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| Main Authors: | , , , |
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| Format: | Preprint |
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2016
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1602.05793 |
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| _version_ | 1866908674216165376 |
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| author | Cordoni, Francesco Di Persio, Luca Maticiuc, Lucian Zălinescu, Adrian |
| author_facet | Cordoni, Francesco Di Persio, Luca Maticiuc, Lucian Zălinescu, Adrian |
| contents | 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 $\mathbbΛ=\mathcal{C}([0,T];\mathbb{R}^{d})$, $(u(\cdot ,ϕ))_{t}:=(u(t+θ,ϕ))_{θ\in[-δ,0]}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1602_05793 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance Cordoni, Francesco Di Persio, Luca Maticiuc, Lucian Zălinescu, Adrian Probability 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 $\mathbbΛ=\mathcal{C}([0,T];\mathbb{R}^{d})$, $(u(\cdot ,ϕ))_{t}:=(u(t+θ,ϕ))_{θ\in[-δ,0]}$ 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. |
| title | A stochastic approach to path-dependent nonlinear Kolmogorov equations via BSDEs with time-delayed generators and applications to finance |
| topic | Probability |
| url | https://arxiv.org/abs/1602.05793 |