B. LAQUERRIÈRE AND N. PRIVAULT

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1 DEVIAION INEQUALIIES FOR EXPONENIAL JUMP-DIFFUSION PROCESSES B. LAQUERRIÈRE AND N. PRIVAUL Abstract. In this note we obtain deviation inequalities for the law of exponential jump-diffusion processes at a fixed time. Our method relies on convex concentration inequalities obtained by forward/backward stochastic calculus. In the pure jump and pure diffusion cases it also improves on classical results obtained by direct application of Gaussian and Poisson bounds. 1. Introduction Deviation inequalities for random variables admitting a predictable representation have been obtained by several authors. When W t ) t R+ is a standard Brownian motion and η t ) t R+ an adapted process using the time change 1) t τt) := on Brownian motion yields the bound ) 2) P η t dw t x provided 3) Σ 2 := t η s 2 ds ) exp x2 2Σ 2 x > η t 2 dt <. On the other hand if Z t ) t R+ is a point process with random intensity λ t ) t R+ and U t ) t R+ is an adapted process we have the inequality ) P U t dz t λ t dt) x exp x 1 2β log + βλ )) 4) x x > provided U t β a.s. for some constant β > and Λ := U t 2 λ t dt < cf. [1] [5] when Z t ) t R+ is a Poisson process and [4] for the mixed point process-diffusion case. Note that although Z t ) t R+ becomes a standard Poisson process N t ) t R+ under the time change t t λ s ds when U t ) t R+ is non-constant the inequality 4) can not be recovered from a Poisson deviation bound in the same way as 2) is obtained from a Gaussian deviation bound. In this paper we consider linear stochastic differential equations of the form 5) ds t S t = σ t dw t + J t dz t λ t dt) 21 Mathematics Subject Classification. Primary 6F99; Secondary 39B62 6H5 6H1. ey words and phrases. Deviation inequalities exponential jump-diffusion processes concentration inequalities forward/backward stochastic calculus. 1

2 2 B. LAQUERRIÈRE AND N. PRIVAUL where W t ) t R+ is a standard Brownian motion Z t ) t R+ is a point process of stochastic) intensity λ t. Here the processes W t ) t R+ and Z t ) t R+ may not be independent but they are adapted to a same filtration F t ) t R+ and σ t ) t R+ J t ) t R+ are sufficiently integrable F t -adapted processes. Clearly the above deviation inequalities 2) and 4) require some boundedness on the integrand processes η t ) t R+ and U t ) t R+ and for this reason they do not apply directly to the solution S t ) t R+ of 5) since the processes η t ) t [ ] = σ t S t ) t [ ] and U t ) t [ ] = J t S t ) t [ ] are not in L Ω L 2 [ ])). his is consistent with the fact that when σ t is a deterministic function S has a log-normal distribution which is not compatible with a Gaussian tail. In this paper we derive several deviation inequalities for exponential jump-diffusion processes S t ) t R+ of the form 5). Our results rely on the following proposition cf. [2] Corollary 5.2 and heorem 1.1 below. Let St ) t R+ be the solution of ds t S t = σ t)dŵt + J t)d ˆN t λ t)dt) where Ŵt) t R+ is a standard Brownian motion ˆN t ) t R+ is a Poisson process of deterministic) intensity λ t) which are assumed to be mutually independent while σ t) and J t) are deterministic functions with J t) t R +. heorem 1.1. Assume that one of the following conditions is satisfied: i) 1 < J t J t) dp dt-a.e. and σ t σ t) J 2 t λ t J t) 2 λ t) dp dt a.e. ii) 1 < J t J t) dp dt-a.e. and σ t 2 + J 2 t λ t σ t) 2 + J t) 2 λ t) dp dt a.e. iii) J t J t) dp dt-a.e. J 2 t λ t J t) 2 λ t) hen we have dp dt-a.e. and σ t 2 + J 2 t λ t σ t) 2 + J t) 2 λ t) dp dt a.e. 6) E[φS t ) S = x] E[φS t ) S = x] x > t R + for all convex function φ such that φ is convex. Note that in the continuous case J = Relation 6) can be recovered by the Doob stopping time theorem and Jensen s inequality applied to the time change 1) since W t := t/2 + logs τ 1 t)/s ) is a standard Brownian motion with respect to a timechanged filtration F) t R+ and letting X t := S e W t t/2 t R + we have E[φS )] = E [ φ )] X τ ) [ [ = E φ E X σ s) 2 ds F ])] σs 2 ds [ [ ) ]] E E φ X σ s) 2 ds F σs 2 ds [ )] = E φ X σ s) 2 ds 7) = E[φS )]. However this time change argument does not apply to the jump-diffusion case and in addition in the pure jump case it cannot be used when J t ) t R+ is non-constant. We refer to [3] for deviation inequalities for exponential stable processes when the number of jumps is a.s. infinite.

