits a great variety of adjustments that can bring the ultimate pricing into line with supply and demand. The Black-Scholes model relies on the assumption that the price of the underlying stock follows equation (1.2). (Shafer and Volk,2001). In the Black-Scholes model, the time interval is a continuum of all real numbers between 0 and T where T is a positive real number. There is a bank account process S 2 (investment into a non-risky asset) with dynamics ds 2 (t) = rs 2 (t)dt, (6) where r > 0 is the rate of interest of the bank (or rate of return on the non-risky asset). There is also one type of stock (or any non-risk asset) with price S 1 that follows (1.2); ds 1 (t) = µs 1 (t)dt + σ R S 1 (t)db R,t (7) where µ is the mean rate of return of the stock, σ R > 0 is the volatility of the stock and B R,t is a Brownian motion or a Wiener process. Note that µ and σ R > 0 are constants.

20 1.5 Statement of the problem The Cramer-Lundberg model or the classical risk model remains the subject of analysis by actuaries and mathematicians. In particular different methods of analysis continue to be used in obtaining bounds or asymptotics of the ruin probability under specific conditions on the claim size distribution. More generalizations of this model exist and more of the research has been inclined to computing ultimate (eventual or infinite) ruin probabilities. In general although there are studies which tried to extend the Lundberg s model to incorporate investments, few of them have incorporated stochastic return on investments even then, in infinite time horizon. Thus, in this research we aim to study finite time ruin probabilities in the classical risk process which is compounded by an independent stochastic return on investments process of the Black and Scholes type. 1.6 Research objectives 1.7 General objective The main objective of this research is to numerically compute the finite horizon ruin probabilities in an insurance portfolio. 1.8 Specific objectives The specific objectives of this study are: (i) To formulate the surplus process compounded by an independent stochastic return on investments process of the Black and Scholes type. (ii) To develop numerical methods to solve the resulting mathematical models in this work. (iii) To investigate the effects of jumps or claims on both the probability of ruin

21 and time to ruin. 1.9 Significance of the study Ruin theory is used by actuaries in order to follow the insurer s surplus and ruin probability which can be explained as the probability of insurer s surplus drops below a specified lower bound such as minus initial capital. Ruin theory can also serve as a useful tool in long run to plan for the use of insurer s funds. This probability is a useful tool for the management since it serves as an indication of the soundness of the insurer s combined premiums and claims process, given the available initial capital. Studying this problem will develop expertise and contribute significantly in the areas of Black-Scholes models, ruin probability which is currently one of the most rapidly expanding directions in mathematical insurance, stochastic return on investments which is very important in Economics and numerical techniques which are very useful mathematical tools. In the next chapter a brief review of literatures closely related to the topic of this study will be discussed.

22 CHAPTER TWO LITERATURE REVIEW In this section a brief review of literatures closely related to the topic of this study are discussed. The areas of research reviewed in this section are risk theory; ruin theory and computation of finite and infinite ruin probabilities. Traditionally, practitioners have approximated the infinite probability of ruin by the expression e Ru, where R is the adjustment coefficient (by some authors called insolvency coefficient or Lundberg s constant) and u the initial surplus. From a technical point of view the need for such an approximation has become less important thanks to the arrival of efficient computers and even personal computers, the exact probability of ruin can be calculated. (Dufresne and Gerber, 1989). Paulsen and Gjessing (1997) considered a risk process with stochastic interest rate, and showed that the probability of eventual ruin and the Laplace transform of the time of ruin can be found by solving certain boundary value problems involving integro - differential equations. These equations were then solved for a number of special cases. They also showed that a sequence of such processes converges weakly towards a diffusion process, and analyzed the above - mentioned ruin quantities for the limit process in some detail. Tang and Tsitsishvili (2004) investigated the finite and infinite - time ruin probabilities in a discrete-time stochastic economic environment. Under the assumption that the insurance risk-the total net loss within one time period-is extended-regularlyvarying tailed or rapidly - varying tailed, various precise estimates for the ruin probabilities were derived. In particular, some estimates obtained were uniform with respect to the time horizon, and so applied in the case of infinite-time ruin. Dufresne and Gerber (1989) presented three methods to calculate the infinite-horizon probability of ruin. The first method, essentially due to Goovaerts and De Vylder, uses the connection between the probability of ruin and the maximal aggregate loss

23 random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distribution. Then the probability of ruin can be calculated in a transparent fashion; the main problem was to determine the non-trivial roots of the equation that defines the adjustment coefficient. For the third method they observed that the probability of ruin was related to the stationery distribution of a certain associated process. Thus it can be determined by a single simulation of the later. For the second and third methods the assumption of only proper (positive) claims was not needed. Slud and Hoesman (1989) introduced a general model for the actuarial- risk reserve process as a superposition of compound delayed renewal processes and related to previous models which have been used in collective risk theory. It was observed that non-stationarity of the portfolio age-structure within this model can have a significant impact upon finite probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationery, the moderate- and large deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csorgo et al. (1987). One consequence is that for non Poisson claim arrival, the large deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with corresponding faster claim - arrivals. Hipp and Schmidli (2000) considered a risk process with the possibility of investment into a risky asset. Their aim was to obtain the asymptotic behaviour of the infinitehorizon ruin probability under the optimal investment strategy in the small claims case. In addition they proved convergence of the optimal investment level as the initial capital tends to infinity. Taylor (1984) examined various aspects of ruin theory from viewpoints which were very much heuristic. He dealt with the renewal equation governing the infinite time ruin probability. It was emphasized as intended to be no more than a pleasant ramble through a few scattered results. An interesting connection between ruin probability and a recursion formula for computation of the aggregate claims distribution was noted and discussed. The relationship between

24 danger of the claim size distribution and ruin probability was re-examined in the light of some recent results on stochastic dominance. Finally, suggestions were made as to how the way in which the formula for ruin probability leads easily to conclusion about the effect on that probability of the long- tailedness of the claim size distribution. Stable distributions, in particular, were examined. Paulsen (2008) treated the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered where presentation of the relevant integro-differential equations, exact and numerical solutions, asymptotic results, bounds on the infinite ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis was on continuous time models, but discreet time model were also covered. A fairly extensive list of references was provided, particularly of papers published after 1998. Brekelmans and De Waegenaere (2000) presented an algorithm to determine both a lower and an upper bound for the finite time probability of ruin for a risk process with constant interest force. They split the time horizon into smaller intervals of equal length and considered the probability of ruin in case premium income for a time interval is received at the beginning of the that interval, which yields a lower bound. For both bounds they presented a renewal equation which depends on the distribution of the present value of the aggregate claim amount in a time interval. This distribution was determined through a generalization of Panjar s (1981) recursive method. Moller (1995) proposed a general approach for finding differential equations to evaluate probability of ruin in finite and infinite time. Attention was given to real value non diffusion processes where a Markov structure is obtainable. Ruin was allowed to occur upon a jump or between the jumps. The key point was to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change

