Some stability results of parameter identification in a jump diffusion model

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Some stability results of parameter identification in a jump diffusion model"

Transcription

1 Some stability results of parameter identification in a jump diffusion model D. Düvelmeyer Technische Universität Chemnitz, Fakultät für Mathematik, Chemnitz, Germany Abstract In this paper we discuss the stable solvability of the inverse problem of parameter identification in a jump diffusion model. Therefore we introduce the forward operator of this inverse problem and analyze its properties. We show continuity of the forward operator and stability of the inverse problem provided that the domain is restricted in a specific manner such that techniques of compact sets can be exploited. Furthermore, we show that there is an asymptotical non-injectivity which causes instability problems whenever the jump intensity increases and the jump heights decay simultaneously. Keywords: Jump diffusion, parameter identification, compact domain, stability, illposedness MSC000 classification scheme numbers: 65J0, 91B84, 6F10, 47H99

2 8 D. Düvelmeyer 1 Introduction In this paper we consider the inverse problem of parameter estimation from return observations in a jump diffusion model. The random price process S t, t [0, satisfying the stochastic differential equation ds t = S t µ λνdt + σdw t + S t dn c t, t 0,, S 0 = ξ, is a jump diffusion process where W t, t [0, is a standard Wiener process and Nt c, t [0, an independent compound Poisson process with intensity λ 0 and jump amplitudes Y j 1, j N. We assume that the random variables log Y j, j N, are independent Gaussian variables with mean µ Y and variance σy. In addition we have ν = e µ Y + 1 σ Y 1 and an initial value ξ. The stochastic behavior of this specific model presented in [] is determined by the parameter vector p D with D := { p = µ,σ,λ,µ Y,σ Y T R 5 : σ > 0,λ 0,σ Y 0. Then the stationary returns r τ := log S t+τ S t for a fixed time lag τ > 0 fulfill the equation N τ r τ = µτ + σw τ + log Y j, where we have µ = µ λν 1 σ. By applying the law of total probability we express the distribution function as Fx,p = Pr τ x = PN τ = j Pr τ x N τ = j = j=1 e λτ λτ j x µτ + jµ Y Φ. j! σ τ + jσy Consequently, the density function attains the form fx,p = e λτ λτ j j! x µτ + jµ Y φ, σ τ + jσy σ τ + jσy where Φx and φx denote distribution and density function of the standard normal distribution. We assume p D to be the exact parameter vector to be determined and analyze the estimation problem by using methods of inverse problem theory in order to find approximate solutions p δ of p, which stably depend on the vector S δ = S δ t 0,S δ t 1,...,S δ t n T of noisy price data and associated returns r δ τ = r δ τ,1,...,r δ τ,n T. Therefore we consider the empirical density function hx,r δ τ of the empirical returns belonging to the data S δ and choose that parameter vector p δ which minimizes the distance between the density function fx,p δ and the empirical density function hx,r δ τ. In this context, we define the operator of the forward problem.

3 Some stability results of parameter identification in a jump diffusion model 9 Definition 1.1 The operator A : p f of the forward problem maps the parameter vector p D to the density function f,p using the following series expansion: [Ap]x = fx,p = e λτ λτ j j! σ τ + jσ Y φ x µτ + jµ Y σ τ + jσ Y x R. 1.1 Throughout this paper AD denotes the range of the operator. Thus, the inverse problem can be written as the nonlinear operator equation Ap = f p D R 5, f AD Y, 1. where Y is a Banach space with norm Y. We analyze properties of equation 1., particularly for Y = CR and Y = L R. Noisy data Instead of the exact density function f we observe the empirical density function hx,r δ τ for solving 1.. This function can be considered as a noisy data function of f and hence we write h,r δ τ = f δ. Assumption.1 We assume that data function f δ fulfills the following conditions 1. R f δ xdx L 1,. f δ x L, x R, 3. f δ f Y δ and 4. f δ x 0, x R, where L 1 and L are positive constants. In general the data function f δ Y need not belong to the range AD of the forward operator A. Therefore we consider the least squares problem Ψp := Ap f δ min for p D..1 L R This leads us to the inverse problem of parameter estimation. Definition. Let the noisy data function f δ Y with noise level δ > 0 satisfy the conditions of assumption.1. Then the problem is to find appropriate approximations p δ of p. We are especially interested in the unique and stable solvability of the inverse problem of parameter estimation. Workshop Stochastische Analysis

4 30 D. Düvelmeyer 3 Properties of the forward operator First of all, the operator A defined by 1.1 is well-defined for all parameters p in the domain D, because each summand is nonnegative and we have [Ap]x = e λτ λτ j j! x µτ + jµ Y φ σ τ + jσy σ τ + jσy e λτ λτ j j! σ τ + jσy π e λτ λτ j j! 1 1 = σ τ π σ < σ > 0. πτ Besides we find [Ap]xdx = R = = R e λτ λτ λτj e j! λτ λτj e j! resulting in Ap L q R 1 q. λτ j j! σ τ + jσy R φ 1 φ σ τ + jσy = e λτ e λτ = 1, x µτ + jµ Y dx σ τ + jσy x µτ + jµ Y dx σ τ + jσy For fixed p the function [Ap]x is an infinite mixture of weighted density functions g j x,p of Gaussian variables with mean µτ + jµ Y and variance σ τ + jσy, that is 1 x µτ + jµ g j x,p = Y φ. 3. σ τ + jσy σ τ + jσy Furthermore, the weights w j = e λτ λτ j correspond with the probability that j jumps j! occur. Such a mixture of bell-shaped curves also appears at considering linear Fredholm integral equations [Fx]s = b a ks, txt dt = ys 3.3 with bell-shaped kernels ks, where k C[c,d] [a,b] see for example [4], example.3. In the pair of Banach spaces X = C[a,b] and Y = C[c,d] the operator F is compact and hence the operator equation 3.3 is ill-posed. Although the operator equation 1. is mapping from the finite dimensional space D R 5 and although it is determined by an infinite sum instead of the integral, the mixture of bell-shaped curves has a similar smoothing character. However, the smoothing character causes some ill-posedness phenomena occurring in solving.1 numerically which we have seen in [].

