Jump Adapted Scheme Of a Non Mark Dependent Jump Diffusion Process with Application to the Merton Jump Diffusion Model

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1 International Journal of Statistics and Probability; Vol. 6, No. 4; July 217 ISSN E-ISSN Published by Canadian Center of Science and Education Jump Adapted Scheme Of a Non Mark Dependent Jump Diffusion Process with Application to the Merton Jump Diffusion Model Renaud Fadonougbo 1 & George O. Orwa 2 1 Pan African University Institute for Basic Sciences Technology and Innovation, Kenya 2 Statistics and Actuarial Department, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya Correspondence: Renaud Fadonougbo, Pan African University Institute for Basic Sciences Technology and Innovation, Kenya. renaudfadonougbo@gmail.com Received: October 11, 216 Accepted: June 1, 217 Online Published: June 26, 217 doi:1.5539/ijsp.v6n4p8 The research is financed by Pan African University Abstract URL: This paper provides a complete proof of the strong convergence of the Jump adapted discretization Scheme in the univariate and mark independent jump diffusion process case. We pun detail and clearly a known and general result for mark dependent jump diffusion process. A Monte-Carlo simulation is used as well to show numerical evidence. Keywords: jump-diffusion process,jump-adapted discretization, strong convergence, non-dependent mark, Monte-Carlo simulation 1. Introduction Since three decades, extensive research have been conducted on the dynamics of asset prices, bond etc, and there is no more doubt about the evidence of presence of jumps, see Johannes, 24; Akgiray & Booth, 1988; Glasserman & kou, 23. The jump diffusion modeling was firsntroduced by Merton, 1976 for option pricing. He showed that asset dynamics present discontinuous path instead of continuous one as claimed in Black-Scholes derivation, see Black & Scholes, 1973, Cox, Ingersoll, & Ross, 1985 and Vasicek, The jump process, now became popular as it drew more attention and being very useful as modeling frame in many area of finance such as credit risk modeling, interest rate, and in the computation of Value at Risk Var, see Johannes, 24; Akgiray & Booth, 1988; Glasserman & Kou, 23 and Aït-Sahalia & Yu, 26. Is also used to describe some stylized facts presented by financial data, see Glasserman, 24, page 134. Regardless of the importance of the jump diffusion process, its implementation remains a challenge. The process is always described as solution of a given stochastic differential equationsde, bus only known in its explicit form jusn few cases, and their probability distribution is not always known see Aït-Sahalia, Fan, & Peng, 29 and Yu, 27. That complexity has led researchers to conduct extensive research of finding appropriate way to approximate the stochastic process. Such approximation is essentially based on discretization approach. As discretization approach there are two range methods of discretization of jump diffusion processes which are the weak convergence approximation and the strong convergence approximation, see details in Bruti-Liberati & Platen, 27. In this work we are interested in one type of strong convergence approximation, called Jump adapted discretization as it offers a simple computation and allows easily an extension of discretization methods known for diffusion processes. Based on Bruti-Liberati & Platen27 convergence theorem, we derived similar result for one dimension Jump adapted approximation with a non dependent Mark jump with a compound Poisson process. 1.1 Jump Diffusion Model The model under study is a univariate parametric jump diffusion process defined on the probability space Ω, F T, F, P as: dx t = µx t, θdt σx t, θdw t JX t dn t 1 with, X t the vector of dimension n = 1 that represents the state, W t a standard Brownian motion of dimension d = 1, µx t, θ : R n R n is the drift, σx t, θ : R n R n d is called diffusion and the variance matrix of the process can be defined as: Vx, θ = σx t, θσx t, θ T 1. N t the pure jump process whose intensity is λθ, J s the jump size with probability density ν., θ, whose suppors C R n and θ Θ, a compact subset of R k, that parameterizes the model. To be consistent we have to differentiate the values of the process before and after the occurrence of a potential jump. Therefore we will denote by X t and J t the value and the size respectively of the process and the jump before the occurrence of jump. Thus 1 can be rewritten as 1 σx t, θ T is the transposition of σx t, θ 8

