Path-differentiability of BSDE driven by a continuous local martingale

Kihun Nam

Published 2016-06-13Version 1

We study the existence and uniqueness, and the path-differentiability of solution for backward stochastic differential equation (BSDE) driven by a continuous local martingale $M$ with $[M,M]_{t}=\int_{0}^{t}m_{s}m_{s}^{*}d{\rm tr}[M,M]_{s}$: \[ Y_{t}=\xi(M_{[0,T]})+\int_{t}^{T}f(s,M_{[0,s]},Y_{s-},Z_{s}m_{s})d{\rm tr}[M,M]_{s}-\int_{t}^{T}Z_{s}dM_{s}-N_{T}+N_{t} \] Here, for $t\in[0,T]$, $M_{[0,t]}$ is the path of $M$ from $0$ to $t$, and $\xi(\gamma_{[0,T]})$ and $f(t,\gamma_{[0,t]},y,z)$ are deterministic functions of $(t,\gamma,y,z)\in[0,T]\times D\times\mathbb{R}^{d}\times\mathbb{R}^{d\times n}$. The path-derivative is defined as a directional derivative with respect to the path-perturbation of $M$ in a similar way to the vertical functional derivative introduced by Dupire (2009), and Cont and Fournie (2013). We first prove the existence and uniqueness of solution in the case where $f(t,\gamma_{[0,t]},y,z)$ is Lipschitz in $y$ and $z$. After proving $Z$ is a path-derivative of $Y$, we extend the results to locally Lipschitz $f$. When the BSDE is one-dimensional, we could show the existence and uniqueness of solution. On the contrary, when the BSDE is multidimensional, we show the existence and uniqueness only when $[M,M]_{T}$ is small enough: we provide a counterexample which the solution blows up otherwise. Lastly, we investigate the applications to the control of SDE driven by $M$ and optimal portfolio selection problem under power and exponential utility function.