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

Francesco Cordoni, Luca Di Persio, Lucian Maticiuc and Adrian Zălinescu

Stochastic Processes and their Applications, 2020, vol. 130, issue 3, 1669-1712

Abstract: We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: where ▪=C([0,T];Rd), (u(⋅,ϕ))t≔(u(t+θ,ϕ))θ∈[−δ,0] and Lu(t,ϕ)≔〈b(t,ϕ),∂xu(t,ϕ)〉+12Tr[σ(t,ϕ)σ∗(t,ϕ)∂xx2u(t,ϕ)].The result is obtained by a stochastic approach. More precisely, 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.

Keywords: Path-dependent partial differential equations; Viscosity solutions; Feynman–Kac formula; Backward stochastic differential equations; Time-delayed generators (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1016/j.spa.2019.05.013

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