2nd ANKARA-ISTANBUL WORKSHOP ON STOCHASTIC PROCESSES JUNE 2015 KOÇ UNIVERSITY SPEAKERS, TITLES & ABSTRACTS

Transcription

1 2nd ANKARA-ISTANBUL WORKSHOP ON STOCHASTIC PROCESSES JUNE 2015 KOÇ UNIVERSITY SPEAKERS, TITLES & ABSTRACTS The speakers are in alphabetical order with respect to their last names. Ceren Vardar Acar, Department of Mathematics, TOBB University of Economics and Technology, Ankara Distribution of Maximum Loss of Fractional Brownian Motion with Drift In this talk, we present bounds on the distribution of the maximum loss of fractional Brownian motion with H 1/2 and derive estimates on its tail probability. Asymptotically, the tail of the distribution of maximum loss over [0, t] behaves like the tail of the marginal distribution at time t. This is joint work with Mine Çağlar. Hülya Acar, Department of Mathematics, Fatih University, Istanbul Loewner Evolution as Itô Diffusion F. Bracci, M.D. Contreras, S. Díaz Madrigal proved that any evaluation family of order d is described by a generalized Loewner chain. G. Ivanov and A. Vasil ev considered randomized version of the chain and found a substitution which transforms it to an Itô diffusion. We generalize their result to vector randomized Loewner chain and prove there are no other possibilities to transform such Loewner chains to Itô diffusions. This is joint work with Alexey Lukashov. 1

2 Savaş Dayanık, Bilkent University, Ankara Bidding for multiple keywords in search-based advertising Search-based advertisement allows small-to-medium size companies to use more effectively their advertisement budgets to target their potential customers. The advertisers manage this by bidding for a place on the result pages for several search keywords relevant to their products and services. The customer flow rates to the advertiser business pages change with the placements of advertisements on the pages, which are determined by the bidding prices of the advertisers in closed second-price auctions. We model the bidding process and calculate optimal dynamic bidding prices for several keywords subject to a single daily advertisement budget. (This is joint work with Semih Onur Sezer, Sabancı University) Hatem Hajri, University of Paris 10, Paris Flows Associated to Tanaka s SDE Revisited The stochastic dynamic associated to a stochastic differential equation is the study of its unique flow of maps when the solution is a strong one and its possibly infinite flows of kernels when the solution is only weak. The first part of this talk will be an overview on the theory of stochastic flows. In a second part, I will consider the example of Tanaka s equation to which the theory of flows of kernels was first applied by Le Jan and Raimond. The main result of Le Jan and Raimond shows that the law of each flow of kernels is uniquely characterized by a probability measure on [0, 1] according to which points are divided at the origin. The purpose of this talk is to present an elementary proof of this result based on skew Brownian motion. 2

3 Ümit Islak, Mathematics Department, University of Minnesota, Minneapolis Stein s Method and Subsequence Problems First, I will briefly discuss the basics of Stein s method which is a technique used for obtaining convergence in distributional approximations. Then, we will focus on the longest common subsequence (LCS) problem, and we will see how a recent approach in Stein s method can be used to study the asymptotics for the LCS of of two independent random words with i.i.d. coordinates (under certain asymmetry assumptions). Time permits, I will also mention some ongoing work on longest increasing subsequences. Erkan Nane, Department of Mathematics and Statistics, Auburn University, Auburn Intermittence and Space-Time Fractional Stochastic Partial Differential Equations I will consider time fractional stochastic heat type equations. The time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. In this talk I discuss: (i) Existence an uniqueness of solutions and existence of a continuous version of the solution; (ii) absolute moments of the solutions of this equation grows exponentially; and (iii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. These results extend the results of Mohammud Foondun and Davar Khoshnevisan, (Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, ) and Conus and Khoshnevisan (On the existence and position of the farthest peaks of a family of stochastic heat and wave equations, Probab. Theory Related Fields 152 (2012), no. 3-4, ) on the parabolic stochastic heat equations. These results are our recent joint work with Jebessa B Mijena. 3

4 Yeliz Yolcu Okur, Institute of Applied Mathematics, METU, Ankara Different Perspectives of Malliavin Calculus in Finance : Greeks in the Clark-Ocone Formula under Change of Measure and Analysis of Volatility Feedback Effect Rate This talk has two parts. In the first part of my talk, I will basically establish the connection between the computation of Greeks and the generalized Clark representation formula in Black-Scholes setting. This work aims to gather and compare the ideas in the integral representation of square integrable functional of Brownian motions with computation of Greeks. One of the practical problem in using the celebrated Clark-Ocone formula under change of measure is to evaluate the conditional expectation of Malliavin derivative of square integrable random variable. This random variable usually represents the discounted payoff of the contingent claim in the applications of Malliavin calculus in finance. In complete financial markets, we show that this value can be easily evaluated by the Delta (the sensitivity with respect to initial risky asset price) of the contingent claim. The second part of the talk is basically on the techniques of Malliavin calculus to analyze the volatility feedback effect for a better understanding of financial market dynamics. First, the definition of feedback effect rate will be introduced in terms of rescaled variation and the connections between market stability and this rate will be discussed in details. (This talk covers joint work with A. İnkaya) 4