5 Some stability results of parameter identification in a jump diffusion model Continuity of the forward operator We know that for every fixed p D the density function f,p is continuous over R. The continuity of the function f : x,p y R + is necessary for the continuity of A. Each summand f j x,p = e λτ λτ j j! x µτ + jµ Y φ σ τ + jσy σ τ + jσy 3.4 of the series 1.1 is for all x,p R D continuous. Moreover, the series 1.1 converges for all x R and p D. The continuity of each series element and the convergence of the series are, nevertheless, not sufficient for the continuity of the limit function fx,p. On the other hand, uniform convergence is sufficient. Theorem 3.1 The function f : x,p R is continuous for all x,p R D. Proof. First of all we consider the series elements 3.4. Together with e λτ 1 1 and σ τ+jσy we get the following upper bound for all x,p R D: 1 σ τ f j x,p λτ λτj = e j! 1 φ σ τ + jσy x µτ + jµ Y σ τ + jσy λτj j! πτσ. 3.5 There exists for every parameter vector p 0 D a δ 0 > 0, so that for the δ 0 ball B δ0 p 0 = {p D : p p 0 δ0 D the relation σ min := σ min B δ0 p 0 = min σ > σ: p=µ,σ,λ,µ Y,σ Y T B δ0 p 0 is fulfilled. We can find lower and upper bounds for every parameter value of the parameter vectors p B δ0 D, in particular λ max := max λ <. λ: p=µ,σ,λ,µ Y,σ Y T B δ0 p 0 Consequently, we have λ λ max < and 0 < σ min σ for all parameter vectors p B δ0 p 0 D and obtain from 3.5 for all p B δ0 p 0 D the estimation The series c j = f j x,p λ maxτ j j! πτσ min =: c j. λ max τ j j! πτσ min = eλmaxτ πτσmin < Workshop Stochastische Analysis

6 3 D. Düvelmeyer converges such that the comparison test implies the uniform convergence of 1.1 in R B δ0 p 0. The uniform convergence of series to the function f ensures that f is continuous for all pairs x,p in R B δ0 see [5, pp ]. Since we have chosen p 0 D arbitrarily, the function fx,p is continuous in R D. Remark 3. We have proven that the function f : x,p y R + is continuous in R D. Additionally, we have the uniformly continuity of f for all pairs x,p in the compact set [ C,C] B δ0 p 0 for an arbitrarily constant C > 0. Theorem 3.3 The operator A mapping from domain D R 5 to Y = CR is continuous for all p D. Proof. We have to show that for a sufficiently small δ = δε > 0 parameter vectors p from the δ ball B δ p 0 satisfy Ap Ap 0 = f,p f,p CR 0 = supfx,p fx,p CR 0 < ε. 3.7 In remark 3. we have just seen that for an arbitrarily chosen constant C > 0 the function f is for all x,p [ C,C] B δ p 0 uniformly continuous. This implies fx,p fx,p 0 < ε p B δ p 0, x [ C,C] 3.8 for a sufficiently small δ = δε. So we still have to show fx,p fx,p 0 < ε p B δ p 0, x R \ [ C,C]. Therefore we analyze the function f for pairs x,p R \ [ C,C] B δ p 0. Every summand f j of the series 1.1 decays exponentially for growing absolute values of x and hence the values of f are going to zero. However, we show that for every ε > 0 a constant C exists such that x R fx,p < ε x,p R \ [ C,C] B δ p is satisfied. We can estimate each summand 3.4 for all pairs x,p R B δ p 0 with f j x,p = e λτ λτ j x µτ + j! σ τ φ jµy e λτ λτ j σ τ j! π σmin τ =: f j λ. The upper bound f j λ converges for all parameter vectors p B δ p 0 very quick to zero when j converges to infinity. Since f j λ = e λτ λτ j j! π σ min τ < εj+1

7 Some stability results of parameter identification in a jump diffusion model 33 is equivalent to j 1 1 λτ = λτj ε j! ε j! ε < j+1 eλτ πσmin τ 3.10 and the left-hand side in 3.10 converges to zero for j, there is a finite ĵ = ĵ ε for every 0 < ε < 1 such that f j x,p < ε j+1 is fulfilled for all x,p R B δ p 0 and j ĵε. Now, we consider the first ĵ summands. For all p B δ p 0 it yields lim f jx,p x ± and hence there exists for each j < ĵ a constant C j with f j x,p < ε j+1 x,p R \ [ C j,c j ] B δ p 0 j = 0,...,ĵ 1. By choosing C = max { C 1,...,Cĵ 1 we get f j x,p < ε j+1 x,p R \ [ C,C] B δ p 0 j = 0, 1,... Then by choosing ε = ε ε+ after all we obtain for x,p R \ [ C,C] B δp 0 fx,p = f j x,p < ε j+1 1 = ε 1 ε = ε, i.e. 3.9 is fulfilled. Accordingly we have Ap Ap 0 = f,p f,p CR 0 = supfx,p fx,p CR 0 x R { = max sup fx,p fx,p 0, max fx,p fx,p 0. x R\[ C,C] x [ C,C] From 3.9 it yields for the first part of the right-hand side sup fx,p fx,p 0 sup fx,p + fx,p 0 < ε x R\[ C,C] x R\[ C,C] + ε = ε. Furthermore, relation 3.8 results in fx,p fx,p 0 < ε x,p [ C,C] B δ p 0 which implies for all p B δ p 0 the relation 3.7 { Ap Ap 0 = max sup fx,p fx,p CR 0, max x R\[ C,C] < max {ε,ε = ε. x [ C,C] fx,p fx,p 0 Workshop Stochastische Analysis

8 34 D. Düvelmeyer With theorem 3.3 we directly get the continuity of the operator A in the pair of spaces R 5 and L q R. Theorem 3.4 The operator A mapping from domain D R 5 to Y = L q R 1 < q < is continuous for all p D. Proof. From theorem 3.3 we know that for all p B δ p 0 we have and which implies R fx,p fx,p 0 dx Ap Ap 0 q = L q R sup fx,p fx,p 0 < ε R fx,p fx,p 0 q dx { q 1 sup fx,p fx,p 0 < ε q 1 =: ε q, fx,p + fx,p 0 dx, R fx,p fx,p 0 dx and hence for all 1 < q < the continuity of A concerning the L q norm. 3. Uniqueness and stability First of all we are going to analyze the uniqueness of the inverse problem 1.. The operator A is obviously not injective on D. To see this, we consider the parameter vectors p 1 = µ,σ,λ, 0, 0 T and p = µ,σ, 0,µ Y,σ Y T. Both vectors map to the same density function [Ap 1 ]x = [Ap ]x = 1 σ τ φ x µτ σ τ x R, which is a normal density function, because the jump part is eliminated. In the case of p 1 the jump size is always zero and in the second case of p jumps do not occur. However, this trivial case is the only example for non-injectivity. The following proposition is proven in [8].

