We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: where [Formula presented]=C([0,T];Rd), (u(⋅,ϕ))t≔(u(t+θ,ϕ))θ∈[−δ,0] and [Formula presented]. 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.
A Stochastic Approach to Path-Dependent Nonlinear Kolmogorov Equations via BSDEs with Time-Delayed Generators and Applications to Finance / Cordoni, F.; Di Persio, L.; Maticiuc, L.; Zalinescu, A.. - In: STOCHASTIC PROCESSES AND THEIR APPLICATIONS. - ISSN 0304-4149. - 2020, 130:3(2020), pp. 1669-1712. [10.1016/j.spa.2019.05.013]
A Stochastic Approach to Path-Dependent Nonlinear Kolmogorov Equations via BSDEs with Time-Delayed Generators and Applications to Finance
Cordoni F.;Di Persio L.;Zalinescu A.
2020-01-01
Abstract
We prove the existence of a viscosity solution of the following path dependent nonlinear Kolmogorov equation: where [Formula presented]=C([0,T];Rd), (u(⋅,ϕ))t≔(u(t+θ,ϕ))θ∈[−δ,0] and [Formula presented]. 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.| File | Dimensione | Formato | |
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