5 Devin Sezer, Institute of Applied Mathematics, METU, Ankara Exit Probabilities and Balayage of Constrained Random Walks Let X be the constrained random walk on Z d + representing the queue lengths of a stable Jackson network and let x Z d + be its initial position. Let τ n be the first time. when the sum of the components of X equals n. The probability p n = Px (τ n < τ 0 ) is one of the key performance measures for the queueing system represented by X and its analysis/computation recieved considerable attention over the last several decades. The stability of X implies that p n decays exponentially n. Currently the only analytic method available to approximate p n is large deviations analysis, which gives the exponential decay rate of p n. Finer results are available via rare event simulation. The present article develops a new, fast and precise method to approximate p n and related expectations. The method has two steps: 1) with an affine transformation, move the origin to a point on the exit boundary associated with τ n ; take limits to remove some of the constraints on the dynamics of the walk; the first step gives a limit unstable constrained random walk Y 2) Construct a new basis of harmonic functions of Y and apply the classical superposition principle of linear analysis (the basis functions can be seen as perturbations of the classical Fourier basis). The basis functions are linear combinations of log-linear functions and come from solutions of harmonic systems; these are graphs with labeled edges whose vertices represent points on the interior characteristic surface of Y ; the edges between the nodes represent conjugacy relations between the points on the characteristic surface, the loops (edges on the same node) represent membership in the boundary characteristic surfaces. Characteristic surfaces are algebraic varities determined by the distribution of the unconstrained increments of X and the boundaries of Z d +. Using our method we derive explicit, simple and almost exact formulas for P x (τ n < τ 0 ) for d-tandem queues, similar to the product form formulas for the stationary distribution of X. The [ same method ] allows us to approximate the Balayage operator mapping f to x E x f(xτn )1 {τn <τ 0 } for a wide range of stable constrained random walks representing the queue lengths of a queueing system with two nodes (i.e., d = 2). We indicate how the ideas of the paper relate to more general processes and exit boundaries. 5

6 Semih Onur Sezer, Faculty of Engineering and Natural Sciences, Sabancı University, Istanbul Sequential Sensor Installation for Wiener Disorder Problem We consider the centralized Bayesian multi-sensor Wiener disorder problem with the additional feature that the observer can install new sensors sequentially before declaring a change. In this talk, we will first review the static version of this problem, and then we discuss how to find an optimal joint sequential sensor installation and detection policy. We will also give some numerical examples, which illustrate that a sequential policy can bring significant improvement, and it can start initially with less number of sensors. The talk is based on a joint work with Savaş Dayanık. Atilla Yılmaz, Department of Mathematics, Boğaziçi University, Istanbul The Stochastic Encounter-Mating Model I will present a recent joint paper of mine with Onur Gn (WIAS Berlin) where we propose a new model of permanent monogamous pair formation in zoological populations comprised of k 2 types of females and males, which unifies and generalizes the encounter-mating models of Gimelfarb (1988). In our model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, which depend on the sex and the type of the animals, we analyze the contingency table Q(t) of permanent pair types at any time t 0. First, we consider definite mating upon encounter and provide a formula for the distribution of Q(t). In particular, at the terminal time T, the so-called mating pattern Q(T ) has a multiple hypergeometric distribution. This implies panmixia which means that female and male types are uncorrelated in the expected mating pattern. Next, when the firing times come from Poisson and Bernoulli point processes, we formulate conditions that characterize panmixia. Moreover, when these conditions are satisfied, the underlying parameters of the model can be changed to yield definite mating upon encounter, and our results for the latter case carry over. Finally, when k = 2, we fully characterize heterogamy/panmixia/homogamy, i.e., negative/zero/positive correlation of same type females and males in the expected mating pattern. We thereby rigorously prove, strengthen and generalize Gimelfarb s results. 6