9 Some stability results of parameter identification in a jump diffusion model 35 Proposition 3.5 The operator A is injective on the restricted domain ˆD = { p D : λ σ Y + µ Y 0. Corollary 3.6 Let p 1,p D be two parameter vectors with Ap 1 = Ap and p 1 p. Then p 1 and p are in D \ ˆD, i.e. λ 1 µ Y 1 + σ Y 1 = λ µ Y + σ Y = 0. We prove now that in case of corollary 3.6 the diffusion parameters µ 1 and µ as well as σ 1 and σ coincide. Lemma 3.7 Let p 1 = µ 1,σ 1,λ 1,µ Y 1,σ Y 1 T D\ ˆD and p = µ,σ,λ,µ Y,σ Y T be two parameter vectors such that Ap 1 = Ap is fulfilled. Then p D \ ˆD and µ 1 = µ as well as σ 1 = σ are fulfilled. Proof. We know from the proof of proposition 3.5 that the logarithms of the characteristic functions log ϕθ,p 1 = i µ 1 θ σ 1 θ + λ 1 exp σ Y 1 exp log ϕθ,p = i µ θ σ θ + λ θ + iµ Y 1θ σ Y θ + iµ Y θ 1 coincide. Together with λ 1 µ Y 1 + σ Y 1 = 0 this implies the equation i µ 1 θ σ 1 θ = i µ θ σ θ + λ exp σ Y θ + iµ Y θ Resulting from 3.11 we get λ µ Y + σ Y = 0 and hence p D \ ˆD. By comparing the coefficients of 3.11 we obtain µ 1 = µ and σ 1 = σ and consequently µ 1 = µ. 1 and The operator equation Ap = f is uniquely solvable whenever a solution p ˆD exists. The parameter vector p = µ,σ,λ,µ Y,σ Y T D \ ˆD is a solution of 1., if and only if µ,σ, 0, ˆµ Y, ˆσ Y T D \ ˆD and µ,σ, ˆλ, 0, 0 T D \ ˆD are solutions of 1. for arbitrarily ˆµ Y R, ˆσ Y 0 and ˆλ 0. The operator equation 1. is not uniquely solvable for all density functions f in range AD. Therefore, the uniqueness condition see [6, p. 10] is not fulfilled and 1. is illposed. Now we construct an example where a sequence {f n n with f n = Ap n converges to f 0 = Ap 0, but the sequence {p n fails to converge to p 0. n Workshop Stochastische Analysis

10 36 D. Düvelmeyer Example 3.8 We consider the sequence {p n n of parameter vectors p D where p n = µ n,σ n,λ n,µ Y n,σ Y n. By choosing µ n µ and σ n σ as well as λ n = 1 n, µ Y n µ Y > 0 and σ Y n σ Y > 0 it yields p n ˆD for every n N. Let {f n n be the sequence of images, i.e. f n = Ap n. We also consider the parameter vector p 0 = µ,σ,λ 0, 0, 0 T D\ ˆD where µ 0 = µ, σ 0 = σ, λ 0 > 0 and σ Y 0 = µ Y 0 = 0. We denote its image by f 0 = Ap 0. The sequence {p n converges to ˆp = µ 0,σ 0, 0,µ Y,σ Y T D \ ˆD. The continuity of A n and the equation Ap 0 = Aˆp imply the convergence of {f n n to f 0 for arbitrarily µ Y and σ Y because µ n = µ 0 = µ, σ n = σ 0 = σ and lim λ 1 n µy n + σ Y n = lim µ n n n Y + σy = By using the Euclidean norm we have for arbitrary λ 0 > 0 as well as µ Y > 0 and σ Y > 0 1 lim p n n p 0 = lim n n λ 0 + µ Y + σy = λ 0 + µ Y + σy > 0, 3.13 such that the convergence of the images {f n n to f 0 does not result in the convergence of the parameter vectors {p n to p 0. By choosing µ Y n = σ Y n = n 1 g g > the distance n 3.13 can be increased without violating the relation 3.1. Example 3.8 shows that the convergence of {f n n to f 0 is not sufficient for the convergence of subsequences {p nk of parameter vectors p nk in the inverse image of f n k Uf n := { p D : Ap = f n to a specific parameter vector p 0 Uf 0. We are interested in the stable solvability of 1.. Due to the non-injectivity of the operator A for parameter vectors in p 0 D\ ˆD the inverse Operator A 1 does not exist for all functions f AD. Nevertheless, the jump parameters λ 0, µ Y 0 and σ Y 0 have no influence on the function Ap 0 = 1 φ x µ 0 τ. σ0 τ σ0 τ So, the existence of sequences {p n as given by example 3.8 is not crucial. Let us consider a sequence {f n n AD which converges in Y to f 0 AD. We are going to show that under certain conditions the diffusion parameters µ and σ of parameter vectors p n Uf n converge to the diffusion parameters of parameter vectors p 0 Uf 0, i.e. lim µ n = µ 0 and lim σ n = σ n n Even under those conditions we cannot ensure the convergence of the jump parameters λ, µ Y and σ Y in every case. However, we obtain the convergence convergence lim λ nµ Y n + σ Y n = λ 0 µ Y 0 + σ Y n

11 Some stability results of parameter identification in a jump diffusion model 37 If the jump intensity λ converges to infinity, we unfortunately cannot ensure stability of the operator equation 1.. Then 3.14 and 3.15 are not necessarily fulfilled. In order to prevent this case, we restrict this parameter by an upper bound λ max. Besides restricting the jump intensity we also restrict the jump heights and consider for sufficiently large constants λ max, µ Y max and σ Y max parameter vectors in the restricted domain D max := { p D : λ λ max <, µ Y µ Y max <,σ Y σ Y max < The boundedness of µ Y and σ Y is not essential, because the jumps and absolute returns r τ increase arbitrarily as µ Y or σ Y. The marginal case λ will be considered separately. In order to show 3.14 and 3.15 for parameter vectors in 3.16, we analyze for a sufficiently small η with 0 < η η 0 and f 0 AD max the lower level sets N η := { p D max : Ap f 0 η Y First of all, we show that for a sufficiently small η the lower level sets 3.17 are compact, i.e. N η is a closed and bounded set in R 5. For this purpose we focus on space Y = CR, i.e. we measure the distance between the densities Ap and f 0 by the maximum norm. Theorem 3.9 The set N η is compact for η > 0 sufficiently small. Proof. In formula 3.4 of section 3.1 we have denoted the series elements of 1.1 by f j. In order to avoid conflicts, we denote in this proof the elements of the sequence {f n n by f {n and the limit function f 0 by f {0. For parameter vectors p D max it yields 0 λ λ max <, µ Y µ Y max and 0 σ Y σ Y max <. Firstly, we show that N η is closed and therefore we consider a convergent sequence {p n of parameter vectors in p n N η which converges to p 0. The parameter vectors p n fulfill Ap n f {0 CR η The operator A is continuous in Y = CR see theorem 3.3 such that lim Ap n Ap 0 = CR n yields. From 3.18 we get for every n N the relation Ap 0 f {0 CR Ap n Ap 0 + Ap CR n f {0 CR Ap n Ap 0 + η, CR and consequently we obtain from 3.19 Ap 0 f {0 CR η. 3.0 n Workshop Stochastische Analysis