3 DEVIAION INEQUALIIES FOR EXPONENIAL JUMP-DIFFUSION PROCESSES 3 2. Deviation bounds We begin with a result in the pure jump case i.e. when σ t = dp dt-a.e. and let gu) = 1 + u log u u u >. Let S t ) t R+ denote the solution of 5) with S = 1. heorem 2.1. Assume that σ t = dp dt-a.e. and that for some and let 1 < J t Λ t = t dp dt a.e. J s 2 λ s ds t R +. hen for any > and all x Λ ) β 1 + )2 1 we have Plog S x) exp Λ 2 g 1 + x ))) β Λ exp 1 x 2 β + Λ )) 2 β 1 log 1 + x ))) 8) β Λ where β = log1 + ). Proof. Let J t) = t R + λ t) = 1 2 J 2 t λ t t and denote by S t = e Λt/ 1 + ) N t t R+ the solution of 9) ds t S t = dn t λ t)dt) with S = 1 where N t ) t R+ is a Poisson process with deterministic intensity λ t)) t R+. Under the above hypotheses heorem 1.1 i) yields the inequality 1) y α PS y) E[S ) α ] = e αλ / E [ ) α ] 1 + ) N = e αλ / e Λ 1+) α 1)/ 2 for the convex function y y α with convex derivative α 2 hence Λ 11) Plog S x) exp )α 1) α Λ ) αx. he minimum in α in the above bound is obtained at α = 1 β log 1 + x )) β Λ which is greater than 2 if and only if 12) x Λ Hence for all x satisfying 12) we have Plog S x) exp = exp ) β 1 + )2 1. Λ 1 + )α 2 Λ 2 g β 1) α x + Λ 1 + x Λ ))) ))

4 4 B. LAQUERRIÈRE AND N. PRIVAUL and Relation 8) follows from the classical inequality 1 2 u log1 + u) g1 + u) u >. Note that an application of the classical Poisson bound 4) only yields Plog S x) = P = P provided ) log1 + J t )dz t J t λ t dt x log1 + J t )dz t λ t dt) x + ) P log1 + J t )dz t λ t dt) x exp x 2β log 1 + β x )) x > Λ J t and ) J t λ t dt log1 + J t )λ t dt log1 + J t ) 2 λ t dt Λ a.s. which is worse than 8) even in the deterministic case since 1 < /β as and Λ Λ. heorem 2.1 admits a generalization to the case of a continuous component when the jumps J t have constant sign. heorem 2.2. Assume that 1 < J t dp dt a.e. or J t dp dt a.e. for some > and assume that hen for all x Λ 13) 14) where β = log1 + ). Λ := ) β 1 + )2 1 Plog S x) exp exp 1 2 σt 2 + J 2 t λ t dt <. Λ x β + Λ 2 we have 2 g β β x ))) Λ )) log β 1 + x ))) Λ Proof. We repeat the proof of heorem 2.1 replacing the use of heorem 1.1 i) by heorem 1.1 ii) and heorem 1.1 iii) and by defining λ t) as λ t) = 1 2 σt 2 + J 2 t λ t t. Letting in 13) or 14) we obtain the following Gaussian deviation inequality in the negative jump case with a continuous component.

5 DEVIAION INEQUALIIES FOR EXPONENIAL JUMP-DIFFUSION PROCESSES 5 heorem 2.3. Assume that 1 < J t dp dt-a.e. and let Σ 2 = σt 2 + J 2 t λ t dt < >. hen we have 15) Plog S x) exp x + ) Σ2 /2)2 x 3Σ 2 /2. Proof. Although this result follows from heorem 2.2 by taking we show that it can also be obtained from heorem 1.1. Let σ t) 2 = σt 2 + Jt 2 λ t t t and denote by St = exp σ s)dw s 1 t ) σ s) 2 ds t R + the solution of 2 dst 16) St = σ t)dw t with initial condition S = 1. By the chebychev inequality and heorem 1.1 ii) applied for = for all positive nondecreasing convex functions φ : R R with convex derivative we have 2Σ 2 17) φy)ps y) E[φS )]. Applying this inequality to the convex function t y α for fixed α 2 we obtain hence y α PS y) E[S ) α ] = exp αα 1)Σ 2 /2 ) 18) PS e x ) exp αx + αα 1)Σ 2 /2 ) x α 2. he function α αx + αα 1)Σ 2 /2 attains its minimum over α 2 at α = x Σ 2 x 3Σ 2 /2 which yields 15). In the pure diffusion case with J = and σ t ) t R+ deterministic the bound 15) can be directly obtained from P log S x) = P exp σ t dw t 1 ) ) σ t 2 dt e x 2 = P exp W Σ 2 1 ) ) 2 Σ2 e x exp x + ) Σ2 19) /2)2 x >. 2Σ 2 On the other hand when J = and σ t ) t R+ is an adapted process the bound 2) only yields Plog S x) = P σ t dw t 1 ) σ t 2 dt x 2 ) P σ t dw t x 2) e x2 /2Σ 2 ) x >

6 6 B. LAQUERRIÈRE AND N. PRIVAUL which is worse than 15) and 19) by a factor expx/2+σ 2 /8). In this case the argument of heorem 2.3 can be based on 7) instead of using heorem 1.1. References [1] C. Ané and M. Ledoux. On logarithmic Sobolev inequalities for continuous time random walks on graphs. Probab. heory Related Fields 1164): [2] J.C. Breton and N. Privault. Convex comparison inequalities for exponential jump-diffusion processes. Communications on Stochastic Analysis 1: [3] S. Cohen and. Mikosch. ail behavior of random products and stochastic exponentials. Stochastic Process. Appl. 1183): [4] h. lein Y. Ma and N. Privault. Convex concentration inequalities via forward/backward stochastic calculus. Electron. J. Probab. 11:no pp. electronic) 26. [5] L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. heory Related Fields 1183): Laboratoire de Mathématiques Université de La Rochelle Avenue Michel Crépeau 1742 La Rochelle Cedex France. address: benjamin.laquerriere@univ-lr.fr Department of Mathematics City University of Hong ong at Chee Avenue owloon ong Hong ong. address: nprivaul@cityu.edu.hk