7 Bilgi Yılmaz, Institute of Applied Mathematics, METU, Ankara Computation of Greeks via Malliavin Calculus As it is well known, one of the main problem in finance is pricing a contingent claim traded in the market. However, while dealing with this problem we have to encounter the unexpected price changes, related to parameters of pricing formula, during the (survival) time of the contingent claim. In the setting of Black-Scholes-Merton model (BSM), one can compute the effect of those parameters on the price, say Greeks, directly or approximately such as finite difference, likelihood and pathwise methods. In recent years, a new method came in the field of computation of Greeks in finance with the benchmark study of Fournie (1999, 2001) using Malliavin calculus. This new method allows obtaining the Greeks of any option as a multiplication of weight and payoff function, which increase computational efficieny in Monte Carlo method. In the light of pioneering studies so many remarkable studies have been done under the assumption of BSM and stochastic volatility models. This study is a review of recent contributions in the field of computation of Greeks via Malliavin calculus. Since this work is a review study, it will be mainly a presentation of the computation of the Greeks and some numerical results of Black-Scholes-Merton and general stochastic volatility models. Moreover, the results will be extended and applied to the general stochastic volatility hybrid models. POSTER TITLES & ABSTRACTS The presenters are in alphabetical order with respect to their last names. Presenter Animoku Abdulwahab, Institute of Applied Mathematics, METU, Ankara Local Volatility Modelling and its Implementation in Bayesian Framework In this study, we investigate advanced numerical techniques applied in Local Volatility (LV) setting in order to characterize the volatility surface. Apart from classical methods, such as, parametric and non-parametric, we study on Bayesian analysis of the (stochastically) parameterized volatility structure in Dupire local volatility equation. With this study, the pros and cons of these methods will be investigated. The most critical limitation of the classical methods is that one can obtain negative local variance values due to ill-posedness of the numerator and/or denominator of Dupire local variance equation. While several numerical techniques, such as Tikhonov regularization, Spline interpolations have been suggested to tackle this problem, we seek a more direct and robust approach. With the Bayesian analysis, we can tackle this challenge by choosing an uninformed prior on the positive plane which eliminates the possibility of negative local variance. 7

8 Presenter Sergazy Nurbavliyev, Department of Mathematics, Boğaziçi University, Istanbul Disordered regimes of directed lattice polymers in random environment and tree polymers in disordered trees Directed lattice polymers on the d+1 dimensional integer lattice are modeled by (random) distributions of graphs of polygonal paths in N Z d with N playing the role of the time and Z d playing the role of the space. We will use the term lattice polymer in reference to the directed polygonal paths on the d + 1 dimensional integer lattice, and tree polymer for the case of polygonal paths of a binary tree. Bolthausen s notion of weak and strong disorder environments will precisely be defined. While the primary focus of polymer research is aimed at low dimensional lattice polymer models, where sharp results are rare, the tree polymer is important for testing lattice methods because sharp results are often possible to obtain for tree paths. We will give some known results for lattice polymers and compare how sharp results we can get for tree polymers. Presenter Mustafa Hilmi Pekalp, Department of Statistics, Ankara University, Ankara Replacement Policies for α-series Processes The optimal replacement policies are studied for a deteriorating system. It is assumed that both sequential working and repair times follows an α-series processes with α 1 > 0 and α 2 < 0, respectively. Two replacement policies are considered. The policy T replaces the system with a new and identical one when the cumulative working time reaches T. The second policy N replaces the system at the N th failure time. We have derived the explicit expressions for the long-run expected cost per unit time by using renewal reward theorem. Then, the optimal T and N are determined by minimizing these expressions analytically or numerically. 8

9 Presenter Sinem Kozpınar Sarı, Institute of Applied Mathematics, METU, Ankara Pricing Equity Options with Stochastic Barrier in Presence of Jumps In this study, we will derive a partial integro-differential equation (PIDE) for an equity option under the assumption that the asset value process follows a double-exponential jump-diffusion model. Here, we consider that the firm defaults when the asset value of the firm hits a stochastic barrier which is defined as the recovery part of the debt. In order to find an approximate value for price function, we will then apply finite difference method to this partial integro-differential equation. Presenter Emel Savku, Institute of Applied Mathematics, METU, Ankara Optimal Control of Stochastic Hybrid Delayed Models Regime switching models may capture not only the sudden changes of behavior of financial markets but also the new dynamics and fundamentals persist for several periods after a change. In this framework, we establish sufficient maximum principle for the optimal control of a time-delayed stochastic hybrid model. The associated adjoint processes are shown to satisfy a time-advanced backward stochastic differential equation (ABSDE). Also, we study on necessary maximum principle for such a system and purpose to apply our results for portfolio optimization problems in finance. Moreover, we study on existence and uniqueness conditions of such ABSDEs. This is joint work with Gerhard-Wilhelm Weber. 9