12 38 D. Düvelmeyer Resulting from of 3.0 we have p 0 N η and hence the closure of this set. Now we show that N η is bounded. Therefore, we have to show the existence of bounds µ max, σ min and σ max, such that all parameter vectors p N η fulfill the following relations µ µ max <, 3.1a 0 < σ min σ σ max <. 3.1b We consider parameter vectors p 0 in the set of inverse images Uf {0 D. If f {0 A ˆD AD, then the set Uf {0 D is a singleton set because of proposition 3.5. Otherwise we have f {0 AD \ ˆD AD. Due to lemma 3.7 the diffusion parameters µ 0 and σ 0 from the parameter vectors p 0 = µ 0,σ 0,λ 0,µ Y 0,σ Y 0 T Uf 0 coincide. For this reason we obtain for all p 0 Uf {0 see 3.1 and { f {0 CR = max x R { f {0 x,p 0 = [Ap 0 ]x max x R e λ 0τ e λ 1 0τ σ 0 τ φ 1 σ 0 πτ x R λ 0 τ j j! x µ 0 τ + jµ φ Y 0 σ0τ + jσ Y 0 σ 0 τ + jσ Y 0 x µ 0 τ σ 0 τ = e λ 0τ 1 σ 0 πτ, consequently 0 < e λ 0τ 1 σ 0 πτ f {0 CR 1 σ 0 πτ <. 3. In case of f {0 AD \ ˆD it yields f {0 CR = 1 σ 0 πτ and therefore we choose in this case λ 0 = 0. Furthermore, we can estimate Ap CR for all p D max with e λmaxτ 1 σ πτ Ap CR 1 σ πτ 3.3 in the same way like 3.. From 3. and 3.3 we get for all p N η e λmaxτ 1 σ πτ Ap Ap f {0 + f {0 1 CR CR CR η +, σ 0 πτ and hence σ e λmaxτ η πτ + 1 σ 0 =: σ min > Now, we show that for a sufficiently small η the parameter σ of parameter vectors in the lower level set 3.17 is bounded above. Together with 3. and 3.3 it yields η Ap f {0 CR f {0 CR Ap 1 1 CR e λ 0τ σ 0 πτ σ πτ

13 Some stability results of parameter identification in a jump diffusion model 39 for p N η, which implies σ 1 e λ 0 τ σ 0 η πτ =: σ max. Consequently, we have shown 3.1b whenever η is sufficiently small. We finally show that the parameter µ is bounded for all parameter vectors in N η. Therefore we consider the case that µ converges to infinity. Along the lines of the proof of theorem 3.3 we can show that for a sufficiently large j the values of the summands 3.4 are marginal. Because of 3.16 and 3.1b see theorem 3.3 there exists for all parameter vectors p N η a finite ĵ = ĵ ε such that for all j ĵ and an arbitrary ε with 0 < ε < 1 the relation f j x,p ε j+1 x R is satisfied. Moreover, the relation f j x,p ε j+1 is fulfilled for the first ĵ summands if and only if x µτ jµ Y σ τ + jσy λτ log λτ j π ε j+1 σ τ + jσy j!. 3.5 The left hand side of 3.5 converges for µ to infinity j < ĵ and hence we obtain for a fixed value x the relation f j x,p ε j+1. This implies with ε = ε for all 1+ε parameter vectors in N η [Ap]x = f j x,p ε j+1 = ε 1 ε = ε. 3.6 We chose x as the point where the function f {0 attains its maximum value. For parameter vectors p N η we have η Ap f {0 CR = max [Ap]x f {0 x,p x R 0 f {0 x,p 0 [Ap]x f {0 CR [Ap]x and consequently e λ 0τ 1 σ 0 πτ [Ap]x [Ap]x e λ 0τ 1 σ 0 πτ η. 3.7 For a sufficiently small η there is a conflict between 3.6 and 3.7 which results in 3.1a. Thus the lower level set N η is compact for a sufficiently small η > 0. We have proven theorem 3.9 in the space Y = CR. If we consider the operator equation 1. in the space Y = L q R, in particular for q =, we can proof theorem 3.9 with similar techniques. By using the compactness of N η for a sufficiently small η we show now the stability of the operator equation for parameter vectors p D max. Workshop Stochastische Analysis

14 40 D. Düvelmeyer Theorem 3.10 Let {f n n AD max be a sequence which converges in Y to the density function f 0 AD max. Then the inverse images p n = µ n,σ n,λ n,µ Y n,σ Y n T Uf n D max and p 0 = µ 0,σ 0,λ 0,µ Y 0,σ Y 0 T Uf 0 D max fulfill 3.14 and Every infinite subsequence {p nk Uf n D max has an accumulation point ˆp Uf 0 D max. Moreover, if k additionally f 0 AD max ˆD, then for a sufficiently large n the sets Uf n D max and Uf 0 are both a singleton and the sequence {p n converges to p 0. n Proof. We consider parameter vectors p n Uf n with bounded jump intensity and jump heights, i.e. p n D max Uf n. The sequence {f n n converges to f 0 and hence due to theorem 3.9 there exists for a sufficiently large n a sufficiently small η η 0 such that p n is in the compact set N η. We begin with the case f 0 AD max ˆD. Due to section 3.5 the set of inverse images is singleton, i.e. {p 0 = Uf 0 D max. Then the function f n is for a sufficiently large n also in 1 AD max ˆD, i.e. {p n = Uf n D max. Due to the theorem on bounded inverse see [3, theorem.6] we obtain the continuity of the inverse mapping A 1 : AN η ˆD N η ˆD, because A is injective on N η ˆD, and consequently the series {p n converges to p 0. So we have shown the second statement n of this theorem. From the convergence of the jump parameters lim λ n = λ 0, lim µ Y n = n n µ Y 0 and lim σ Y n = σ Y 0 we directly get lim λ n µ Y n + σ Y n = λ 0 µ Y 0 + σ Y 0. Now we n n consider the case f 0 A D max D \ ˆD. We know from corollary 3.6 and lemma 3.7 that the diffusion parameters of all parameter vectors p 0 Uf 0 D max coincide and the jump parameters fulfill λ 0 µ Y 0 + σ Y 0 = 0. Let {p nk N η be an arbitrary subsequence of the set of inverse images. Then there is an accumulation point ˆp N η because N η is compact. So we have Ap nk Aˆp which implies Aˆp = f 0 because of f nk f 0. Consequently, we get ˆµ = µ 0, ˆσ = σ 0 and ˆλ ˆµ Y + ˆσ Y = 0 and hence lim µ n = µ 0, n lim σ n = σ 0 as well as lim λ n µ Y n + σ Y n = 0 = λ 0 µ Y 0 + σ Y 0. n n If we accept solutions only in D max, the operator equation 1. is stably solvable in terms of properties formulated in theorem Nevertheless, in [] we have shown that there are instability effects in the numerical solution of the least squares problem.1 even the noise level δ is very small. These ill-posedness phenomena are caused by an illconditioned system of nonlinear equations which we have obtained after discretization. 1 The existence of a subsequence {f nk k with f nk A D max D \ ˆD implies for p nk Uf nk the relation λ nk µy nk + σ Y nk = 0. Because Nη is a compact set there exists a subsequence {p nkl l with p nkl ˆp D \ ˆD. Due to the continuity of A it yields f nkl = Ap nkl Aˆp AD \ ˆD which contradicts f nkl f 0. Thus, the functions f n is for a sufficiently large n also in AD max ˆD. k