25 of variable formula or the martingale representation theorem for point processes, the differential equation for evaluating the probability of ruin was obtained. Numerical illustrations were given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon. Constantinescu and Thomann (2003) studied the classical result of Cramer - Lundberg and they stated that if the rate of premium income, c, exceeds the average of the claims paid per unit time λu, then the infinite-horizon probability of ruin of an insurance company decays exponentially fast as the initial capital, u. In this note, the asymptotic behaviour of the probability of ruin was derived by means of infinitesimal generators and Laplace transforms. Using these same tools, it was shown that the probability of ruin has an algebraic decay rate if the insurance company invests its capital in risky assets with a price which follows a geometric Brownian motion. The latter result was shown to be valid not only for exponentially distributed claim amounts, as in Frolova et al. (2002), but more generally, for any claim amount distributed that has a moment generating function defined in a neighbourhood of the origin. Kasozi and Paulsen (2005) addressed the issue of ruin of an insurer whose portfolio is exposed to insurance risk arising from the classical surplus process. They stated that availability of a positive interest rate in the financial world forces the insurer to invest into a risk free asset. They derived a linear Volterra integral equation of the second kind and applied order four Block - by- block method in conjunction with the Simpson rule to solve the Volterra equation for ultimate ruin. This probability was arrived at by taking a linear combination of some two solutions to the Volterra integral equation. Then several numerical examples were given which showed that their results were excellent and reliable. Cai and Yang (2005) studied ruin in a perturbed compound Poisson risk process under stochastic interest force and constant interest force. By using the technique of stochastic control, they showed that the infinite-horizon ruin probability in the

26 perturbed risk model is always twice continuously differentiable provided that claim sizes have continuous density functions. In the perturbed risk model, ruin may be caused by a claim or by oscillation. They decomposed the ruin probability into the sum of two ruin probabilities; one is the probability that ruin is caused by a claim and the other is the probability that ruin is caused by Oscillation. Integro differential equations for these ruin probability were derived when the interest force was constant. When the claim sizes were exponentially distributed, explicit solutions of the ruin probabilities were derived from the integro differential equations. They gave numerical examples to illustrate the effects of diffusion volatility and interest force on the ruin probabilities. Tang (2005) established a simple asymptotic formula for the finite-time ruin probability of the compound Poisson model with constant interest force and sub-exponential claims in the case that the initial surplus is large. The formula was consistent with known results for the ultimate ruin probability and, in particular was uniform for all time horizons when the claim size distribution is regularly varying tailed. Moller (1996) introduced some techniques, based on the change of variable formula for processes of finite variation, for establishing (integro) differential equations for evaluating the distribution of jump processes for a fixed period of time. This is of interest in insurance mathematics for evaluating the distribution of the total amount of claims occurred over some period of time, and attention was given to such issues. Firstly he studied some techniques when the process had independent increments, and then a more refined martingale technique was discussed. The building blocks were delivered by the theory of marked point processes and associated martingale theory. A simple numerical example was given. Dickson and Waters (2002) studied the distribution of the time to ruin in the classical risk model. They considered some methods of calculating this distribution, in particular by using algorithms to calculate finite time ruin probabilities. They also discussed calculation of the moments of this distribution.

27 Asmussen and Rubinstein (1999) showed how, from a single simulation run, to estimate the infinite ruin probabilities and their sensitivities (derivatives) in a classic insurance risk model under various distribution of the number of claims and the claim size. Similar analysis was given for the tail probabilities of the accumulated sensitivity analysis with respect to both distributional and structural parameters of the underlying risk model. In the former case, they used the score function method and in the latter a combination of the push-out method and the score function. They finally showed how, from the same sample path, to derive a consistent estimator of the optimal solution in an optimization problem associated with excess-of- loss reinsurance. In conclusion to this section, we notice that almost every researcher has designed numerical methods to solve problems arising from the Cramer-Lundberg model. However, none of these methods belong to the standard numerical methods developed by numerical analysts to solve the relevant equations. Also very few papers have incorporated stochastic return on investments of the Black and Scholes type in finite time horizon. We contribute towards this area by designing alternative methods that can be used to solve the models in this study, with the aim of computing finite horizon ruin probabilities in an insurance portfolio. In the next chapter the surplus process that incorporates stochastic return on investments will be formulated.

28 CHAPTER THREE MODEL FORMULATION AND THEORETICAL RESULTS 4 Introduction. In Lundberg s model, the company did not earn any investment on its capital (surplus). An obvious reason for this assumption, although there may be other reasons as well, is that the mathematics is easier, and back in the first half of the 20th century the theory of stochastic processes was far less developed, and also far less known, than it is today. In general although there are studies which tried to extend the Lundberg s model to incorporate investments, few of them have incorporated stochastic return on investments even then, in infinite time horizon. It is for this reason that in this chapter the surplus process that incorporates stochastic return on investments will be formulated. 4.1 Model formulation All processes and random variables are defined on the stochastic basis (Ω, F, P, {F} t 0 ) satisfying the usual conditions, i.e. F t is right continuous and P-complete. Here, we introduce in mathematical terms a general risk process that includes (stochastic) return on investments. This process or specializations of it, provide the basic for the research in this study. The basic insurance process is modeled as: N P,t P t = pt + σ P B P,t Z P,i ; t 0, (8) Where N P is a Poisson process, B P is a standard Brownian motion independent of the compound Poisson process N P,t i=1 Z P,i. Let the intensity of N P be λ P, and F P be the distribution of the claim Z P. We shall assume that F P (0) = 0, and that at i=1

29 least one of σ P or λ P is nonzero. The interpretation of equation (8) is that p is the premium income per unit time,n P is the claim number process representing the number of claims up to time t. The compound Poisson process N P,t i=1 Z P,i are claims paid with {Z P,i } i N assumed to be an i.i.d sequence. The assumption F P (0) = 0 assures that they are positive. The Brownian term σ P B P represents fluctuations in the other two terms so that P is a classical risk process perturbed a by diffusion. where X 0 is the stock price at t = 0. The process X is a geometric Brownian motion. The solution to (??) is the value of the stock at time t and is given by X t = X 0 exp(r 1 2 σ2 R )t + σ RB R,t. Making λ R > 0 ensures that there are sudden changes in the value of the stock. This makes the model realistic although we do not pursue this in later chapters. To get the impact of a jump in R right, we must write (??) as In this case, the solution is X t = X 0 + t 0 X s dr s X t = X 0 exp {(r 12 } N σ2r)t R,t + σ R B R,t (1 + Z R,i ). A combination of equation (8) and (??) leads to the promised risk process which we shall denote Y = {Y t } t R+. This process represent the insurance portfolio, that is, premium income plus return on investments (which may be negative) minus claims paid and has been studied extensively for ultimate ruin probability, see e.g Paulsen et al (2005), Paulsen (1998), Paulsen and Gjessing (1997), Paulsen and Rasmussen (2003), Sundt and Teugels (1995), and many others. In other words, i=1 dy t = dp t + Y t dr t. (9) Mathematically, this means that Y is the solution of the linear stochastic differential equation Y t = y + P t + t 0 Y s dr s (10)

30 where Y 0 = y is the initial capital of the company. The solution to (10) is given in Paulsen (1993) as where Y t = R t (y + t 0 R 1 s dp s ) (11) R t = exp {(r 12 } N σ2r)t R,t + σ R B R,t (1 + Z R,i ), t 0 By equations (8) and (??), equation (9) can be written as which simplifies to N P,t dy t = d(pt + σ P B P,t Z P,i ) + Y t d(rt + σ R B R,t + Z R,i ) dy t = (p + ry t )dt + i=1 σ 2 P + σ2 R Y 2 i=1 N P,t t db t d( i=1 N R,t i=1 N R,t Z P,i ) + Y t d( i=1 Z R,i ) (12) where B is a new standard Brownian motion independent of the compound Poisson processes involved. By excluding the return on investments model and setting σ P = 0, we recover Lundberg s model. Using Itô s formula, the infinitesimal generator for Y is given by the following quite complicated integro-differential operator: Ag(y) = 1 2 (σ2 Ry 2 + σp 2 )g (y) + (ry + p)g (y) + λ P (g(y x) g(y))df P (x) 0 + λ R g(y(1 + x)) g(y))df R (x). 1 Use the process Y to compute the probability of ruin (or survival) for cases (λ P = 0 and λ P 0) (13) Throughout, ignore the sudden jumps into R so that λ R = 0. We leave this as an extension of this research work. Definition 4.1. Let T y be the time of ruin defined as follows: inf t : Y t < 0, T y =, if Y t 0 for all t > 0.

