A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance

Abstract

We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: \[ \begin{cases} \partial_{t}u(t,ϕ)+\mathcal{L}u(t,ϕ)+f(t,ϕ,u(t,ϕ),\partial_{x}u(t,ϕ) σ(t,ϕ),(u(\cdot,ϕ))_{t})=0,\;t\in[0,T),\;ϕ\in\mathbbΛ\, ,u(T,ϕ)=h(ϕ),\;ϕ\in\mathbbΛ, \end{cases} \] where $\Lambda =\mathcal{C}([0,T];{\mathbb{R}}^d)$, $(u(\cdot ,\varphi ){)}_t:=(u(t+\theta ,\varphi ){)}_{\theta \in [-\delta ,0]}$ and \[ \mathcal{L}u(t,ϕ):=\langle b(t,ϕ),\partial_{x}u(t,ϕ)\rangle+\dfrac {1}{2}\mathrm{Tr}\big[σ(t,ϕ)σ^{\ast}(t,ϕ)\partial_{xx} ^{2}u(t,ϕ)\big]. \] The result is obtained by a stochastic approach. In particular we prove a new type of nonlinear Feynman-Kac representation formula associated to a backward stochastic differential equation with time-delayed generator which is of non-Markovian type. Applications to the large investor problem and risk measures via $g$-expectations are also provided.

45 pages

References in corpus (2)

• Pricing, Hedging and Optimally Designing Derivatives Via Minimization of Risk Measures
• Applications of time-delayed backward stochastic differential equations to pricing, hedging and portfolio management

Cited by in corpus (4)

• Feynman-Kac formula for BSDEs with jumps and time delayed generators associated to path-dependent nonlinear Kolmogorov equations
• Time-Delayed Generalized BSDEs
• BSDEs and SDEs with time-advanced and -delayed coefficients
• A nonlinear Kolmogorov equation for stochastic functional delay differential equations with jumps