15 Some stability results of parameter identification in a jump diffusion model Asymptotic considerations In the domain D = {p R 5 : σ > 0, λ 0, σ Y 0 we have excluded parameter vectors with σ = 0. The reason for this is that in these cases the return N τ r τ = µτ + log Y j is not an absolutely continuous random variable and consequently no density function exists. In the above section we have shown that the operator equation 1. is stably solvable for all parameter vectors in the restricted domain D max D. Therefore, instead of.1 we consider the least squares problem j=1 Ψp = Ap f δ L R min for p D max. 3.8 A drawback of the exclusion is that the domains D and D max are not closed. So we have to ask whether there exists a solution p of the least squares problem for every empirical density function f δ which fulfills the conditions of assumption.1. We are going to analyze the case σ 0 and its consequences for the least squares problem 3.8. For this purpose we consider the first summand f 0 see 3.4 of the series 1.1 which satisfy the relation lim f 0x,p = lim e λτ 1 x µτ σ 0 σ 0 σ τ φ σ τ if x = µτ = 0 else. Accordingly, the density function has at least one peak the maximum of which increases to infinity as σ 0 and consequently parameter vectors with an arbitrarily small σ cannot be a solution of 3.8. In the proof of theorem 3.9 we have shown that for all parameter vectors in D max the inequality 3.3 is satisfied, i.e. it yields e λmaxτ 1 σ πτ Ap CR 1 σ πτ. The upper as well as the lower bound converge for σ 0 to infinity such that Ap CR converges to infinity. In a way similar to 3.3 we can show e λmaxτ 1 σ τπ Ap L R 1 σ πτ, 3.9 which also implies Ap when σ converges to zero. Moreover, we have shown L R for all parameter vectors in the lower level set N η that there is an upper bound for the Workshop Stochastische Analysis

16 4 D. Düvelmeyer values of σ. Because of 3.3 and 3.9 the norms Ap CR and Ap L R converge to zero when σ increases arbitrarily. Since f δ Y Ap Y Ap f δ Y f δ Y + Ap Y, the expression Ap f δ becomes arbitrarily small whenever p is a solution of the Y least squares problem 3.8 and we can exclude parameter vectors with very small or very large values of σ. In the proof of theorem 3.9 we have further shown that there is a bound µ max with µ µ max. The essential nonnegative part of the density function tends to plus or minus infinity whenever µ increases arbitrarily and hence f δ Ap Y could not be arbitrarily small for such parameter vectors. These considerations show that we can exclude parameter vectors as solutions of 3.8 when the diffusion parameters µ and σ converge to the boundary of D max. We want to analyze this more in detail. The function f δ is a approximation of the density function f = Ap, i.e. f δ f Y δ. In order to state something about the solutions of 3.8 we introduce for a sufficiently small η with η δ the set M η := { p D max : Ap f δ Y η. We have shown that the similar set N η is a compact set for η > 0 sufficiently small. The function f δ, which satisfies the conditions from assumption.1, can be seen as an approximation of the limit function f {0 in 3.17 and hence we obtain with similar arguments the compactness of M η whenever η is arbitrarily small. By restricting the intensity parameter λ we particularly get the existence of a positive lower bound of the diffusion parameter σ. Along the lines of 3.4 we get for all parameter vectors in M η the relation σ e λmaxτ δ + L πτ > 0 and σ e λmaxτ δ + L πτ > 0, respectively. However, if the jump intensity λ converges to infinity these bounds converge to zero. But what happens if λ increases? The case that λ converges to infinity only makes sense, if the mean µ Y and variance σy of the jumps converges to zero. Otherwise the absolute returns r τ would converge to infinity for every time lag τ. Particularly, the jump part Nτ log Y j of the returns converges under certain conditions see [1] to a Brownian motion when the jump intensity increases and µ Y as well as σ Y converge to zero. Then the return is a mixture of two Brownian motions which is again a Brownian motion. The case σ 0 can be compensated when λ converges to infinity and µ Y + σ Y to zero. Furthermore there is an asymptotical non-injectivity because parameter vectors with unbounded jump intensity and arbitrarily small jump heights could have the same image like parameter vectors in set D max ˆD. By restricting the domain to D max we avoid this asymptotical non-injectivity which causes instability problems.

17 Some stability results of parameter identification in a jump diffusion model 43 References [1] S. Asmussen, J. Rosiński. Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Prob., 38, , 000. [] D. Düvelmeyer, B. Hofmann. Ill-Posedness of parameter estimation in jump diffusion processes. In J. vom Scheidt, editor, Tagungsband zum Workshop Stochastische Analysis , pages 5-0, Chemnitz, 004. [3] B. Hofmann. Regularization for Applied Inverse and Ill-posed Problems. Teubner, Leipzig, [4] B. Hofmann. Mathematik Inverser Probleme. B. G. Teubner, Leipzig, [5] H. Jeffreys, B.S. Jeffreys, Methods of Mathematical Physics. 3rd ed., Cambridge University Press, Cambridge, [6] A. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems. Springer, New York, [7] A. N. Shiryaev. Probability. Springer, New York, [8] H.-J. Starkloff, D. Düvelmeyer, B. Hofmann, A note on uniqueness of parameter identification in a jump diffusion model. In this issue, pages 51-56, Chemnitz, 005. Workshop Stochastische Analysis

18 44

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia

Estimating the Degree of Activity of jumps in High Frequency Financial Data. joint with Yacine Aït-Sahalia Estimating the Degree of Activity of jumps in High Frequency Financial Data joint with Yacine Aït-Sahalia Aim and setting An underlying process X = (X t ) t 0, observed at equally spaced discrete times

More information

BANACH AND HILBERT SPACE REVIEW

BANACH AND HILBERT SPACE REVIEW BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but

More information

Metric Spaces. Chapter 7. 7.1. Metrics

Metric Spaces. Chapter 7. 7.1. Metrics Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some

More information

1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm

1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 005-06-15 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

More information

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails

A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails 12th International Congress on Insurance: Mathematics and Economics July 16-18, 2008 A Uniform Asymptotic Estimate for Discounted Aggregate Claims with Subexponential Tails XUEMIAO HAO (Based on a joint

More information

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics

No: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results

More information

1. R In this and the next section we are going to study the properties of sequences of real numbers.

1. R In this and the next section we are going to study the properties of sequences of real numbers. +a 1. R In this and the next section we are going to study the properties of sequences of real numbers. Definition 1.1. (Sequence) A sequence is a function with domain N. Example 1.2. A sequence of real

More information

Metric Spaces. Chapter 1

Metric Spaces. Chapter 1 Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence

More information

P (A) = lim P (A) = N(A)/N,

P (A) = lim P (A) = N(A)/N, 1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or non-deterministic experiments. Suppose an experiment can be repeated any number of times, so that we