Then the finite time ruin probability ψ(t, y) C 1,2 (R+, R) defined as 31 ψ(t, y) = P(T y t) It follows from Paulsen and Gjessing (1997) and Norberg (1999) that the propability of ruin before time t = T satisfies ψ(t, y) t + Aψ(t, y) = 0 (14) (where the generator A acts on the y variable in ψ(t, y)) with with extra condition φ(t, y) = 0, y 0, 0 < t T, φ(t, y) = 1, y > 0, σ 2 P > 0. φ(t, y) = 1, 0 < t T. 5 The case λ P = 0 (i.e. in the absence of jumps or claims) Equation (??) yields t φ(t, y) + 1 2 (σ2 P + σry 2 2 ) 2 y φ(t, y) + (ry + p) φ(t, y) = 0 (15) 2 y We need a finite- difference scheme to solve (15). Finite-difference methods constitute a very powerful and flexible technique of obtaining (many times very accurate) numerical solutions to partial differential equations like (15). The idea underlying finite-difference methods is to replace the partial derivatives occurring in (15) by approximations based on Taylor series expansions of function near the point or points of interest. To progress the solution from t = 0 to t = T, values of φ(0, y), 0 y y need to be known. Unfortunately, these values are unknown complicating the use of finitedifference methods. Since we know φ(0, y), 0 < y y, we let t = T t so that φ(t, y) = φ(0, y).then φ(t, y) t = φ(t t, y). t

t = T 32 1 φ(t, y) 1 0 unknown y Figure 5.1: Domain 1. Therefore (with φ := φ) t φ(t, y) + 1 2 (σ2 P + σry 2 2 ) 2 y φ(t, y) + (ry + p) φ(t, y) = 0 (16) 2 y to be solved on the domain φ(t, y) = 0, y 0, 0 < t T, φ(t, y) = 1, 0 < t T, φ(t, y) = 1, y > 0, σp 2 > 0. To use numerical schemes to solve the partial differential equation (16), we must address three fundamental issues: Consistency: A numerical scheme is said to be consistent if the finite-difference representation converges to the partial differential equation at hand as the time and space steps tend to zero; Stability: A numerical scheme is said to be stable if the difference between the numerical solution and the exact solution remains bounded as the number of time steps tends to infinity;

t = 0 33 unknown φ(t t, y) 1 T = 1 1 y Figure 5.2: Domain 2. Convergence: A scheme is said to be convergent if the difference between the numerical solution and the exact solution tends to zero uniformly as the time and space discretisations tend to zero. A powerful statement links these issues together. This statement is the Lax Equivalence. Theorem:Given a properly posed problem and a consistent finite-difference scheme, stability is the only requirement for convergence. The Lax Equivalence Theorem is formally stated as Theorem 5.1 on page 139 in Morton and Mayers (2002). The implicit finite-difference scheme named after Crank and Nicolson who in (1947) applied the scheme successfully to problems in dyeing of textiles. This scheme is convergent and stable for all values of k h 2 40. The general theory on singularities has concentrated on the special problem which arise in scientific contexts. In particular, we have no simple expression for the local truncation error near the point of discontinuity and no idea of the function of h which governs the difference between the (unknown) true solution and the approximate finite-difference solution.

34 We have tried a favored but somewhat inelegant method of coping with the discontinuity which is to ignore it, for as long as the finite difference method employed is convergent and stable, the error introduced by the discontinuity decay. When the exact solution is known, Milne (1953) used convergent and stable methods to examine the area of infection. He compared the local analytical solution with the solution of the five point difference formula and showed that the difference rapidly got smaller as he moved away from the singularity. Some subroutines are found in Press et al (1992). The following section gives an important understanding of continuous and discontinuous functions. This will help us discuss the concept of discontinuity and how to find a way of copying with the discontinuities: 6 Limits and Continuity of Functions The concepts of limit and continuity are very closely related. An intuitive understanding of these concepts can be obtained through the following section. 6.1 Limits of Functions In this section we shall consider real valued functions whose domain is an interval. Consider a real valued function defined in the open interval (a δ, a + δ) except possibly at the point y=a itself. Definition 6.1. The function φ(y) is said to approach L R as y approaches a, if given ɛ > 0, there exists a δ > 0 such that φ(y) L < ɛ whenever 0 < y a < δ. We denote this by lim y a φ(y) = L or φ(y) L as y a. Note that the point a need not be in the domain of φ. The δ in the above definition is not unique. The definition is valid for any δ such that 0 < δ < δ. Note 1. In the definition of the limit, the condition that o < y a < δ excludes

35 the point a. Hence we consider the values of y in some interval a δ, a + δ other than a so the value of φ(y) at y = a is immaterial and the function φ(y) need not even be defined at y = a. Further even if φ is defined at a, it need not happen that lim φ(y) = φ(a) as shown in the Example 3.2 below. y a Note 2. For the existence of limit of a function φ at y = a, it is necessary that the domain of definition of φ must contain (a δ, a + δ) a for some δ > 0. A subset of R containing an interval of the form (a δ, a + δ) for some δ > 0 is called a neighbourhood of a. A neighbourhood of a without a is called a deleted neighbourhood of a. Hence for the existence of limit of φ as y a, φ must be defined in a deleted neighbourhood of a. We shall illustrate the definition with examples below. Example 3.1 P rove that lim y 2 (2x 1) = 3. Proof. Let φ(y) = (2y 1), L = 3 and a = 2. Given ɛ > 0, we have to find a δ such that φ(y) 3 < ɛ whenever 0 < y 2 < δ. Now φ(y) 3 = 2y 1 3 = 2y 4 = 2 y 2. This will be less than ɛ, if y 2 < ɛ/2. Hence we can take δ = ɛ/2. So if 0 < y 2 < δ = ɛ/2, then φ(y) 3 = 2 y 2 < 2δ = ɛ. Hence given ɛ > 0, we can choose δ = ɛ/2 so that lim y 2 (2x 1) = 3 Example 3.2 Prove that lim y 2 φ(y) = 11 Proof. 4y + 3, if y 2 Let φ(y) = 12, if y = 2.

36 Let ɛ > 0 be given; we want to find a δ > 0 such that φ(y) 11 < ɛ whenever y 2 < δ. Now φ(y) 11 = 4y + 3 11 = 4y 8 = 4 y 2 we can choose δ = ɛ 4. Then we have φ(y) 11 < ɛ whenever 0 < y 2 < ɛ. Hence lim φ(y) = 11. But by 4 y 2 definition, at y = 2, φ(y) = 12 so that lim φ(y) φ(2). y 2 Definition 6.2. A function φ(y) is said to approach L as y a from the right or from above if given ɛ > 0, there exists a δ > 0 such that φ(y) L < ɛ whenever a < y < a + δ. The number L is called the right hand limit of φ at a. We write this as lim φ(y) = L. y a+ Note that for the existence of the right hand limit φ(y) should be defined in (a, a+δ). Definition 6.3. A function φ(y) is said to approach M as y a from the left or from below if given ɛ > 0, there exists a δ > 0 such that φ(y) M < ɛ whenever a δ < y < a. The number M is called the left hand limit of φ at a. We write this as lim φ(y) = M. y a Note that for the existence of the left hand limit φ(y) should be defined in (a δ, a). 6.2 Continuous Functions In this section, we shall give some necessary preliminaries of continuous real valued functions which we need for the study of our work. Intuitively a continuous function means a function whose gragh is without a break or interrruption. When we defined the limit of a function at the point y = a, we have not considered the behaviour of the function φ at y = a. For the definition of the limit at y = a, the function need not be defined at y = a. Even if φ is defined at y = a, its value at y = a need not be equal to the value of the limit L at y = a. If it happens that φ is defined at y = a and if it also happens that φ(a) = L, then φ is said to be continuous at y = a. This is made precise in the following definition.

