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Asymptotic behavior of the transition density for jump type processes in small time. (English) Zbl 0818.60074
Let \(x_ t\) be the Markov process of pure jump type which satisfies the following stochastic differential equation \[ x_ t(x) = x + \sum_{s \leq t} \gamma \bigl( x_{s-} (x),\;\Delta z(s) \bigr), \] where \(z(t)\) is an \(R^ d\)-valued Lévy process of pure jump type with the Lévy measure \(h(d \zeta)\), \(\Delta z(t) = Z(t) - z(t-)\) and \(\gamma(x,\zeta)\) is a nondegenerate bounded function from \(R^ d \times R^ d\) to \(R^ d\). Under some assumptions on the Lévy measure \(h(d \zeta)\), this paper provides an estimate on the density \(p_ t (x,y)\) of the Markov process \(x_ t\) when the time parameter is small. More precisely, it is proven that as \(t \to 0\), \[ p_ t (x,y) \sim C \bigl( x,y, \alpha (x,y) \bigr) t^{\alpha (x,y)}, \] where \(\alpha (x,y)\) can be interpreted roughly as the minimum number of jumps by which the trajectory can reach \(y\) from \(x\) \((x \neq y)\). This result is a refinement of an earlier results by R. Léandre [Sémin. probabilités XXI, Lect. Notes Math. 1247, 81-99 (1987; Zbl 0616.60078)]. This paper makes use of the theory of Malliavin calculus of jump type and the technique of Léandre’s paper.
Reviewer: R.Song (Ann Arbor)

60J75 Jump processes (MSC2010)
60J35 Transition functions, generators and resolvents
60H07 Stochastic calculus of variations and the Malliavin calculus
Full Text: DOI
[1] K. BICHTELER, J. -B. CRAVEREAUX AND J. JACOD, Malliavin Calculus for Processes with Jumps, Gordon and Breach Science Publishers, New York, 1987. · Zbl 0706.60057
[2] J. -M. BISMUT, Calcul des variations stochastiques et processus de sauts, Z. Wahrsch. Verw. Gebiet 63 (1983), 147-235. · Zbl 0494.60082 · doi:10.1007/BF00538963
[3] M. DUFLO, Representations de semi-groupes de measures sur un groupe localment compact, Ann Inst. Fourier 28 (1978), 225-249. · Zbl 0368.22006 · doi:10.5802/aif.712 · numdam:AIF_1978__28_3_225_0 · eudml:74374
[4] P. GRACZYK, Malliavincalculus for stable processes onhomogeneous groups, Studia Math. 100 (1991), 183-205. · Zbl 0745.60055 · eudml:215882
[5] N. IKEDA AND S. WATANABE, Stochastic differential equations and diffusion processes, North Holland/Kodansha, Tokyo, 1981. · Zbl 0495.60005
[6] R. LEANDRE, Plot d’une equation differentielle stochastique avec semimartingale directricediscontinue, in Seminaire de Probabilites XIX (J. Azema and M. Yor, eds.), Lecture Notes in Math. 1123, Springer-Verlag, Berlin, 1984, pp. 271-275. · Zbl 0567.60058 · numdam:SPS_1985__19__271_0 · eudml:113522
[7] R. LEANDRE, Regularite de processus de sauts degeneres, Ann. Inst. H. Poincare Probabilites 21 (1985), 125-146. · Zbl 0567.60056 · numdam:AIHPB_1985__21_2_125_0 · eudml:77252
[8] R. Leandre, Densite en temps petit d’un processus de sauts, in Seminaire de Probabilities XXI (J. Azema, P. A. Meyer and M. Yor, eds.), Lecture Notes in Math. 1247, Springer-Verlag, Berlin, 1987, pp. 81-99. · Zbl 0616.60078 · numdam:SPS_1987__21__81_0 · eudml:113619
[9] R. LEANDRE, Minoration en temps petit de la densite d’une diffusion degeneree, J. Funct. Anal. 7 (1987), 399^14. · Zbl 0637.58034 · doi:10.1016/0022-1236(87)90031-0
[10] R. LEANDRE, Regularite de processus de sauts degeneres (II), Ann. Inst. H. Poincare Probabilites 2 (1988), 209-236. · Zbl 0669.60068 · numdam:AIHPB_1988__24_2_209_0 · eudml:77324
[11] J. P. LEPELTIER AND R. MARCHAL, Problemes de martingales associees a un operateur integro differentiel, Ann. Inst. H. Poincare Probabilites 12 (1976), 43-103. · Zbl 0345.60029 · numdam:AIHPB_1976__12_1_43_0 · eudml:77035
[12] J. R. NORRIS, Integration by parts for jump processes, in Seminaire de Probabilites XXII (J. Azema, P. A. Meyer and M. Yor, eds.), Lecture Notes in Math. 1321, Springer-Verlag, Berlin, 1988, pp. 271-315. · Zbl 0649.60080 · numdam:SPS_1988__22__271_0 · eudml:113640
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