More information

Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems

Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems Weierstrass Institute for Applied Analysis and Stochastics Coagulation equations and particle systems Wolfgang Wagner Mohrenstrasse 39 10117 Berlin Germany Tel. +49 30 20372 0 www.wias-berlin.de WIAS workshop,

More information

The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation. Lectures INF2320 p. 1/88 The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

More information

Pricing of an Exotic Forward Contract

Pricing of an Exotic Forward Contract Pricing of an Exotic Forward Contract Jirô Akahori, Yuji Hishida and Maho Nishida Dept. of Mathematical Sciences, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga 525-8577, Japan E-mail: {akahori,

More information

Lecture 13 Linear quadratic Lyapunov theory

Lecture 13 Linear quadratic Lyapunov theory EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

More information

Chapter 2: Binomial Methods and the Black-Scholes Formula

Chapter 2: Binomial Methods and the Black-Scholes Formula Chapter 2: Binomial Methods and the Black-Scholes Formula 2.1 Binomial Trees We consider a financial market consisting of a bond B t = B(t), a stock S t = S(t), and a call-option C t = C(t), where the

More information

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS

INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem

More information

Pacific Journal of Mathematics

Pacific Journal of Mathematics Pacific Journal of Mathematics GLOBAL EXISTENCE AND DECREASING PROPERTY OF BOUNDARY VALUES OF SOLUTIONS TO PARABOLIC EQUATIONS WITH NONLOCAL BOUNDARY CONDITIONS Sangwon Seo Volume 193 No. 1 March 2000

More information

Week 1: Introduction to Online Learning

Week 1: Introduction to Online Learning Week 1: Introduction to Online Learning 1 Introduction This is written based on Prediction, Learning, and Games (ISBN: 2184189 / -21-8418-9 Cesa-Bianchi, Nicolo; Lugosi, Gabor 1.1 A Gentle Start Consider

More information

Lecture 13: Martingales

Lecture 13: Martingales Lecture 13: Martingales 1. Definition of a Martingale 1.1 Filtrations 1.2 Definition of a martingale and its basic properties 1.3 Sums of independent random variables and related models 1.4 Products of

More information

Section 3 Sequences and Limits, Continued.

Section 3 Sequences and Limits, Continued. Section 3 Sequences and Limits, Continued. Lemma 3.6 Let {a n } n N be a convergent sequence for which a n 0 for all n N and it α 0. Then there exists N N such that for all n N. α a n 3 α In particular

More information

Another Example: the Hubble Space Telescope

Another Example: the Hubble Space Telescope 296 DIP Chapter and 2: Introduction and Integral Equations Motivation: Why Inverse Problems? A large-scale example, coming from a collaboration with Università degli Studi di Napoli Federico II in Naples.

More information

Fuzzy Differential Systems and the New Concept of Stability

Fuzzy Differential Systems and the New Concept of Stability Nonlinear Dynamics and Systems Theory, 1(2) (2001) 111 119 Fuzzy Differential Systems and the New Concept of Stability V. Lakshmikantham 1 and S. Leela 2 1 Department of Mathematical Sciences, Florida

More information

1 Norms and Vector Spaces

1 Norms and Vector Spaces 008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)

More information

Riesz-Fredhölm Theory

Riesz-Fredhölm Theory Riesz-Fredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A Ascoli-Arzelá Result 18 B Normed Spaces

More information

Course 221: Analysis Academic year , First Semester

Course 221: Analysis Academic year , First Semester Course 221: Analysis Academic year 2007-08, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................

More information

Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

More information

x if x 0, x if x < 0.

x if x 0, x if x < 0. Chapter 3 Sequences In this chapter, we discuss sequences. We say what it means for a sequence to converge, and define the limit of a convergent sequence. We begin with some preliminary results about the

More information

{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,...

{f 1 (U), U F} is an open cover of A. Since A is compact there is a finite subcover of A, {f 1 (U 1 ),...,f 1 (U n )}, {U 1,... 44 CHAPTER 4. CONTINUOUS FUNCTIONS In Calculus we often use arithmetic operations to generate new continuous functions from old ones. In a general metric space we don t have arithmetic, but much of it

More information

ALMOST COMMON PRIORS 1. INTRODUCTION

ALMOST COMMON PRIORS 1. INTRODUCTION ALMOST COMMON PRIORS ZIV HELLMAN ABSTRACT. What happens when priors are not common? We introduce a measure for how far a type space is from having a common prior, which we term prior distance. If a type

More information

MATH 425, PRACTICE FINAL EXAM SOLUTIONS.

MATH 425, PRACTICE FINAL EXAM SOLUTIONS. MATH 45, PRACTICE FINAL EXAM SOLUTIONS. Exercise. a Is the operator L defined on smooth functions of x, y by L u := u xx + cosu linear? b Does the answer change if we replace the operator L by the operator

More information

LECTURE 15: AMERICAN OPTIONS

LECTURE 15: AMERICAN OPTIONS LECTURE 15: AMERICAN OPTIONS 1. Introduction All of the options that we have considered thus far have been of the European variety: exercise is permitted only at the termination of the contract. These

More information

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties

SOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces

More information

Ri and. i=1. S i N. and. R R i

Ri and. i=1. S i N. and. R R i The subset R of R n is a closed rectangle if there are n non-empty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an

More information

RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis

RANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied

More information

1 if 1 x 0 1 if 0 x 1

1 if 1 x 0 1 if 0 x 1 Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

More information

Probability and Statistics

Probability and Statistics CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

More information

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL

EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL EXIT TIME PROBLEMS AND ESCAPE FROM A POTENTIAL WELL Exit Time problems and Escape from a Potential Well Escape From a Potential Well There are many systems in physics, chemistry and biology that exist

More information

Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x

Example 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written

More information

CHAPTER IV - BROWNIAN MOTION

CHAPTER IV - BROWNIAN MOTION CHAPTER IV - BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time

More information

Parabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation

Parabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation 7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity

More information

ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME

ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME ON SOME ANALOGUE OF THE GENERALIZED ALLOCATION SCHEME Alexey Chuprunov Kazan State University, Russia István Fazekas University of Debrecen, Hungary 2012 Kolchin s generalized allocation scheme A law of

More information

arxiv:math/0202219v1 [math.co] 21 Feb 2002

arxiv:math/0202219v1 [math.co] 21 Feb 2002 RESTRICTED PERMUTATIONS BY PATTERNS OF TYPE (2, 1) arxiv:math/0202219v1 [math.co] 21 Feb 2002 TOUFIK MANSOUR LaBRI (UMR 5800), Université Bordeaux 1, 351 cours de la Libération, 33405 Talence Cedex, France

More information

Metric Spaces Joseph Muscat 2003 (Last revised May 2009)

Metric Spaces Joseph Muscat 2003 (Last revised May 2009) 1 Distance J Muscat 1 Metric Spaces Joseph Muscat 2003 (Last revised May 2009) (A revised and expanded version of these notes are now published by Springer.) 1 Distance A metric space can be thought of

