Locally Lipschitz BSDE driven by a continuous martingale: path-derivative approach

TL;DR

This paper introduces a new path-derivative concept to analyze well-posedness of locally Lipschitz BSDEs driven by continuous martingales, establishing existence, uniqueness, and counterexamples, with applications to utility maximization.

Contribution

It develops a novel path-derivative framework for BSDEs driven by continuous martingales, proving existence and uniqueness results and providing counterexamples in multidimensional cases.

Findings

Existence and uniqueness for one-dimensional BSDEs.

Counterexamples with blow-up solutions in multidimensional cases.

Application to utility maximization problems.

Abstract

Using a new notion of path-derivative, we study well-posedness of backward stochastic differential equation driven by a continuous martingale $M$ when $f(s,\gamma,y,z)$ is locally Lipschitz in $(y,z)$: \[Y_{t}=\xi(M_{[0,T]})+\int_{t}^{T}f(s,M_{[0,s]},Y_{s-},Z_{s}m_{s})d{\rm tr}[M,M]_{s}-\int_{t}^{T}Z_{s}dM_{s}-N_{T}+N_{t}\] Here, $M_{[0,t]}$ is the path of $M$ from $0$ to $t$ and $m$ is defined by $[M,M]_{t}=\int_{0}^{t}m_{s}m_{s}^{*}d{\rm tr}[M,M]_{s}$. 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 existence and uniqueness only when $[M,M]_{T}$ is small enough: otherwise, we provide a counterexample that has blowing-up solution. Then, we investigate the applications to utility maximization problems.