Dept Banner
Dept Banner

Calendar

Download as iCal file

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.

Special Note to All Travelers

Directions: map and driving directions. If you need information on public transportation, you may want to check the New Jersey Transit page.

Unfortunately, cancellations do occur from time to time. Feel free to call our department: 848-445-6969 before embarking on your journey. Thank you.

Contact Us

HillCenter small

Department of Mathematics

Department of Mathematics
Rutgers University
Hill Center - Busch Campus
110 Frelinghuysen Road
Piscataway, NJ 08854-8019, USA

Phone: +1.848.445.2390
Fax: +1.732.445.5530