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

2016/02/18 by Cordoni, Francesco, Di Persio, Luca, Maticiuc, Lucian +1 · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1602.05793

Abstract

We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: \begincases ∂_tu(t,ϕ)+Lu(t,ϕ)+f(t,ϕ,u(t,ϕ),∂_xu(t,ϕ) σ(t,ϕ),(u(⋅,ϕ))_t)=0, t∈[0,T), ϕ∈\mathbbΛ ,u(T,ϕ)=h(ϕ), ϕ∈\mathbbΛ, \endcases where \mathbbΛ=C([0,T];ℝ^d), (u(⋅ ,ϕ))_t:=(u(t+θ,ϕ))_θ∈[-δ,0] and Lu(t,ϕ):=⟨ b(t,ϕ),∂_xu(t,ϕ)⟩+\dfrac 12Tr[σ(t,ϕ)σ^∗(t,ϕ)∂_xx ²u(t,ϕ)]. 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.

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