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학술대회/행사

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제출번호(No.) 0636
분류(Section) Plenary Lecture
분과(Session)
영문제목
(Title(Eng.))
Bridge: Backward stochastic differential equation, nonlinear expectation and path-dependent PDE
저자(Author(s))
Shige Peng1
Shandong University, China1
초록본문(Abstract) Backward stochastic differential equation (BSDE) theory is a powerful tool in quantitative analysis and calculation in financial risk measurements and many other situations. The corresponding nonlinear Feynman-Kac formula tells us that, once the coefficients depend only on the state of the Brownian path, then the BSDE is a quasi linear partial differential equation (PDE) of parabolic types. This reveals that, a general (non-Markovian) BSDE is in fact a PDE in which the Brownian path and/or other paths of stochastic process plays the role of state variable.
For a fully nonlinear parabolic PDE, can one still establish the corresponding BSDE, nonlinear expectation and path-dependence PDE (PPDE)? The answer of this deep problem is: in general the Wiener probability measure and/or any other probability measures corresponding can no longer be the reference probability space of the corresponding nonlinear expectation since the latter is essentially ‘fully nonlinear’. A direct solution to this problem is to use a fully nonlinear PDE to construct the corresponding nonlinear expectation, called G-expectation. The nonlinear martingale under this expectation, called G-martingale, is the solution of the BSDE which can be regarded as the corresponding fully nonlinear path-dependence PDE. In this framework, the corresponding canonic process is called G-Brownian motion which is a continuous process with independent and stable increments.
It turns out that a new notion of G-expectation-weighted Sobolev spaces, or in short, G-Sobolev spaces, is naturally introduced and each Ito's process under G-expectation can be written as a generalized Ito's formula. Consequently, a backward SDEs driven by G-Brownian motion is in fact a PPDEs in the corresponding G-Sobolev spaces. For the linear case of G corresponding the classical Wiener probability measure, we have established 1-1 correspondence between BSDE and such new type of quasi linear PPDE in the corresponding Sobolev space of paths. In nonlinear stuation, also provide a 1-1 correspondence between a fully nonlinear PDE in the corresponding G-Sobolev space and BSDE driven by G-Brownian.
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강연 형태
(Language of Session (Talk))
English