2 International Journal of Statistics and Probability Vol. 6, No. 4; 217 follows: dx t = µx t, θdt σx t, θdw t JX t dn t 2 where X t = lim X u. Let us made the following conditions on the coefficient functions. u <t - Lipschitz conditions µxt, θ µy t, θ 2 C 1 x t y t 2, σx t, θ σy t, θ 2 C 2 x t y t 2, Jx t Jy t 2 C 3 x t y t 2, 3 for all t [, T] and x, y R d - Linear growth conditions µx t, θ 2 K 1 1 x 2, σx t, θ 2 K 2 1 x 2, Jx t, 2 K 3 1 x 2, 4 for all t [, T] and x, R d. Therefore under conditions3 and 4 the SDE1admits a unique strong solution see Bruti-Liberati-Platen, 27. Here are some examples of univariate jump diffusion process. 1 One of the basic example of univariate jump diffusion process is as follows: dx t = µdt σdw t J t dn t 5 where, N s a Poisson process with arrival rate λ, J t, i.i.d. Nµ j, σ 2 j the jump size, µx t, θ = µ; σx t, θ = σ and θ = µ, σ, λ, µ j, σ j. The conditional distribution is well known and is as follows: X t X s = x Nx µt s jµ j, σ 2 t s jσ 2 j. 2 The following is the Merton jump diffusion process described as follows: ds t = µs t dt σs t dw t S t dj t 6 with J t = N t j=1 Y j 1, where Y1, Y 2... are i.i.d. random variables. The solution of the SDE6is known and is S t = S e µ 1 2 tσwt Π N t j=1 Y j see Merton, Discretization Review Method The transition density has an analytic expression for a few cases, and in general no, many approaches have proposed to get an approximation. In fact most of the method have been proposed for diffusion process. Since the jump diffusion process in the absence of jump remains a diffusion process is better to have an overlook of these different methods. As first method, we have the Euler scheme approximation for which the diffusion process X t solution of the following SDE: dx t = µx t, θdt σx t, θdw t 7 is approximated by the following discrete time model: X ih = X i 1h µx i 1h, θ σx i 1h, θ hε i 8 where ε i i.i.dn, 1. Meanwhile of how easy is to implement the closed form expression of the transition, the accuracy of this approximation depends on the interval observation h. When is small, the approximation is accurate, but when is large the approximation can be poor. Merton, 198; Lo, 1988 have found out that the approach is not consistent when h is fixed. A closely approach has been suggested by Gollnick, Houthakker, & Taylor, 1968 and Bergstrom, 1966, and is defined on integrating the SDE and the use of the following trapezoidal rule approximation: ih µx t, θdt = h { µxih, θ µx i 1h, θ } 9 i 1h 2 81

3 International Journal of Statistics and Probability Vol. 6, No. 4; 217 The two approximations are equivalent when h is small. Unfortunately the consistency of the approach is measured in term of h and some authors have showed the asymptotic bias in the estimate is Oh 2. Therefore many works have been done to reduce the discretization bias on the Euler s approximation, see Elerian, 1998 and Tse, Zhang, & Yu, 24 that have improved the method of Milstein, The first used a second order term in a stochastic Taylor series expansion has been used to refine the approximation 8, while the later in a Bayesian context improved the Milstein s scheme. Merener, 24 argued that the Euler scheme converges weakly with order one while the Milstein scheme with order two. They found that between the time of ocurence of jumps, the dynamics of process are purely diffusive and is possile to be simulated using standard discretization methods. Under conditions on the coefficient functions of the jump-diffusion process, they showed that the method used conserves the same weak convergence order for both jump diffusion and pure diffusion processes and the construction of jumps does not degrade the convergence of the method. The following section describes the discretization approach, futher the proof of convergence is presented in section and finally the section 3 has been consecrated to some example and numerical results using Monte-Carlo approach. 2. Jump Adapted Scheme The discretization approach used is called Jump-diffusion adapted approximationjad that consists of separating the diffusion part from the jump one. An appropriate discretization is used for the diffusion part where the effect of the jump is added at due time. This method is based on Itô-Taylor expansion of the drift and diffusion term regardless of the jump part, see Bruti-Liberati & Platen27. Since we assumed that the jump occurs at discrete point time and between two discretization points the process has a diffusion dynamics. We consider a jump time dicretization = < t 1 <... < t N = T; constructed by a superposition of the jump times {,...} to a deterministic equidistant grid with maximum step size >. From the SDE2 and for t [, 1 we have : i1 dx t = i1 µx t, θdt i1 σx t, θdw t i1 J t dn t 1 As X t denotes the value of the process just before the occurrence of a potential jump, it may be then expressed as a solution of the following SDE: dx s = µx s, θdt σx s, θdw s, and applying the Itô formula, this leads to: and µx t, θ = µx ti, θ µ x X s, θµx s, θds 1 2 µ x X s, θσx s, θdw s σx t, θ = σx ti, θ σ x X s, θµx s, θds 1 2 Subtituting µx t, θ and σx t, θ into 1 we have: σ x X s, θσx s, θdw s µ xx X s, θσ 2 X s, θds σ xx X s, θσ 2 X s, θds X ti1 = X ti i1 i1 i1 µx ti, θdt σx ti, θdw t σ x X s, θσx s, θdw s dw t [ µ x X s, θµx s, θ 1 ] 2 µ xxx s, θσ 2 X s, θ dsdt i1 i1 i1 = X ti1 [ σ x X s, θµx s, θ 1 ] 2 σ xxx s, θσ 2 X s, θ dsdw t µ x X s, θσx s, θdw s dt i1 J t dn t i1 J t dn t 82