More information

The Dirichlet Unit Theorem

The Dirichlet Unit Theorem Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

More information

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )

6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, ) 6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a non-empty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points

More information

Probability Generating Functions

Probability Generating Functions page 39 Chapter 3 Probability Generating Functions 3 Preamble: Generating Functions Generating functions are widely used in mathematics, and play an important role in probability theory Consider a sequence

More information

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression

Logistic Regression. Jia Li. Department of Statistics The Pennsylvania State University. Logistic Regression Logistic Regression Department of Statistics The Pennsylvania State University Email: jiali@stat.psu.edu Logistic Regression Preserve linear classification boundaries. By the Bayes rule: Ĝ(x) = arg max

More information

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh

Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Modern Optimization Methods for Big Data Problems MATH11146 The University of Edinburgh Peter Richtárik Week 3 Randomized Coordinate Descent With Arbitrary Sampling January 27, 2016 1 / 30 The Problem

More information

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces

Geometrical Characterization of RN-operators between Locally Convex Vector Spaces Geometrical Characterization of RN-operators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg

More information

Chapter 5: Application: Fourier Series

Chapter 5: Application: Fourier Series 321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1

More information

Adaptive Online Gradient Descent

Adaptive Online Gradient Descent Adaptive Online Gradient Descent Peter L Bartlett Division of Computer Science Department of Statistics UC Berkeley Berkeley, CA 94709 bartlett@csberkeleyedu Elad Hazan IBM Almaden Research Center 650

More information

Lectures on Stochastic Processes. William G. Faris

Lectures on Stochastic Processes. William G. Faris Lectures on Stochastic Processes William G. Faris November 8, 2001 2 Contents 1 Random walk 7 1.1 Symmetric simple random walk................... 7 1.2 Simple random walk......................... 9 1.3

More information

4. Expanding dynamical systems

4. Expanding dynamical systems 4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,

More information

APPLICATIONS OF THE ORDER FUNCTION

APPLICATIONS OF THE ORDER FUNCTION APPLICATIONS OF THE ORDER FUNCTION LECTURE NOTES: MATH 432, CSUSM, SPRING 2009. PROF. WAYNE AITKEN In this lecture we will explore several applications of order functions including formulas for GCDs and

More information

ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS

ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS ON NONNEGATIVE SOLUTIONS OF NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS FOR TWO-DIMENSIONAL DIFFERENTIAL SYSTEMS WITH ADVANCED ARGUMENTS I. KIGURADZE AND N. PARTSVANIA A. Razmadze Mathematical Institute

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

More information

Chapter 2 Limits Functions and Sequences sequence sequence Example

Chapter 2 Limits Functions and Sequences sequence sequence Example Chapter Limits In the net few chapters we shall investigate several concepts from calculus, all of which are based on the notion of a limit. In the normal sequence of mathematics courses that students

More information

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida

ARBITRAGE-FREE OPTION PRICING MODELS. Denis Bell. University of North Florida ARBITRAGE-FREE OPTION PRICING MODELS Denis Bell University of North Florida Modelling Stock Prices Example American Express In mathematical finance, it is customary to model a stock price by an (Ito) stochatic

More information

p Values and Alternative Boundaries

p Values and Alternative Boundaries p Values and Alternative Boundaries for CUSUM Tests Achim Zeileis Working Paper No. 78 December 2000 December 2000 SFB Adaptive Information Systems and Modelling in Economics and Management Science Vienna

More information

Continuity of the Perron Root

Continuity of the Perron Root Linear and Multilinear Algebra http://dx.doi.org/10.1080/03081087.2014.934233 ArXiv: 1407.7564 (http://arxiv.org/abs/1407.7564) Continuity of the Perron Root Carl D. Meyer Department of Mathematics, North

More information

2.3 Convex Constrained Optimization Problems

2.3 Convex Constrained Optimization Problems 42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions

More information

CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí

CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian. Pasquale Candito and Giuseppina D Aguí Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANT-SIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE p-laplacian Pasquale

More information

E3: PROBABILITY AND STATISTICS lecture notes

E3: PROBABILITY AND STATISTICS lecture notes E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................

More information

Notes on weak convergence (MAT Spring 2006)

Notes on weak convergence (MAT Spring 2006) Notes on weak convergence (MAT4380 - Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p

More information

Numerical methods for American options

Numerical methods for American options Lecture 9 Numerical methods for American options Lecture Notes by Andrzej Palczewski Computational Finance p. 1 American options The holder of an American option has the right to exercise it at any moment

More information

5. Convergence of sequences of random variables

5. Convergence of sequences of random variables 5. Convergence of sequences of random variables Throughout this chapter we assume that {X, X 2,...} is a sequence of r.v. and X is a r.v., and all of them are defined on the same probability space (Ω,

More information

Lecture 10. Finite difference and finite element methods. Option pricing Sensitivity analysis Numerical examples

Lecture 10. Finite difference and finite element methods. Option pricing Sensitivity analysis Numerical examples Finite difference and finite element methods Lecture 10 Sensitivities and Greeks Key task in financial engineering: fast and accurate calculation of sensitivities of market models with respect to model

More information

On ADF Goodness of Fit Tests for Stochastic Processes. Yury A. Kutoyants Université du Maine, Le Mans, FRANCE

On ADF Goodness of Fit Tests for Stochastic Processes. Yury A. Kutoyants Université du Maine, Le Mans, FRANCE On ADF Goodness of Fit Tests for Stochastic Processes Yury A. Kutoyants Université du Maine, Le Mans, FRANCE e-mail: kutoyants@univ-lemans.fr Abstract We present several Goodness of Fit tests in the case

More information

Exact shape-reconstruction by one-step linearization in electrical impedance tomography

Exact shape-reconstruction by one-step linearization in electrical impedance tomography Exact shape-reconstruction by one-step linearization in electrical impedance tomography Bastian von Harrach harrach@math.uni-mainz.de Institut für Mathematik, Joh. Gutenberg-Universität Mainz, Germany

More information

Notes on metric spaces

Notes on metric spaces Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.

More information

An example of a computable

An example of a computable An example of a computable absolutely normal number Verónica Becher Santiago Figueira Abstract The first example of an absolutely normal number was given by Sierpinski in 96, twenty years before the concept

More information

MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...

MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators... MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................

