# Some stability results of parameter identification in a jump diffusion model

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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

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