Limit theorem for the statistical solution of Burgers equation

AbstractIn this work we study limit theorems for the Hopf–Cole solution of the Burgers equation when the initial value is a functional of some Gaussian processes. We use the Gaussian chaos decomposition, and we get “Gaussian scenario” with new normalization factors

On limiting values of stochastic differential equations with small noise intensity tending to zero

AbstractWhen the right-hand side of an ordinary differential equation (ODE in short) is not Lipschitz, neither existence nor uniqueness of solutions remain valid. Nevertheless, adding to the differential equation a noise with nondegenerate intensity, we obtain a stochastic differential equation which has pathwise existence and uniqueness property. The goal of this short paper is to compare the limit of solutions to stochastic differential equation obtained by adding a noise of intensity ε to the generalized Filippov notion of solutions to the ODE. It is worth pointing out that our result does not depend on the dimension of the space while several related works in the literature are concerned with the one dimensional case

On the strict value of the non-linear optimal stopping problem

We address the non-linear strict value problem in the case of a general filtration and a completely irregular pay-off process (ξt). While the value process (Vt) of the non-linear problem is only right-uppersemicontinuous, we show that the strict value process (V+t) is necessarily right-continuous. Moreover, the strict value process (V+t) coincides with the process of right-limits (Vt+) of the value process. As an auxiliary result, we obtain that a strong non-linear f-supermartingale is right-continuous if and only if it is right-continuous along stopping times in conditional f-expectation

Optimal stopping with f-expectations: The irregular case

We consider the optimal stopping problem with non-linear f-expectation (induced by a BSDE) without making any regularity assumptions on the payoff process ξ and in the case of a general filtration. We show that the value family can be aggregated by an optional process Y. We characterize the process Y as the Ef-Snell envelope of ξ. We also establish an infinitesimal characterization of the value process Y in terms of a Reflected BSDE with ξ as the obstacle. To do this, we first establish some useful properties of irregular RBSDEs, in particular an existence and uniqueness result and a comparison theorem