4 International Journal of Statistics and Probability Vol. 6, No. 4; 217 From 11 we deduce the following approximation Y = { Y ti, i =,..., N } of X, where: where Y ti1 = Y ti1 i1 i1 i1 i1 Y ti1 = Y ti µy ti, θdt σy ti, θdw t JY ti1 dn t 11 σ x Y s, θσy s, θdw s dw t = Y ti µy ti, θh σy ti, θ hε i 1 2 σ xy ti, θσy ti, θhε 2 i 1 12 Y is a jump adapted scheme and it converges strongly at time t with order one to the solution X of the SDE2,the proof is given in the following section. 3. Convergence Before we state the result of the convergence, we have to define a certain compact notation that we use to make explicit the proof. For m N, we define the set of all multi-indices α as follows: M m = {α = j 1, j 2,..., j l ; j i {, 1,..., m}; i {1,..., l}, for l N} {v} where v is the multi-index of length zero. nα the number of component of α that equals to, α denotes the multi index obtained by deleting the last component of α, α is the one obtained by the deleting the first component of α. Let L be the set of functions f t, x : [, T] R d R d from C 1,2 and L k the set of f t, x with partial derivatives x i f t, x, i {1,..., d}, we also introduce the following operator for a function f t, x L k and L f t, x := t f t, x d L k f t, x := d i=1 f α t, x := σ i,k i=1 µ i t, x x i f t, x 1 2 d m i,r=1 j=1 σ i, j σ r, j 2 x i f t, x 13 x j x i f t, x, k {1,..., m} t [, T]andx Rd 14 The Itô coefficient function f α for all α M m and for f t, x : [, T] R d R d is defined by: f t, x for lα = f t, µt, x for lα = 1, j 1 = f t, σ j 1 t, x for lα = 1, j 1 {1,..., m} L j 1 f α t, x for lα 2, j 1 {,..., m} We also require to define some compact notation to represent the multiple stochastic integral obtained from the Itô s Taylor expansion. So is fundamental to define those integral on an appropriate set of integrable function. So let define: { } H v = g : sup E gt < H = H j = { g : E { g : E t [,T] T T } gs ds < } gs 2 ds <, for j {1, 2,... m}, and H α define the set of all predictable process g for which the following multiple stochastic integral is defined gρ I α [g] ρ,τ := if l = and α = v τ I ρ α [g] ρ,z dz if l 1 and j l = τ ρ I α [g] ρ,z dw j l z if l 1 and j l {1, 2,... m} 17 83