37 Definition 6.4. Let φ be a real valued function defined on R. φ is said to be continuous at y = a, if lim y a φ(y) = φ(a). Using the definition of the limit of a function at a point, the above definition can be formulated equivalently as follows. Definition 6.5. φ is said to be continuous at y = a, given any ɛ > 0, there exists a δ > 0 such that φ(y) φ(a) < ɛ whenever y y 0 < δ. From the definition, we note that the continuity of a function φ at a point y = a has the following three implications. (i) φ should be defined at y = a. That is φ(a) exists. (ii) lim y a φ(y) exists. (iii) The value of φ at y = a viz. φ(a) and lim y a φ(y) are equal. The function φ is said to be continuous on a subset A R if and only if it is continuous at each point of A. Example 1. Constant functions are continuous everywhere. If φ(y) = c for all y R, then lim φ(y) = lim c = c = φ(a) for every y a y a a R. So φ is continuous everywhere. Example 2. The identity function is continuous everywhere. If φ(y) = y for all y R, then lim φ(y) = lim x = a = φ(a) for every a R. So the identity function y a y a is continuous everywhere. Example 3. Consider φ(y) = y 2 + 1 defined on R. Now lim y 2 φ(y) = 5 and φ(y) = 5. Hence φ satisfies the conditions (i), (ii) and (iii) of a continuous function given under its definition. Hence φ is continuous at y = 2. If any one of the three conditions (i), (ii) and (iii) fails at a point, then the function φ is said to be discontinuous at that point and the point is called a point of discontinuity of the function. The points of discontinuities may arise in any of the following manner.

(a) The condition (ii) of the definition holds good but (i) and (iii) fail. 38 (b) φ(y) is defined at y = a and lim y a φ(y) exists but lim φ(y) φ(a) y a (c) (i) holds good but lim φ(y) does not exist. It may happen that lim φ(y) and y a y a lim φ(y) exist and the are not equal so that lim φ(y) does not exist. y a+ y a Definition 6.6. In the case of (c), the function φ is said to have a jump discontinuity and in case of (a) and (b), the discontinuity is said to be removable. But there are discontinuities other than removable discontinuity and jump discontinuity. If a discontinuity is either removable or or a jump discontinuity at y = a, then the function φ is said to have a simple discontinuity or a discontinuity of the first kind at y = a. Every other discontinuity is called a discontinuity of the second kind. In the case of a discontinuity of the second kind, if at least one of the two one-sided limits at y = a fails to exist. Example 4. Let φ(y) = y2 9 y 3.Let φ(y) = y2 9 y 3. φ is a real valued function defined on R. Though 3 is in the domain of definition of φ, φ(3) is not defined. But y 2 9 lim φ(y) = lim y 3 y 3 y 3 = lim y + 3 = 6 y 3 But if we define φ(3) = 6, then φ satisfies the conditions (i)(ii) and (iii). Thus φ is made continuous. The point y = 3 is a removable discontinuity. Example 5. T he function φ(y) = y y lim y 0+ for y 0 and φ(0) = 0 is discontinuous at y = 0. For lim φ(y) = 1 and y 0 φ(y) = 1. Thus the function has a jump discontinuity. A jump discontinuity cannot be removed by redefining φ(a). Note that the points of discontinuity

39 in Examples 4 and 5 are examples of simple discontinuity or a discontinuity of first kind. Remark. Thus if the point of discontinuity is apparent, we may apply a simple Taylor expansion of φ around the point of discontinuity to check the type of discontinuity whether removable, infinite or jump discontinuity. But in our case the function is not known. So how can we cope with the discontinuities? 7 Singularities Estimates of the errors between the solution of the differential equation and that of finite difference approximation depend on boundedness of partial derivatives of some order. These estimates cannot be valid at or near a singular point on the boundary or in the interior of the integration domain. In rectangular coordinates the most common form of boundary singularity occurs at the corners of the integration domain where the initial line meets the vertical boundary. Weaker types of discontinuities can also occur and cause computational problems. Finite difference methods have questionable value in the neighbourhood of singular points since a region of infection lies adjacent to such points. The effect of such discontinuities does not penetrate deeply into the field of integration provided the methods we use are stable, in which case the errors introduced by the discontinuities decay. There are serious difficulties in obtaining accurate solutions near the the points of discontinuity and we must often abandon finite difference processes to obtain this accuracy. Crandal explored the accuracy obtained by approximating the linear diffusion equation on coarse nets and found the resulting solution very sensitive to values assigned at the singular points. Some problems are such that singularities occur at points p i inside the integration domain. Interior singularities occur when one, or more than one coefficient of the partial differential equation becomes singular. In such problems the solution will

40 ordinarily also have a singularity at p i, and the finite difference scheme will not be applicable without modification. Physical problems where such singularities occur are source-sink and concentrated point-load problems. The accepted technique in interior singularity problems is to subtract the singularity where possible, thereby generating a new problem with different boundary conditions but with a well-behaved solution. In practice, this procedure work well with some linear problems. The more difficult none- linear problems require individual treatment. The generation of local solutions can sometimes be useful if the character of the singularity is known. Most of the effort on singularities has been concentrated on scientific problems and currently no general theory is available. We shall examine several methods for treating singularities in some problems. 1. Subtracting out the singularity. Suppose u has a discontinuity at a point p. The goal is to calculate a solution U of the same partial differential equation such that u U is well behaved at the point p. 2. Mesh refinement. One very common method of dealing with discontinuities is effectively to ignore them and attempt to diminish their effect by using a mesh refinement in the region near to and surrounding the singularity. The mesh refinement procedure has the effect of minimising the area of infection created by the singularity. The procedure of subtracting out the singularity appears to be preferable to the refined net method, if the analysis can be performed. 3. The method of Motz and Woods. See Ames (1977) pg 244. 4. Removal of singularity. In some problems removal of a singularity is possible by the adoption of a new independent variable. The transformation is to be so devised that the singular point expanded into a line or curve. Thus, a sudden change will become a smooth change along the line or curve in the new

41 independent variables. For a general background to singularities, we refer to Ames (1977). Parameters p = 0.1, r = 0.1, σ P = σ R = 0.2, T = 1. 8 The case λ P 0 (i.e. in the presence of jumps or claims) Here, (16) modifies to t φ(t, y) + 1 2 (σ2 P + σry 2 2 ) 2 y φ(t, y) + (ry + p) φ(t, y) 2 y y +λ P (φ(t, y x) φ(t, y))df P (x) = 0 0 from which on the same domain t φ(t, y) + 1 2 (σ2 P + σry 2 2 ) 2 y φ(t, y) + (ry + p) φ(t, y) 2 y y λ P φ(t, y)f P (y) + λ P φ(t, y x)df P (x) = 0 0 (17) φ(t, y) = 0, y 0, 0 < t T, φ(t, ỹ) = 1, 0 < t T, φ(t, y) = 1, y > 0, σp 2 > 0. Therefore, we add integral term to (16). Take Exp(α) claims so that F P (x) = 1 e αx, α > 0. is the distribution for light tailed Exponential claims and F P (x) = ( ) α α 1, α > 0. α 1 + x is the distribution for heavy tailed Pareto claims. In the next chapter we are going to use numerical methods to simulate the models that we have formulated in this chapter.

