Continuous dependence estimates for viscosity solutions of integro-PDEs. (English) Zbl 1082.45008
The authors consider the following general nonlinear degenerate parabolic integro-partial differential equation (PDE):
\[
\begin{matrix} u_t (t,x)+ F\left( t,x,u(t,x),Du(t,x),D^2 u(t,x), u(t,.)\right)=0,\quad\text{ in } Q_T, \\ u(0,x) = u_0 (x), \quad \text{ in }\mathbb{R}^N,\end{matrix}\tag{1}
\]
where \( Q_T:= (0,T)\times\mathbb{R}^N, F \;:\overline {Q}_T \times\mathbb{R} \times\mathbb{R}^N \times\mathbb{S}^N\times C^{2}_p (\mathbb{R}^N) \rightarrow \mathbb{R} \) is a given functional, \(\mathbb{S}^N \) is the space of symmetric \(N\times N\) real valued matrices, \(C^{2}_p (\mathbb{R}^N)\) is the space of \( C^{2} (\mathbb{R}^N)\) functions with polynomial growth of order \(p\geq 0\) at infinity.
The authors formulate and prove abstract continuous dependence estimates for viscosity solutions of problem (1). Then they apply their results to the Black-Scholes model as well as to a singular perturbation problem by J. L. Lions [Perturbations singulières dans les problèmes aux limites et en contrôle optimal. (1973; Zbl 0268.49001)] and S. Koike [J. Math. Anal. Appl. 157, 243–253 (1991; Zbl 0747.35001)].
The authors formulate and prove abstract continuous dependence estimates for viscosity solutions of problem (1). Then they apply their results to the Black-Scholes model as well as to a singular perturbation problem by J. L. Lions [Perturbations singulières dans les problèmes aux limites et en contrôle optimal. (1973; Zbl 0268.49001)] and S. Koike [J. Math. Anal. Appl. 157, 243–253 (1991; Zbl 0747.35001)].
Reviewer: Mohamed O. El-Doma (Beirut)
MSC:
| 45K05 | Integro-partial differential equations |
| 91B24 | Microeconomic theory (price theory and economic markets) |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
Keywords:
nonlinear degenerate parabolic integro-partial differential equation; Bellman equation; Isaacs equation; viscosity solution; continuous dependence estimates; obstacle problem; regularity; vanishing jump viscosity method; vanishing viscosity method; Black-Scholes model; option pricing; Lévy model; singular perturbationReferences:
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