5 International Journal of Statistics and Probability Vol. 6, No. 4; 217 where ρ and τ are two stopping times with ρ τ T, a.s. In the more explicit case i.e in dimension one where m = 1 and d = 1, we have: 13 = L f t, x := t f t, x µt, x x f t, x 1 2 2σ2 f t, x x 2 and 14 = L 1 f t, x := σ x f t, x, t [, T]andx R. Then considering the function f t, x = x we have: 15 = f α t, x := x for lα = µt, x for lα = 1, j 1 = σt, x for lα = 1, j 1 = 1 L j 1 f α t, x for lα 2, j 1 {, 1} 18 and gρ if l = and α = v τ 17 = I α [g] ρ,τ := I ρ α [g] ρ,z dz if l 1 and j l = τ I ρ α [g] ρ,z dw z if l 1 and j l = 1 19 Let s define now the set for γ = 1, the hierachical set we also define the remainder set BA γ of A γ by A γ := {α M m : lα nα 2γ }, BA γ = α M m A, α A For illustration we found that: A γ = {v; ; 1; 1, 1} and BA γ = {, ;, 1; 1, }. Using 19 and 18 we can write Y ti1 and X ti1 as follows: Y ti1 = Y ti X ti1 = X ti α A γ \v α A γ \v 2 and 21 are called Wagner-Platen expansion, see I α [ f α, Y ti ] ti,1 = I α [ f α t, Y ti ] ti,1 2 α A γ I α [ f α, Y ti ] ti,1 α BA γ I α [ f α, X ti ] ti,1 21 Definition The scheme Y is said to converge strongly with order γ > at time T to the solution X of a given SDE if there exists a positive constant C such that: ε = E X T Y T 2 C γ Assumption 1. For γ = 1, let Y = {Y t, t [, T]} be the order γ jump adapted time discretization with step size. Let consider the following assumptions: EX and E X Y 2 K 2γ. 22 For α A γ, t [, T] and x, y R d, the Itô coefficients satisfy the following Lipschitz condition f α t, x f α t, y K 1 x y. 23 For all γ A γ BA γ for t [, T] and x R d f α C 1,2 and f α H α, 24 f α t, x 2 K 2 1 x

6 International Journal of Statistics and Probability Vol. 6, No. 4; 217 Theorem 1. Under the Assumptions1, the following estimate E sup X s Y s 2 /A K 3 γ 26 s T holds where the constant K 3 does not depend on Before to prove the theorem above we have to get some preliminaries results. The following lemma is about the boundedness of the Itô multi-stochastic integral. Lemma 1. For α M m \ {v}, g H α and for two stopping times and τ 2 such that F τ1 -measurable and satisfying the conditions τ 2 T, a.s Then, Φ α τ 2 ; = E sup I α [g.] τ1,s τ2 2 Fτ1 4 lα nα lαnα 1 P τ1,z, 27 sτ 2 where for z [, τ 2 ]. P τ1,z := E sup gt Fτ1 < 28 tz Proof: The proof of 27 is done by induction on the different cases of α, applying the Cauchy-Schwarz inequality we get: -When α = with lα = 1 and nα = 1 2 gzdz s τ 1 gz 2 dz. 29 That leads to: Φ α τ 2 = E sup s 2 gzdz sτ 2 F E sup s gz 2 dz sτ 2 F τ2 = E τ 2 gz 2 dz F τ2 E gz 2 dz F τ2 Φ α τ 2 E gz 2 Fτ1 dz since τ 2 is F τ1 measurable = 4 lα nα lαnα 1 τ2 -When α = 1 with lα = 1 and nα =. The process 3 P τ1,z, 31 is a martingale. We have { Iα [g.] τ1,t, t [, T] } { } = gsdw s, t [, T] 32 Φ α τ 2 = E sup sτ 2 4E τ2 = 4E 2 gzdw z F 2 gzdw z F from the application of Doob s inequality gz 2 dz F from the isometry formula 85

