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

Author

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

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.

Suggested Citation

  • Cordoni, Francesco & Di Persio, Luca & Maticiuc, Lucian & Zălinescu, Adrian, 2020. "A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1669-1712.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:3:p:1669-1712
    DOI: 10.1016/j.spa.2019.05.013
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    References listed on IDEAS

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    1. Maticiuc, Lucian & Rascanu, Aurel, 2010. "A stochastic approach to a multivalued Dirichlet-Neumann problem," Stochastic Processes and their Applications, Elsevier, vol. 120(6), pages 777-800, June.
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    3. Delong, Lukasz & Imkeller, Peter, 2010. "On Malliavin's differentiability of BSDEs with time delayed generators driven by Brownian motions and Poisson random measures," Stochastic Processes and their Applications, Elsevier, vol. 120(9), pages 1748-1775, August.
    4. Rosazza Gianin, Emanuela, 2006. "Risk measures via g-expectations," Insurance: Mathematics and Economics, Elsevier, vol. 39(1), pages 19-34, August.
    5. Kazmerchuk, Yuriy & Swishchuk, Anatoliy & Wu, Jianhong, 2007. "The pricing of options for securities markets with delayed response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 75(3), pages 69-79.
    6. Uwe Küchler & Eckhard Platen, 2007. "Time Delay and Noise Explaining Cyclical Fluctuations in Prices of Commodities," Research Paper Series 195, Quantitative Finance Research Centre, University of Technology, Sydney.
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