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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 - 11:45AM

Kihun Nam, Rutgers University: 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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Department of Mathematics

Department of Mathematics
Rutgers University
Hill Center - Busch Campus
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