42 CHAPTER FOUR MODEL SIMULATIONS CHAPTER FOUR MODEL SIMULATIONS 5 Introduction Solutions of some partial differential equations cannot be expressed as a linear combination of known simple functions. In the few cases where such a representation is possible the resulting expressions may be very complicated and too involved for direct evaluation even with the aid of modern calculating devices hence for the need of numerical solutions. Most analytical results are for the infinite time horizon. In the finite time horizon case analytical solutions are hard to come by. 6 Numerical Methods In this section we will discuss numerical solutions of the survival probability φ(t, y) using a fixed grid y = 0, h, 2h,...1 using some methods of numerical solutions based on finite differences. Basically there are two types of finite difference methods:explicit finite difference method and implicit finite difference method. Other types are just the deviation of these two types. These methods are by far the most popularly used in solving numerically partial differential equations. They are based on the principle of converting a given partial differential equation into an approximating algebraic equation, usually in the form of a difference (recurrence) equation which is then solved numerically to give approximate values of the solution of the original partial differential equation. The convertion process is done by replacing all partial derivatives appearing in the differential equation with approximating partial difference expressions. In this study we will use the Implicit-finite difference scheme or the Crank-Nicolson Implicit Method. This method is a finite difference method used

43 T t h k φ i,j+1 jk φ i 1,j φ i,j φ i+1,j 0 y ymin y min+ih y max Figure 7.1: PDE solution grid. for numerically solving the heat equation and similar partial differential equations. It is a second order method in time, implicit in time, and is numerically stable. The method was developed by John Crank and Phylis Nicolson in the mid 20th century. 7 Implicit- finite difference scheme This section provides a brief introduction to finite difference methods for solving partial differential equations. We focus on the case of a PDE in one state variable plus time. We wish to find the function φ(t, y) satisfying the pde (16) subject to the given initial conditions and boundary conditions. Obviously one cannot calculate the entire function φ(t, y). What we shall consider a solution is the numerical values that φ(t, y) takes on a grid of t, y values placed over some domain of interest. Once we have these values, since φ(t, y) is assumed to be smooth almost everywhere, we can interpolate within this grid to get values for arbitrary t, y. To this end, suppose the domain we will work on is rectangular with y ranging from y min to y max and t ranging from 0 to T. Divide [0, T ] into J equally spaced intervals at t values indexed

44 by j = 0, 1,..., J, and [y min, y max ] into I intervals at y values indexed by i = 0, 1,..., I. The length of these intervals is k in the time direction and h in the y direction. We seek an approximation to the true values of φ(t, y) at the (J +1) (I +1) gridpoints. Let φ i,j denote our approximation at the gridpoint where y = y min + ih, t = jk. The next step and this is what makes this procedure a finite difference method is to approximate the partial derivatives of φ(t, y) at each gridpoint by difference expressions in the as yet unknown φ i,j s. We can calculate φ i,0 for each i directly from the initial value condition. Thus it is natural to start from this boundary and work outward, calculating the φ i,j+1 s from φ i,j. Focussing on an arbitrary internal gridpoint ij, one could approximate the partial derivatives at that point by the following: t φ(t, y) = φ i,j+1 φ i,j k y φ(t, y) = φ i+1,j φ i 1,j 2h 2 y φ(t, y) = φ i+1,j 2φ i,j + φ i 1,j 2 h 2 The differences in the y, or state, direction have been centered around the point ij to give second order accuracy to the approximation. These expressions could then be substituted into the pde (16). Solving the resulting equation for φ i,j+1 gives the explicit solution (18) φ i,j+1 = (kσ 2 P + kσ 2 Ry 2 hkry hkp)φ i 1,j (2kσ 2 P 2kσ R y 2 2h 2 )φi, j (19) +(kσp 2 + kσry 2 2 + hkry + hkp)φ i+1,j One could then proceed to calculate all the φ i,j+1 s from the φ i,j s and recursively obtain φ(t, y) for the entire grid. Since equation (18) applies only to the interior gridpoints, we at each time step would have to make use of some other boundary conditions (e.g., at y min and y max ) to identify all the φ(t, y) s. More will be said about this later. The result of this is called an explicit finite difference solution for φ(t, y). It is second order accuracy in the y direction, though only first order accurate in the t direction, and easy to implement. Unfortunately the numerical solution is unstable

45 unless the ratio k/h 2 is sufficiently small. By unstable is meant that small errors due to either to arithmetic inaccuracies or to the approximate nature of the derivatrive expressions will tend to accumurate and grow as one proceeds rather than dampen out. Loosely speaking, this will occur when the difference equation system described by (18) has eigenvalues greater than one in absolute value. See a numerical analysis book such as Vemuri and Karplus (1981) or Lapidus and Pinder (1982) for discussion of stability issues. The implicit finite difference scheme is an alternative method which uses a central difference approach and is second order accurate in both the y and t directions (i.e., one can get a given level of accuracy with a coarser grid in the time direction, and hence less computation cost). It also has the advantage of being stable for all values of k/h 2 (the same ratio as in the explicit method); this does no mean that it is more accurate in all cases than the explicit method, but it usually is. The price that has to be paid for the general improvement is that more computation is required at each stage, but it is generally worth it. Although k/h 2 is no longer restricted, smaller k/h 2 will still give better results. In practice, one chooses a k by which one can save a considerable amount of work, without making k/h 2 too large. For instance, often a good choice is k/h 2 = 1 (which would be impossible in the previous direct method). The algorithm entails proceeding as before, but using difference expressions for the partial derivatives which are centered around t + k/2 rather than around t. Thus the expressions for φ(t, y), 2 φ(t, y) and φ(t, y) are averages of what we had in t y y 2 (18) for times j and j + 1: t φ(t, y) = φ i,j+1 φ i,j k y φ(t, y) = φ i+1,j φ i 1,j + φ i+1,j+1 φ i 1,j+1 4h 2 y φ(t, y) = φ i+1,j 2φ i,j + φ i 1,j + φ i+1,j+1 2φ i,j+1 + φ i 1,j+1 2 2h 2 (20) Substituting the above into the pde (16), multiplying through by 4h 2 k to eliminate the denominators, and collecting all the terms involving the unknown φ.,j+1 s on the

46 left hand side results in (kσ 2 P + kσ 2 Ry 2 hkp hkry)φ i 1,j+1 (2kσ 2 P + 2kσ 2 Ry 2 + 4h 2 )φ i,j+1 + (kσ 2 P + kσ 2 Ry 2 + hkp + hkry)φ i+1,j+1 = (kσ 2 P + kσ 2 Ry 2 hkp hkry)φ i 1,j + (2kσ 2 P + 2kσ 2 Ry 2 + 4h 2 )φ i,j (21) (kσp 2 + kσry 2 2 + hkp + hkry)φ i+1,j for each i = 1,..., I 1. It is apparent that that the φ.,j+1 s cannot individually be written as simple linear combinations of the φ.,j s, but are simultaneously determined as the solution to this system of linear equations. However this system has a very convenient structure. The biggest advantage of this method is that there is no longer any need to put restrictions on the value of k/h 2. We say that the Crank Nicolson method is unconditionally stable. For diffusion equations (and many other equations) it can be shown that the Crank-Nicolson method is unconditionally stable. However, this significant gain in stability has been obtained at the cost of having to consider at one go three unknown values of the solution, which are φ i 1,j+1,φ i,j+1 and φ i+1,j+1. A pictorial representation of the scheme suggested by equation (21) is as shown in the figure 4.2.

