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

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 $(y,z)$: \[Y_{t}=ξ(M_{[0,T]})+\int_{t}^{T}f(s,M_{[0,s]},Y_{s-},Z_{s}m_{s})d{\rm tr}[M,M]_{s}-\int_{t}^{T}Z_{s}dM_{s}-N_{T}+N_{t}\] Here, $M_{[0,t]}$ is the path of $M$ from $0$ to $t$ and $m$ is defined by $[M,M{]}_t={\int }_0^t{m}_s{m}_s^{\ast }d\mathrm{tr}[M,M{]}_s$. 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.

36 pages

References in corpus (2)

• Functional Itô calculus and stochastic integral representation of martingales
• Differentiability of quadratic BSDEs generated by continuous martingales

Cited by in corpus (1)

• Global Well-posedness of Non-Markovian Multidimensional Superquadratic BSDE