More information

DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER. 1. Introduction

DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER. 1. Introduction DIRICHLET S PROBLEM WITH ENTIRE DATA POSED ON AN ELLIPSOIDAL CYLINDER DMITRY KHAVINSON, ERIK LUNDBERG, HERMANN RENDER. Introduction A function u is said to be harmonic if u := n j= 2 u = 0. Given a domain

More information

Perpetual barrier options in jump-diffusion models

Perpetual barrier options in jump-diffusion models Stochastics (2007) 79(1 2) (139 154) Discussion Paper No. 2006-058 of Sonderforschungsbereich 649 Economic Risk (22 pp) Perpetual barrier options in jump-diffusion models Pavel V. Gapeev Abstract We present

More information

Asymptotics of discounted aggregate claims for renewal risk model with risky investment

Asymptotics of discounted aggregate claims for renewal risk model with risky investment Appl. Math. J. Chinese Univ. 21, 25(2: 29-216 Asymptotics of discounted aggregate claims for renewal risk model with risky investment JIANG Tao Abstract. Under the assumption that the claim size is subexponentially

More information

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand

Notes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such

More information

Convergence Rates for Tikhonov Regularization from Different Kinds of Smoothness Conditions

Convergence Rates for Tikhonov Regularization from Different Kinds of Smoothness Conditions Convergence Rates for Tikhonov Regularization from Different Kinds of Smoothness Conditions Albrecht Böttcher 1, Bernd Hofmann 2, Ulrich Tautenhahn 3 and Masahiro Yamamoto 4 28 November 2005 Abstract.

More information

Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks

Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks 1 Asymptotics for ruin probabilities in a discrete-time risk model with dependent financial and insurance risks Yang Yang School of Mathematics and Statistics, Nanjing Audit University School of Economics

More information

Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization

Research Article Stability Analysis for Higher-Order Adjacent Derivative in Parametrized Vector Optimization Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 510838, 15 pages doi:10.1155/2010/510838 Research Article Stability Analysis for Higher-Order Adjacent Derivative

More information

Statistical Machine Learning

Statistical Machine Learning Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes

More information

Let H and J be as in the above lemma. The result of the lemma shows that the integral

Let H and J be as in the above lemma. The result of the lemma shows that the integral Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;

More information

t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d).

t := maxγ ν subject to ν {0,1,2,...} and f(x c +γ ν d) f(x c )+cγ ν f (x c ;d). 1. Line Search Methods Let f : R n R be given and suppose that x c is our current best estimate of a solution to P min x R nf(x). A standard method for improving the estimate x c is to choose a direction

More information

VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS

VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS VERTICES OF GIVEN DEGREE IN SERIES-PARALLEL GRAPHS MICHAEL DRMOTA, OMER GIMENEZ, AND MARC NOY Abstract. We show that the number of vertices of a given degree k in several kinds of series-parallel labelled

More information

LINEAR PROGRAMMING WITH ONLINE LEARNING

LINEAR PROGRAMMING WITH ONLINE LEARNING LINEAR PROGRAMMING WITH ONLINE LEARNING TATSIANA LEVINA, YURI LEVIN, JEFF MCGILL, AND MIKHAIL NEDIAK SCHOOL OF BUSINESS, QUEEN S UNIVERSITY, 143 UNION ST., KINGSTON, ON, K7L 3N6, CANADA E-MAIL:{TLEVIN,YLEVIN,JMCGILL,MNEDIAK}@BUSINESS.QUEENSU.CA

More information

1 The Brownian bridge construction

1 The Brownian bridge construction The Brownian bridge construction The Brownian bridge construction is a way to build a Brownian motion path by successively adding finer scale detail. This construction leads to a relatively easy proof

More information

THE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties:

THE DYING FIBONACCI TREE. 1. Introduction. Consider a tree with two types of nodes, say A and B, and the following properties: THE DYING FIBONACCI TREE BERNHARD GITTENBERGER 1. Introduction Consider a tree with two types of nodes, say A and B, and the following properties: 1. Let the root be of type A.. Each node of type A produces

More information

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...

n k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +... 6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence

More information

Lecture 12 Basic Lyapunov theory

Lecture 12 Basic Lyapunov theory EE363 Winter 2008-09 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12

More information

Trading activity as driven Poisson process: comparison with empirical data

Trading activity as driven Poisson process: comparison with empirical data Trading activity as driven Poisson process: comparison with empirical data V. Gontis, B. Kaulakys, J. Ruseckas Institute of Theoretical Physics and Astronomy of Vilnius University, A. Goštauto 2, LT-008

More information

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)

Limit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x) SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f

More information

0 ( x) 2 = ( x)( x) = (( 1)x)(( 1)x) = ((( 1)x))( 1))x = ((( 1)(x( 1)))x = ((( 1)( 1))x)x = (1x)x = xx = x 2.

0 ( x) 2 = ( x)( x) = (( 1)x)(( 1)x) = ((( 1)x))( 1))x = ((( 1)(x( 1)))x = ((( 1)( 1))x)x = (1x)x = xx = x 2. SOLUTION SET FOR THE HOMEWORK PROBLEMS Page 5. Problem 8. Prove that if x and y are real numbers, then xy x + y. Proof. First we prove that if x is a real number, then x 0. The product of two positive

More information

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

More information

Mila Stojaković 1. Finite pure exchange economy, Equilibrium,

Mila Stojaković 1. Finite pure exchange economy, Equilibrium, Novi Sad J. Math. Vol. 35, No. 1, 2005, 103-112 FUZZY RANDOM VARIABLE IN MATHEMATICAL ECONOMICS Mila Stojaković 1 Abstract. This paper deals with a new concept introducing notion of fuzzy set to one mathematical

More information

Convergence of Feller Processes

Convergence of Feller Processes Chapter 15 Convergence of Feller Processes This chapter looks at the convergence of sequences of Feller processes to a iting process. Section 15.1 lays some ground work concerning weak convergence of processes

More information

Lecture 7: Finding Lyapunov Functions 1

Lecture 7: Finding Lyapunov Functions 1 Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 7: Finding Lyapunov Functions 1

More information

ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE. 1. Introduction

ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE. 1. Introduction SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17) (2009), 55 62 ON A MIXED SUM-DIFFERENCE EQUATION OF VOLTERRA-FREDHOLM TYPE B.. PACHPATTE Abstract. The main objective of this paper is to study some basic properties

More information

Stationary random graphs on Z with prescribed iid degrees and finite mean connections

Stationary random graphs on Z with prescribed iid degrees and finite mean connections Stationary random graphs on Z with prescribed iid degrees and finite mean connections Maria Deijfen Johan Jonasson February 2006 Abstract Let F be a probability distribution with support on the non-negative

More information

PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION

PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume L, Number 3, September 2005 PROPERTIES OF SOME NEW SEMINORMED SEQUENCE SPACES DEFINED BY A MODULUS FUNCTION YAVUZ ALTIN AYŞEGÜL GÖKHAN HIFSI ALTINOK Abstract.

More information

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein) Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

More information

Advanced Microeconomics

Advanced Microeconomics Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68 Part A. Basic decision and preference theory 1 Decisions

More information

Adaptive Search with Stochastic Acceptance Probabilities for Global Optimization

Adaptive Search with Stochastic Acceptance Probabilities for Global Optimization Adaptive Search with Stochastic Acceptance Probabilities for Global Optimization Archis Ghate a and Robert L. Smith b a Industrial Engineering, University of Washington, Box 352650, Seattle, Washington,

More information