Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs
Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs
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DOI：
10.1016/j.spa.2017.10.011
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发表时间：
2017-02
影响因子：
1.4
通讯作者：
Juan Li
中科院分区：
文献类型：
--
作者：
Juan Li
本文考虑了由布朗运动和独立Poisson随机测度驱动的平均场倒向随机微分方程。翻译Buckdahn等人介绍的分裂方法。（2014），证明了分裂方程解（Y t，x，P，Z t，x，P，H t，P），（Y t，x，P，H t，x，P）的存在唯一性.过程的一阶和二阶导数（Y t，x，P <$，Z t，x，P <$，H t，x，P <$）对x的导数，过程的导数（Y t，x，P <$，Z t，x，P <$，H t，x，P <$）关于测度P <$，以及过程的导数在适当的系数正则性假设下，分别研究了（<$μ Yt，x，P <$（y），<$μ Zt，x，P <$（y），<$μ Ht，x，P <$（y））关于y的解.这些导数在L2上是有界的和连续的.二阶导数的连续性的证明是特别复杂的，需要精细的估计。这个正则性保证了值函数V（t，x，P <$）<$Y t t，x，P <$是正则的，并且允许借助于一个新的Itô公式证明它是相关的非局部拟线性平均场型积分-偏微分方程（PDE）的唯一经典解。
In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al.(2014) to BSDEs, the existence and the uniqueness of the solution (Y t, ξ, Z t, ξ, H t, ξ),(Y t, x, P ξ, Z t, x, P ξ, H t, x, P ξ) of the split equations are proved. The first and the second order derivatives of the process (Y t, x, P ξ, Z t, x, P ξ, H t, x, P ξ) with respect to x, the derivative of the process (Y t, x, P ξ, Z t, x, P ξ, H t, x, P ξ) with respect to the measure P ξ, and the derivative of the process (∂ μ Y t, x, P ξ (y),∂ μ Z t, x, P ξ (y),∂ μ H t, x, P ξ (y)) with respect to y are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in L 2. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function V (t, x, P ξ)≔ Y t t, x, P ξ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.