# Calendar

Mathematical Finance and Probability Seminars

## Path differentiability of BSDE driven by a continuous martingale

Location:  Hill 705
Date & time: Tuesday, 13 September 2016 at 11:45AM -

### "Path differentiability of BSDE driven by a continuous martingale"

Time: 11:45 AM
Location: Hill 705
Abstract: We study existence, uniqueness, and path-differentiability of solution for backward stochastic differential equation (BSDE) driven by a continuous 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 $$tin[0,T]$$, $$M_{[0,t]}$$ 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 $$(t,gamma,y,z)in[0,T]times DtimesbbR^{d}timesbbR^{dtimes n}$$. 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, uniqueness, and path-differentiability 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 existence and uniqueness only when $$[M,M]_{T}$$ is small enough: otherwise, we provide a counterexample that has a blowing-up solution. Lastly, we investigate the applications to utility maximization problems under power and exponential utility function.

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