Geiss Christel, Steinicke Alexander
Department of Mathematics and Statistics, University of Jyvaskyla, Jyvaskyla, Finland.
Department of Applied Mathematics and Information Technology, Montanuniversitaet Leoben, Leoben, Austria.
Stochastics (Abingdon). 2019 Jun 12;92(3):418-453. doi: 10.1080/17442508.2019.1626859. eCollection 2020.
We investigate conditions for solvability and Malliavin differentiability of backward stochastic differential equations driven by a Lévy process. In particular, we are interested in generators which satisfy a local Lipschitz condition in the and variable. This includes settings of linear, quadratic and exponential growths in those variables. Extending an idea of Cheridito and Nam to the jump setting and applying comparison theorems for Lévy-driven BSDEs, we show existence, uniqueness, boundedness and Malliavin differentiability of a solution. The pivotal assumption to obtain these results is a boundedness condition on the terminal value and its Malliavin derivative . Furthermore, we extend existence and uniqueness theorems to cases where the generator is not even locally Lipschitz in . BSDEs of the latter type find use in exponential utility maximization.
我们研究由 Lévy 过程驱动的倒向随机微分方程的可解性条件和 Malliavin 可微性。特别地，我们关注在(y)和(z)变量中满足局部 Lipschitz 条件的生成元。这包括那些变量中线性、二次和指数增长的情形。将 Cheridito 和 Nam 的一个想法推广到跳跃情形，并应用 Lévy 驱动的 BSDE 的比较定理，我们证明了解的存在性、唯一性、有界性和 Malliavin 可微性。获得这些结果的关键假设是终端值(\xi)及其 Malliavin 导数(D\xi)的有界性条件。此外，我们将存在性和唯一性定理推广到生成元在(y)中甚至不是局部 Lipschitz 的情形。后一种类型的 BSDE 在指数效用最大化中有应用。