7 International Journal of Statistics and Probability Vol. 6, No. 4; 217 Since τ 2 is F τ1 -measurable we have τ2 Φ α τ 2 E = 4 4 τ 1 τ2 τ2 gz 2 dz F E gt 2 Fτ1 dz E sup gt 2 F τ1 dz tz = 4 lα nα lαnα 1 τ2 P τ1,z, 33 -When α = j 1, j 2 with j 2 = then j 1 we have by applying the Cauchy-Schwarz inequality we get: Φ α τ 2 = E sup s 2 I sτ 2 α [g.] τ1,zdz F E sup s I α [g.] τ1,z 2 dz sτ 2 F τ2 = E τ 2 I α [g.] τ1,z 2 dz F τ2 E I α [g.] τ1,z 2 dz F τ2 E sup I α [g.] τ1,s 2 dz sτ 2 F τ2 = E dz sup I α [g.] τ1,s 2 sτ 2 F 2 E sup I α [g.] τ1,s 2 sτ 2 F = 2 Φ α τ 2 34 Considering the previous result we conclude that, Φ α τ lα nα lα nα 1 τ2 P τ1,z τ2 = 4 lα nα lαnα 1 P τ1,z 35 Since lα = lα 1 and nα = nα 1 -When α = j 1, j 2 with j 2 = 1 then j 1, using the fact that the process {I α [g.] τ1,t, t [, T]} 36 is a martingale we obtain through the application of Doob s inequality and Itô s isometry we get the expected result as 86

8 International Journal of Statistics and Probability Vol. 6, No. 4; 217 follows Φ α τ 2 = E sup s 2 I sτ 2 α [g.] τ1,zdw z F τ2 2 4E I α [g.] τ1,zdw z F Doob s inequality τ2 = 4E I α [g.] τ1,z 2 dz F Itô s isometry τ2 4 sup I α [g.] τ1,s 2 dz sτ 2 F τ2 = 4 dz sup I α [g.] τ1,s 2 sτ 2 F = 4 τ 2 sup I α [g.] τ1,s 2 sτ 2 F 4 sup I α [g.] τ1,s 2 sτ 2 F = 4 lα 1 nα lα 1nα 1 τ2 = 4 lα nα lαnα 1 τ2 P τ1,z Since lα = lα 1 and nα = nα and that completes the proof of the lemma1. P τ1,z 37 Lemma 2. For α a given multi-index in M{v}, a time discretization {, i =... N} with step size, 1, and g H α P t,u := E sup gz 2 F t < 38 zu and where Φ α t := E sup i z 1 2 I α [g.] ti,1 I α [g.] tiz,z zt F i= 39 i t = max{i {, 1...} : t} 4 Then Proof: Φ α t := t 2lα 1 P t,udu when : lα = nα 4 lα nα2 lαnα 1 P t,udu when : lα nα From the definition of i z, for z [, 1, the relation z = holds. Then, for a multi index α = j 1, j 2 with j 2 =, we have = = = i z 1 I α [g.] ti,1 I α [g.] tiz,z i= i z 1 i1 i= i z 1 i1 I α [g.] ti,sds z i= z z I α [g.] tis,sds z z I α [g.] tiz,sds I α [g.] tis,sds I α [g.] tis,sds 42 87

9 International Journal of Statistics and Probability Vol. 6, No. 4; 217 For α = j 1, j 2 with j 2 = 1, we have = = = i z 1 I α [g.] ti,1 I α [g.] tiz,z i= i z 1 i1 i= i z 1 i1 I α [g.] ti,sdw s z i= z z I α [g.] tis,sdw s z z I α [g.] tiz,sdw s I α [g.] tis,sdw s I α [g.] tis,sdw s Let consider the case lα = nα Φ α t = E sup z 2 I zt α [g.] tis,sds F z E sup z I α [g.] tis,s 2 ds zt F Cauchy-Schwarz inequality t E I α [g.] tis,s 2 ds F = t E I α [g.] tis,s 2 Ft ds t E sup I α [g.] tis,z 2 F t ds = t E s zs Since s and then F t F tis for s [, t]. From Lemma1 we subtitute the value of Φ α s E sup I α [g.] tis,z 2 F tis s zs } {{ } =Φ α s into 44 and yields F t Φ α t t 4 lα nα E lα nα 1 P tis,zdz F ds since s s. Given that F t F tis for s [, t] s ds 44 t 4 lα nα E lα nα 1 s s P tis,s F t ds t 4 lα nα lα nα E P tis,s F t ds 45 E P tis,s F t = E E sup gu 2 F tis s us F = E sup gu 2 F t s us E sup gu 2 F t us = P t,s 46 88

