arXiv 1506.022492015-06-07math.PR

随机测度驱动的倒向随机微分方程解的存在唯一性

Existence and uniqueness for backward stochastic differential equations driven by a random measure

Elena Bandini

摘要：

本文研究了有限时域上由$\mathbb R_+\times E$上的整数值随机测度$μ$驱动的倒向随机微分方程，其中$E$是一个Lusin空间，具有补偿器$ν(dt,dx)=dA_t\,ϕ_t(dx)$：\[ Y_t = ξ+ \int_{(t,T]} f(s,Y_{s-},Z_s(\cdot))\, d A_s - \int_{(t,T]} \int_E Z_s(x) \, (μ-ν)(ds,dx),\qquad 0\leq t\leq T. \]生成元$f$对于它的后两个参数，通常满足一致的利普希茨条件.在文献中，只有当$A$是连续的或确定的时，才建立了上述方程在目前一般情况下的存在唯一性。本文讨论了一般情况，即$A$是右连续非减可料过程.这些结果是相关的，例如，在分段确定性马尔可夫过程（PDMP）的控制问题的研究。实际上，当$μ$是PINK的跳跃度量时，则$A$是可预测的（但不是确定性的）和不连续的，跳跃的大小等于1。

英文摘要：

We study the following backward stochastic differential equation on finite time horizon driven by an integer-valued random measure $μ$ on $\mathbb R_+\times E$, where $E$ is a Lusin space, with compensator $ν(dt,dx)=dA_t\,ϕ_t(dx)$: \[ Y_t = ξ+ \int_{(t,T]} f(s,Y_{s-},Z_s(\cdot))\, d A_s - \int_{(t,T]} \int_E Z_s(x) \, (μ-ν)(ds,dx),\qquad 0\leq t\leq T. \] The generator $f$ satisfies, as usual, a uniform Lipschitz condition with respect to its last two arguments. In the literature, the existence and uniqueness for the above equation in the present general setting has only been established when $A$ is continuous or deterministic. The general case, i.e. $A$ is a right-continuous nondecreasing predictable process, is addressed in this paper. These results are relevant, for example, in the study of control problems related to Piecewise Deterministic Markov Processes (PDMPs). Indeed, when $μ$ is the jump measure of a PDMP, then $A$ is predictable (but not deterministic) and discontinuous, with jumps of size equal to 1.

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摘要：

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