Nonlinear Brownian motion and nonlinear Feynman-Kac formula of path-functions

We consider a typical situation in which probability model itself has non-negligible cumulated uncertainty. A new concept of nonlinear expectation and the corresponding non-linear distributions has been systematically investigated: cumulated nonlinear i.i.d random variables of order \(1/n\) tend to a maximal distribution according a new law of large number, whereas, with a new central limit theorem, the accumulation of order \(1/\sqrt{n}\) tends to a nonlinear normal distribution. The continuous time uncertainty accumulation derives a nonlinear Brownian motion as well as the corresponding OU-process driven by this nonlinear Brownian motion which converges to a nonlinear invariant measure of Gaussian type. The related stochastic calculus provides us a powerful tools to introduce time-space derivatives for functional of paths. The corresponding Feynman-Kac formula for gives one to one correspondence between fully nonlinear parabolic partial differential equations and backward stochastic differential equations driven by the nonlinear Brownian motion.

Date

Speakers

Shige Peng

Affiliation

Shandon University

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