10 International Journal of Statistics and Probability Vol. 6, No. 4; 217 It then follows Φ α t t 4 lα nα lα nα P t,sds = t 2lα 1 47 That completes then the proof of the case lα = nα. 3. Let consider the case for the multi-index α = j 1, j 2 such that lα nα and j 2 = 1. For that we know that the multiple stochastic integral { z } I t α [g.] tis,sdw s is a martingale. We then have z Φ α t = E sup I α [g.] tis,sdw s 2 zt F z 4E I α [g.] tis,sdw s 2 F Doob s inequality z = 4E I α [g.] tis,s 2 ds F Itô s isometry z E I α [g.] tis,s 2 Ft ds 48 Φ α t And as the previous proof it leads to z E E I α [g.] tis,s 2 F tis Ft ds 4E E sup I α [g.] tis,z 2 F tis F s zs t ds } {{ } 44 lα nα =Φ α s E lα nα 1 P tis,zdz F ds s 4 lα nα 1 lα nα E P tis,s F t ds 49 since lα = lα 1 and nα = nα. Φ α t 4 lα nα 1 lα nα P t,sds = 4 lα nα lαnα 1 P t,sds 5 4 To complete the proof we finally consider the case of α = j 1, j 2 such that lα nα and j 2 =. Applying Cauchy Schwarz inequality on Φ α t, we get Φ α t = E sup i z 1 2 I α [g.] ti,1 I α [g.] tiz,z zt i= F i 2E sup z 1 2 I α [g.] ti,1 F t zt i= 2E sup I α [g.] tiz,z 2 F t 51 zt 89

11 International Journal of Statistics and Probability Vol. 6, No. 4; 217 ni= I α [g.] ti,1 I α [g.] tiz,z is a discrete time martingale, see Kloeden & Platen, 1992 Then considering the first term of 51 we have, i E sup z 1 2 I α [g.] ti,1 F t zt i= i t 1 2 4E I α [g.] ti,1 F t i= Doob s Inequality i t 2 2 = 4E I α [g.] ti,1 I α [g.] tit 1,t F t i= i t 2 i t 2 4E I α [g.] ti,1 2 2 I α [g.] ti,1 E I α [g.] tit 1,t Ftit 1 i= i= E I α [g.] tit 1,t 2 Ftit 1 F t i t 2 4E I α [g.] ti,1 2 i= E I α [g.] tit 1,t 2 Ftit 1 Ft 52 since from the discrete martingale property of the stochastic integral I α [g.] tit 1,t we have E I α [g.] tit 1,t Ftit 1 =. By applying the Lemma1 on E I α [g.] tit 1,t 2 Ftit 1 we get i E sup z 1 2 I α [g.] ti,1 F t zt i= i t 2 4E I α [g.] ti,1 2 i= it 4 lα nα lαnα 1 P tit 1,udu t 1 F 9

12 International Journal of Statistics and Probability Vol. 6, No. 4; 217 Expanding i t 2 i= I α[g.] ti,1 as i t 1 i= I α[g.] ti,1 we obtain i E sup z 1 2 I α [g.] ti,1 F t zt i= i t 3 4E I α [g.] ti,1 2 i= it 4 lα nα lαnα 1 1 P tit 2,udu t 2 it 4 lα nα lαnα 1 P tit 1,udu t 1 F i t 3 4E I α [g.] ti,1 2 i= it 4 lα nα lαnα 1 1 P tit 2,udu t 2 it 4 lα nα lαnα 1 P tit 2,udu t 1 F since P tit 1,u P tit 2,u i t 3 4E I α [g.] ti,1 2 i= it 4 lα nα lαnα 1 P tit 2,udu t 2 F 53 and repetitively it leads to i E sup z 1 2 I α [g.] ti,1 F t zt i= 4 lα nα1 lαnα 1 E = 4 lα nα1 lαnα 1 P t,udu P t,udu F Considering now the secoond term of equation 51 using Cauchy Schwarz inequality we have E sup I α [g.] tiz,z 2 F t = E sup z 2 I zt zt α [g.] tiz,udu z F z E sup z z I α [g.] tiz,u 2 du zt F z Similarly as in the proof of Lemma1 we obtain E sup I α [g.] tiz,z 2 F t zt 4 lα nα lαnα 1 P t,udu. Therefore, from the equations 54 and 54 we finally get Φ α t 2 4 lα nα1 lαnα 1 4 lα nα2 lαnα 1 and that completes the proof of Lemma2. P t,udu 4 lα nα lαnα 1 P t,udu P t,udu, 54 The following Lemma is an intermediary result required to handle the moment of the solution of the SDE2. 91