t 47 φ i 1,j+1 φ i,j+1 φ i+1,j+1 φ i 1,j φ i,j φ i+1,j y Figure 7.2: Pictorial representation of the scheme. The coefficients of the interior nodes are A i = (kσp 2 + kσry 2 2 hkp hkry) i = 2, 3,..., I 1 B i = (2kσP 2 + 2kσRy 2 2 + 4h 2 ) C i = (kσp 2 + kσ 2 Ry 2 + hkp + hkry) D i = (kσp 2 + kσry 2 2 hkp hkry)φ i 1,j + (2kσ 2 P + 2kσ 2 Ry 2 + 4h 2 )φ i,j (kσ 2 P + kσ 2 Ry 2 + hkp + hkry)φ i+1,j For the Dirichlet boundary conditions in equation (22) B 1 = 1, C 1 = 0,D 1 = φ 0 A I = 0, B I = 1, D I = φ L Written in matrix form, (21) provides the interior equations of MU = R (22) where B 1 C 1 0 0... 0 A 2 B 2 C 2 0... 0 M = 0 A 3 B 3 C 3... 0........ 0 0... A I B I

U =. 48 R =. φ 1,j+1 φ 2,j+1 φ 3,j+1 φ I,j+1 This sort of system is most efficiently solved by Gaussian elimination to determine the unknown φ.,j+1 s (e.g., see Press et al., 1986, p.40). D 1 D 2 D 3 D I 8 Boundary conditions We almost have a procedure for recursively determining the entire grid of φ i,j s starting from the given initial values. Substitution of the the difference expressions into the differential equation only gave us a linear equation for each interior point in the grid. That gives I 1 equations at each time step, which is not sufficient to determine the I + 1 unknowns. The missing two equations must be provided by boundary conditions applied at each time step. It would be desirable for these to be representable in a form that preserves the tridiagonal form of the system and thus the efficiency of solution.the most convenient to apply would be knowledge of the value of φ(t, y) on the boundaries y max and y min. In (21), this could be embodied by setting the coefficients B 1 = 1, C 1 = 0, A I = 0 and B I = 1, then setting D 1 and D I on the right hand side of (21) equal to the known values φ(y min, t) and φ(y max, t) respectively. The elements D i (i = 1, 2, 3..., I) constituting the righthand side vector R are the sums of the known quantities on the right hand side of each respective equation. The matrix M has a particularly simple form known as

49 tri-diagonal. But the situation at hand in our model is abit different from what has been discussed above. What makes it different is in that in our model φ(t, y) is equal to - φ(t t t t, y). It is for this reason that we will now calculate the values of φ(t t, y). This implies that we know the values of the φ.,j+1 s and not those of the φ.,j s. This can be clearly seen from the given initial and boundary conditions. Therefore we will use the known values of the φ.,j+1 s to calculate the unkown values of the φ.,j s. Having this in mind, the pictorial representation of our scheme will look different from the one we have in figure 4.2 in terms of nodal points. Consider the division of the interval [0, 1] using I so that the interval width is h = 1 I 1 shows this difference nodes. Figure 4.3 below

t 50 φ i 1,j φ i,j φ i+1,j φ i 1,j+1 φ i,j+1 φ i+1,j+1 y Figure 8.1: Pictorial representation of the scheme in our model. The molecule is applied at each of the mid-points M 1, M 2,...M I 1, to give the matrix equation below. We also apply the boundary conditions φ 0,j = φ 0,j+1 = 0 and φ I,j = φ I,j+1 = 1. Mφ j+1 = Rφ j + b where B 1 C 1 0 0... 0 A 2 B 2 C 2 0... 0 M = 0 A 3 B 3 C 3... 0........ 0 0... A I B I B 1 C 1 0 0... 0 A 2 B 2 C 2 0... 0 R = 0 A 3 B 3 C 3... 0........ 0 0... A I B I

51 1 1 φ j+1 = 1. 1 0 0 b = 0. 1 This is the Implicit method; as it stands the full set of equations have to be set up to give the row of probabilities at the new time level. It can be made into an explicit form φ j+1 = M 1 Rφ j but then the molecule is more complicated. In our case we are looking for the numerical values of the φ.,j s in the equation below φ j = R 1 Mφ j+1 + b where φ j = [φ 1,j, φ 2,j, φ 3,j,...φ I 1,j ] T, T is the transporse of the matrix and b is the boundary conditions. By analysing the eigenvalues of the matrix R 1 M it can be shown that the scheme is stable for all k/h 2. A proof is given in most texts on the subject. 9 Numerical integration Numerical integration is the approximate computation of an integral using numerical techniques. The numerical computation of an integral is sometimes called quadrature. Ueberhuber (1997, p. 71) uses the word quadrature to mean numerical computation of a univariate integral, and cubature to mean numerical computation of a multiple integral. When the integrand or some of its low-order derivative

52 is infinite at some point in or near the interval of integration, standard quadrature rules will not work well. It is not uncommon that a single step taken close to such a discontinuity point will give a larger error than all other steps combined. In some cases a point of discontinuity can be completely missed by the quadrature rule. If the points of discontinuity are known, then the integral should first be broken up into several pieces so that all the points of discontinuity are located at one (or both) ends of the interval [a, b].many integrals can then be treated by weighted quadrature rules, i.e., the points of discontinuity are incorporated into the weight function. 9.1 The case λ P 0 (i.e. in the presence of jumps or claims) In this section we wish to find the function φ(t, y) satisfying the model (17) subject to the same initial and boundary conditions as in (16). Inorder to progress we first need to approximate the integral in the model (17) by numerical integration.the primary purpose of numerical integration (or quadrature) is the evaluation of integrals which are either impossible or else very difficult to evaluate analytically. Numerical integration is also essential for the evaluation of integrals of functions available only at discrete points. Such functions often arise in the numerical solution of differential equations or from experimental data taken at discrete intervals. In this study we prefer to use the Simpson s rule because the error is proportional to ( y) 4. 9.2 Simpson s Rule In numerical analysis, Simpson s rule is a method for numerical integration, the numerical approximation of definite integrals. Simpson s rule is a numerical integration technique which is based on the use of parabolic arcs to approximate the function say, φ(y) instead of the straight lines employed on the trapezoid rule. Even higher order polynomials such as cubics can also be used to obtain more accurate results but in our case its impossible or else very difficult to evaluate the integral analytically. If the interval of integration [a, b] is in some sense small, then Simpson s

53 rule will provide an adequate approximation to the exact integral. By small, what we really mean is that the function being integrated is relatively smooth over the interval [a, b]. For such a function, a smooth quadratic interpolant like the one used in Simpson s rule will give good results. However, it is often the case that the function we are trying to integrate is not smooth over the interval. Typically, this means that either the function is highly oscillatory, or it lacks derivatives at certain points. In these cases, Simpson s rule may give very poor results. One common way of handling this problem is by breaking up the interval [a, b] into a number of small subintervals. Simpson s rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. Consider an integrable function φ(y) over an interval [a, b]. We wish to evaluate the intergral I = b a φ(y)dy We divide the interval [a, b] into n equal subintervals each of width y, where h = y = b a n Each of the subintervals will be referred to as a panel. Simpson s 1/3 rule is obtained when a second-order interpolating polynomial is substituted for φ(y) I = b a φ(y) b a φ 2 (y)dy where φ 2 (y) is a second-order Lagrange interpolating polynomial using the three points y i 1, y i and y i+1 φ 2 (y) = (y y i )(y y i+1 ) (y i 1 y i )(y i 1 y i+1 ) φ(y i 1) + (y y i 1)(y y i+1 ) (y i y i 1 )(y i y i+1 ) φ(y i) + (y y i 1)(y y i ) (y i+1 y i 1 )(y i+1 y i ) φ(y i+1) Consider an expanded view of a general region including one panel where the points φ(y i 1 ), φ(y i ), and φ(y i+1 ) have been connected by a parabola. This parabola approximates the function φ 2 (y) between y i 1 and y i+1. Approximating the area of the

