Events (Since covid these events are taking place online.)
Path differentiability of BSDE driven by a continuous martingale.
Tuesday, September 13, 2016 at 11:45am - 12:45pm
Kihun Nam, Rutgers University
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.
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.
Location Hill 705