13 International Journal of Statistics and Probability Vol. 6, No. 4; 217 Lemma 3. Assuming conditions 3 and 4 satisfied and let Then the solution X t of 2 satisfies E X t 2 <. 55 E sup X s 2 F t C1 E X t 2 56 st for every t [, T], with C a positive constant that depends only on T and the linear growth bound. Proof of Theorem1 Considering the Wagner-Platen expansion for the diffusion process as described in equations2 and 21 the solution of the SDE2 at time t [, T] can be written as follows: i t 1 X t = X I α [ f α, X ti ] ti,1 I α [ f α t, X ti ] tit,t and the jump adapted scheme is α A γ \v α BA γ i t 1 i=o i=o I α [ f α, X ti ] ti,1 I α [ f α t, X ti ] tit,t JX tis dn s, 57 Y t = Y i t 1 I α [ f α, Y ti ] ti,1 I α [ f α t, Y ti ] tit,t JY tis dn s, 58 α A γ \v i=o for t [, T]. From Lemma3 we have E sup X s 2 F t C1 E X t 2 st Now let s prove the same result for the jump adapted scheme. E sup Y s 2 F t st E sup E st 1 Y s 2 F sup 1 Y st i t 1 { α A γ \v i=o I α [ f α, Y ti ] ti,1 I α [ f α t, Y ti ] tit,t JX tiu dn u 2 F E sup 1 2 Y 2 i t 1 4 { I st α [ f α, Y ti ] ti,1 α A γ \v i=o I α [ f α t, Y ti ] tit,t 2 4 JX tiu dn u 2 F E sup 1 Y 2 st 4 α A γ \{v} E sup I α [ f α t, Y ti ] tit,t 2 F 4E sup i t 1 st i=o st I α [ f α, Y ti ] ti,1 JX tiu dn u 2 F 92

14 International Journal of Statistics and Probability Vol. 6, No. 4; 217 Then from the Lema2 and the Linear growth condition25 we have E sup i t 1 2 I α [ f α, Y ti ] ti,1 I α [ f α t, Y ti ] tit,t α A γ \{v} st i=o F T K 2 E sup f α s, Y s 2 F du su K 2 C 2 α A γ \{v} T α A γ \{v} T E sup su E sup 1 Y s 2 su F 1 Y s 2 F du du 59 E sup 2 JX tiu dn u F st E sup 2 JX tiu dñ u λ JX where Ñ is the compensated tiu du F, st poisson measure 2E sup 2 JX tiu dñ u F 2E sup 2 JX tiu du F st st T 2 8E JX tiu dñ u F 2E sup 2 JX tiu du F st Using the Itô s isometry for jump and the linear growth condition 4 we get E sup 2 T JX tiu dn u F C 3 E st Subtituting 59 and 6 into 59 we have E sup Y s 2 F C 1 1 E Y 2 C 4 st Then by applying the Gronwall inequality, we obtain E sup Y s 2 F st T sup su 1 Y s 2 F du E sup 1 Y s 2 su F du 6 61 C 1 E Y 2, 62 where C is a positive constant. The final step of proof consists of analyzing the mean square given by Zt := E sup X s Y s 2 F st E sup i s 1 st X Y { I α [ f α, X ti f α, Y ti ] ti,1 α A γ \{v} I α [ f α s, X tis f α s, Y tis ] tis,s i t 1 I α [ f α, X ti ] ti,1 I α [ f α s, X tis ] tis,s α BA γ i=o i= {JX tiu JY tiu }dn u 2 F Cste x Y 2 α A γ \{v} A α t α BA γ B α t D t 63 93