54 panel by the area under the parabola yields yi+1 y i 1 φ 2 (y)dy = yi+1 [ y i 1 (y y i )(y y i+1 ) (y i 1 y i )(y i 1 y i+1 ) φ(y i 1) + (y y i 1)(y y i+1 ) (y i y i 1 )(y i y i+1 ) φ(y i) + (y y i 1)(y y i ) (y i+1 y i 1 )(y i+1 y i ) φ(y i+1)]dy This expression can be integrated and simplified to yi+1 y i 1 φ 2 (y)dy = y 3 [φ(y i 1) + 4φ(y i ) + φ(y i+1 )] Therefore (with φ 2 := φ) the Simpson s rule approximation to the integral over the entire interval is b a φ(y)dy y 3 (φ 0 + 4φ 1 + 2φ 2 + 4φ 3 +... + 2φ n 2 + 4φ n 1 + φ n ) = y 3 (φ n 1 n 2 0 + 4 φ i + 2 φ i + φ n ) i=1 where φ(y j ) = φ(a + jh) for j = 0, 1,..., n 1, n with y = b a ; in particular, n φ 0 = φ(y 0 ) = φ(a) and φ n = φ(y n ) = f(b). The truncation error from the application of the Simpson s rule over [a, b] is i=2 E = ( y)4 180 (b a)φiv (y) where φ (y) is the average second derivative over the interval. So, we have where b a φ(y)dy = y 3 (φ n 1 n 2 0 + 4 φ i + 2 φ i + φ n ) ( y)4 180 (b a)φiv (y) i=1 i=2 n 1 i=1 φ i = φ 1 + φ 3 +... + φ n 1, n 2 i=2 φ i = φ 2 + φ 4 +... + φ n 2 are sums over odd and even indices, respectively. This shows that we have gained two orders of accuracy compared to the trapezoidal rule, without using more function evaluation. This is why Simpson s rule is such a popular general-purpose quadrature rule. Simpson s rule is termed a fourth order method of numerical integration because the error is proportional to ( y) 4. Example: Demonstrate the use of the Simpson s rule with n = 4 to evaluate I = π 0 sin(y)dy

55 Simpson s rule with n = 4 yields I = π/4 3 {φ 0 + 4 [φ(π/4) + φ(3π/4)] + 2φ(π/2) + φ(π)} = π {sin(0) + 4 [sin(π/4) + sin(3π/4)] + 2 sin(π/2) + sin(π)} 12 = 0.261799 [0 + 4(0.707107 + 0.707107) + 2(1.0) + 0] = 2.00456 Here, the exact solution is 2 and the error is 0.000228. From these results, we can see that the difference between the exact and approximate solutions is very minimal. Also the error is very small and this result gives us confidence to use the Simpson s rule to approximate the integral appearing in the model of our study. We are now in the position to demonstrate the use of Simpson s rule with n = 10 to evaluate the integral in (17) as shown below. y I = λ P φ(t, y x)df P (x) 0 From the given initial and boundary conditions, we may consider the function φ(t, y x) = 0.001. Since F P (x) = 1 e αx the integral above reduces to I = λ P α The general formula for this is given by λ P α b a where, t,y x 0 e αx dx φ(t, y x)dy y 3 [φ(y 0) + 4φ(y 1 ) + 2φ(y 2 ) + 4φ(y 3 ) + 2φ(y 4 ) +... + 4φ(y n 1 ) + φ(y n )] y 0 = a = 0 y 1 = a + y = 0.1 y 2 = a + 2 y = 0.2 y 3 = a + 3 y = 0.3 y 4 = a + 4 y = 0.4 (23)

56 y 5 = a + 5 y = 0.5 y 6 = a + 6 y = 0.6 y 7 = a + 7 y = 0.7 y 8 = a + 8 y = 0.8 y 9 = a + 9 y = 0.9 y n = a + y y = 0.1y Substituting these values in (23) we obtain λ P α y 0 e αx dx 0.1 3 = 0.1 3 [ 1 + 4e 0.1α + 2e 0.2α + 4e 0.3α + 2e 0.4α +... + 4e 0.9α + e αy] [ 18.59574785 + e y ] = 0.6199 + 0.0333e y We can now multiply this result by 0.001 and incorporate it to the rest of the model and use MATLAB to obtain the results as shown in the table below. In the next chapter we will report some numerical results obtained using the methods described in this chapter.

57 CHAPTER FIVE NUMERICAL RESULTS

58 CHAPTER FIVE NUMERICAL RESULTS 6 Introduction In this chapter we will report some numerical results obtained using the methods described in chapter 4. Most of our calculations uses k = 0.001, h = 0.1 and the number of time step n = 100 for observation domain T in [0, 1]. For a given y, let φ(t, y) be the survival probability. This implies that the ruin probability will be given by ψ(t, y) = 1 φ(t, y). All the calculations were done on a PC and the parameters commonly used in this literature are p = 0.1, r = 0.1, σ P = σ R = 0.2, T = 1, λ P = 1 and α = 1. The program language was MATLAB used to perform the simulation. Of course other programs like FORTRAN, C++ and Mathematica could have been, but only at the expense of considerable longer computing time. 7 The case λ P = 0 (i.e. in the absence of jumps or claims) Here we present the results of the survival probabilities as shown in Table 5.1. We shall also make use of the survival probability function φ(t, y) = 1 ψ(t, y) to calculate the ruin probability ψ(t, y) = 1 φ(t, y) shown in Table 5.2.

59 Table 7.1: Calculated survival probabilities φ(t, y) with no jumps with p = 0.1, r = 0.1, σ P = σ R = 0.2, T = 1 Values of y t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.001 0.9945 0.9890 0.9836 0.9782 0.9729 0.9676 0.9623 0.9570 0.9518 0.002 0.9917 0.9835 0.9753 0.9672 0.9592 0.9512 0.9433 0.9354 0.9276 0.003 0.9913 0.9827 0.9742 0.9657 0.9573 0.9490 0.9408 0.9326 0.9245 0.004 0.9908 0.9816 0.9725 0.9636 0.9547 0.9458 0.9371 0.9284 0.9199 0.005 0.9900 0.9802 0.9704 0.9608 0.9512 0.9418 0.9324 0.9231 0.9139 0.006 0.9892 0.9785 0.9679 0.9574 0.9470 0.9368 0.9266 0.9166 0.9067 0.007 0.9881 0.9764 0.9649 0.9534 0.9421 0.9310 0.9199 0.9090 0.8982 0.008 0.9870 0.9741 0.9514 0.9489 0.9365 0.9243 0.9122 0.9003 0.8886 0.009 0.9853 0.9707 0.9563 0.9421 0.9280 0.9141 0.9003 0.8867 0.8732 1.000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

60 From Table 5.1 we see that the survival probability is much higher during the initial stages of time. As time increases the this probability decreases. This could be due to the reason that with the increase in time, the expenses in an insurance portfolio also increases. It is these expenses that reduces the survival probability which in turn increases the ruin probability.

61 Table 7.2: Calculated ruin probabilities ψ(t, y) with no jumps with p = 0.1, r = 0.1, σ P = σ R = 0.2, T = 1 Values of y t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.001 0.0055 0.0110 0.0164 0.0218 0.0271 0.0324 0.0377 0.0430 0.0482 0.002 0.0083 0.0165 0.0247 0.0328 0.0408 0.0488 0.0567 0.0646 0.0724 0.003 0.0087 0.0173 0.0258 0.0343 0.0427 0.0510 0.0592 0.0674 0.0755 0.004 0.0092 0.0187 0.0275 0.0364 0.0453 0.0542 0.0629 0.0716 0.0801 0.005 0.0100 0.0198 0.0296 0.0392 0.0488 0.0582 0.0676 0.0769 0.0861 0.006 0.0108 0.0215 0.0321 0.0426 0.0530 0.0632 0.0734 0.0834 0.0933 0.007 0.0119 0.0236 0.0351 0.0466 0.0579 0.0690 0.0801 0.0910 0.1018 0.008 0.0130 0.0259 0.0386 0.0511 0.0635 0.0757 0.0878 0.0997 0.1114 0.009 0.0147 0.0293 0.0437 0.0579 0.0720 0.0859 0.0997 0.1133 0.1268 1.000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 In Table 5.2 we see that the ruin probability is high in the initial stages of time and the distribution of the time to ruin is concentrated on the first few time periods. This is expected as the surplus from good experience has not been built up over the first few time periods to absorb any adverse experience over that time. Again, the time to ruin tends to increase as the initial capital increases, as the initial capital is sufficient to offset any adverse experience over the first few time periods. This means that the surplus process built up over the first few years may not be sufficient to cover the high number of claims.