15 International Journal of Statistics and Probability Vol. 6, No. 4; 217 t [, T] where A α t, Bα t and D t are defined as follows A α t := E sup i s 1 I α [ f α, X ti f α, Y ti ] ti,1 st i= I α [ f α s, X tis f α s, Y tis ] tis,s 2 F, 64 and B α t := E sup i t 1 2 I α [ f α, X ti ] ti,1 I α [ f α s, X tis ] tis,s st F, 65 i=o D t := E sup st {JX tiu JY tiu }dn u 2 F 66 Now by using the Lemma2and Lipschitz23 we get A α t := E sup i s 1 I α [ f α, X ti f α, Y ti ] ti,1 st i= I α [ f α s, X tis f α s, Y tis ] tis,s 2 F cste cste E sup X tiu Y tiu su 2 F du Zudu 67 In the same way using the Lemma2and Linear growth condition25 we get B α t := E sup i t 1 st i=o cste E sup I α [ f α, X ti ] ti,1 I α [ su f α s, X s 2 F du cste ψα E sup 1 X s 2 F du cste ψα t su E sup X s 2 F du su 2 f α s, X tis ] tis,s F 68 { 2lα 2 : lα = nα where ψα = So for α BA lα nα 1 : lα nα γ, we have lα γ1 if lα = nα and lαnα 2γ if lα nα so that ψα 2γ. Therefore applying Lemma3 together with the above result we have B α t Cste 2γ 1 X

16 International Journal of Statistics and Probability Vol. 6, No. 4; 217 Now let us analyze the last term D t := E sup 2 {JX tiu JY tiu }dn u F s = E sup {JX tiu JY tiu }dñ u s 2 {JX tiu JY tiu }du F 2E sup 2 {JX tiu JY tiu }dñ u F s 2E sup 2 {JX tiu JY tiu }du F s 8E JX tiu JY tiu 2 applying Doob s inequality du F and Itô s isometry for jump 2λE JX tiu JY tiu 2 du F Cste Then from the results 67, 69 and 7 we have for t [o, T] Therefore for t = T we have Zudu. 7 Zt cste{e x Y 2 2γ 1 X 2 Zudu 71 T ZT cste{e x Y 2 2γ 1 X 2 Zudu 72 Using the Assumptions22 and applying the Gronwall inequality we obtain ZT K 2γ T Zudu 73 K 2γ where K is a positive constant 74 Finally we get the desired result that concludes the proof as E sup X s Y s 2 F = ZT K γ 75 st 4. Numerical Results In this section we present some numerical results based on the Merton jump diffusion model as defined in Eq6, where the jump s magnitude Y j, j = 1... is assumed i.i.d and log-normal distributed with mean µ j and standard deviation σ j. We use a Monte-Carlo approach, we simulate the process, using the JAD scheme, defined in Eq11, and compare it with the exact path. Considering the discretization time = < t 1 <... < t N = T, combined with the jumps time τ i, i = 1..., the jump adapted scheme Y = { Y ti, i =,..., N } of the process X is : where i1 Y ti1 = Y ti1 JY ti1 dn t 76 Y ti1 = Y ti µy ti h σy ti hεi 1 2 σ2 Y ti hε 2 i 1 77 For the following set parameters values µ = 2, sigma = 1; λ = 1, µ j =, sigma j = 2, T = 1; we obtained: 95

17 International Journal of Statistics and Probability Vol. 6, No. 4; 217 Number of iterations N=1 and step size h=1/1 Number of iterations N=1 and step size h=1/1 5. Conclusion Through this paper we have presented the Jump adapted discretization scheme for jump diffusion with numerical on Merton Jump diffusion process. This jump discretization is more tractable since it avoids the computation of multi stochastic integral involving Poisson measure and Wiener process. The jump adapted discretization scheme has been proved to converge strongly with order one to the exact solution for a non dependent mark jump. Acknowledgements The completion of this research thesis has been possible through the help of many individuals who gave me their support. I thank all of them and a special thank to my wife and my son. 96

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