62 Table 8.1: Calculated survival probabilities φ(t, y) with Exponentially distritributed jumps with p = 0.1, r = 0.1,σ P = σ R = 0.2,T = 1. Values of y t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.001 0.9952 0.9899 0.9845 0.9792 0.9740 0.9687 0.9635 0.9584 0.9532 0.002 0.9925 0.9845 0.9765 0.9685 0.9607 0.9528 0.9451 0.9374 0.9297 0.003 0.9922 0.9838 0.9756 0.9674 0.9592 0.9511 0.9431 0.9352 0.9273 0.004 0.9917 0.9829 0.9741 0.9655 0.9569 0.9483 0.9399 0.9315 0.9232 0.005 0.9911 0.9816 0.9722 0.9629 0.9537 0.9446 0.9356 0.9266 0.9178 0.006 0.9903 0.9800 0.9698 0.9598 0.9498 0.9400 0.9302 0.9206 0.9110 0.007 0.9893 0.9781 0.9670 0.9560 0.9451 0.9344 0.9238 0.9133 0.9029 0.008 0.9882 0.9758 0.9636 0.9516 0.9397 0.9280 0.9164 0.9050 0.8937 0.009 0.9865 0.9725 0.9587 0.9450 0.9314 0.9180 0.9048 0.8916 0.8786 8 The case λ P 0 (i.e. in the presence of jumps or claims)

63 Table 8.2: Calculated ruin probabilities ψ(t, y) with Exponentially distributed jumps with p = 0.1, r = 0.1, σ P = σ R = 0.2, T = 1. Values of y t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.001 0.0048 0.0101 0.0155 0.0208 0.0260 0.0313 0.0365 0.0416 0.0468 0.002 0.0075 0.0155 0.0235 0.0315 0.0393 0.0472 0.0549 0.0626 0.0703 0.003 0.0078 0.0162 0.0244 0.0326 0.0408 0.0489 0.0569 0.0648 0.0727 0.004 0.0083 0.0171 0.0259 0.0345 0.0431 0.0517 0.0601 0.0685 0.0768 0.005 0.0089 0.0184 0.0278 0.0371 0.0463 0.0554 0.0644 0.0734 0.0822 0.006 0.0097 0.0200 0.0302 0.0402 0.0502 0.0600 0.0698 0.0794 0.0890 0.007 0.0107 0.0219 0.0330 0.0440 0.0549 0.0656 0.0762 0.0867 0.0971 0.008 0.0118 0.0242 0.0364 0.0484 0.0603 0.0720 0.0836 0.0950 0.1063 0.009 0.0135 0.0275 0.0413 0.0550 0.0686 0.0820 0.0952 0.1084 0.1214

64 Table 8.3: Calculated survival probabilities φ(t, y) with Pareto distritributed jumps with p = 0.1, r = 0.1, σ P = σ R = 0.2, T = 1. Values of y t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.001 0.9952 0.9899 0.9845 0.9792 0.9740 0.9687 0.9635 0.9584 0.9532 0.002 0.9925 0.9845 0.9765 0.9685 0.9607 0.9528 0.9451 0.9374 0.9297 0.003 0.9922 0.9838 0.9756 0.9674 0.9592 0.9511 0.9431 0.9352 0.9273 0.004 0.9917 0.9829 0.9741 0.9655 0.9569 0.9483 0.9399 0.9315 0.9232 0.005 0.9911 0.9816 0.9722 0.9629 0.9537 0.9446 0.9356 0.9266 0.9178 0.006 0.9903 0.9800 0.9698 0.9598 0.9498 0.9400 0.9302 0.9206 0.9110 0.007 0.9893 0.9781 0.9670 0.9560 0.9451 0.9344 0.9238 0.9133 0.9029 0.008 0.9882 0.9758 0.9636 0.9516 0.9397 0.9280 0.9164 0.9050 0.8937 0.009 0.9865 0.9725 0.9587 0.9450 0.9314 0.9180 0.9048 0.8916 0.8786

65 Table 8.4: Calculated ruin probabilities ψ(t, y) with Pareto distributed jumps with p = 0.1, r = 0.1, σ P = σ R = 0.2, T = 1. Values of y t 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.001 0.0048 0.0101 0.0155 0.0208 0.0260 0.0313 0.0365 0.0416 0.0468 0.002 0.0075 0.0155 0.0235 0.0315 0.0393 0.0472 0.0549 0.0626 0.0703 0.003 0.0078 0.0162 0.0244 0.0326 0.0408 0.0489 0.0569 0.0648 0.0727 0.004 0.0083 0.0171 0.0259 0.0345 0.0431 0.0517 0.0601 0.0685 0.0768 0.005 0.0089 0.0184 0.0278 0.0371 0.0463 0.0554 0.0644 0.0734 0.0822 0.006 0.0097 0.0200 0.0302 0.0402 0.0502 0.0600 0.0698 0.0794 0.0890 0.007 0.0107 0.0219 0.0330 0.0440 0.0549 0.0656 0.0762 0.0867 0.0971 0.008 0.0118 0.0242 0.0364 0.0484 0.0603 0.0720 0.0836 0.0950 0.1063 0.009 0.0135 0.0275 0.0413 0.0550 0.0686 0.0820 0.0952 0.1084 0.1214

66 CHAPTER SIX CONCLUSION CHAPTER SIX CONCLUSION In this dissertation, the problem of ruin probabilities in finite time horizon has been dealt with. Main emphasis has been on the classical risk model in an insurance process compounded with an independent stochastic return on investments of Black and Scholes type. Very little is known analytically in dealing with numerical methods for calculating the ruin probability in finite time horizon. We have dealt with two cases one with no jumps and the other one with jumps. In cases where we have jumps we have considered the light tailed Exponentially distributed jumps and the heavy tailed Pareto distributed distributed jumps. In both cases analytical solutions seemed so hard to get due to complexity of the equations. The complexity of the equations increases with the complexity of the model, and explicit formulas quickly become unattainable. Instead we put emphasis on numerical procedures and give some numerical results. The complexity of the models can be due to the stochastic nature of the portfolio (the intesities) which the company cannot control, or the policy of the company itself such as reinsurance schemes and premium settings. One numerical method has been suggested and this is an implicit finite difference scheme. Our results confirm intuitive thinking that when the surplus process is compounded by an independent stochastic return on investments the resulting surplus process has a minimum finite horizon ruin probabilities. Obviously this means that the ruin probability is reduced in the presence of positive interest. However we have noticed that the presence of exponetially distributed jumps make the finite horizon ruin probability less than in the case where there are no jumps. Similarly the number of exponetial claims happening in a short period of time reduces the finite horizon ruin probability contrary to what was predicted earlier that the model with jumps will have a bigger finite horizon ruin probability as compared to the model with no claims. This could be due to the reason that exponential claims do not deteriorate with time. By this we mean that if the time of the claim is expo-

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