Stochastic differential equations with non-lipschitz coefficients: I. Pathwise uniqueness and large deviation

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence a...

Generalized Large deviation principles was developed for Colombeau-Ito SDE with a random coefficients. We is significantly expand the classical theory of large deviations for randomly perturbed dynamical systems developed by Freidlin and Wentzell.Using SLDP approach, jumps phenomena, in financial markets, also is considered. Jumps phenomena, in financial markets is explained from the first principles, without any reference to Poisson jump process. In contrast with a phenomeno...

In this paper, we study the asymptotic behavior of randomly perturbed path-dependent stochastic differential equations with small parameter $\vartheta_{\varepsilon}$, when $\varepsilon \rightarrow 0$, $\vartheta_\varepsilon$ goes to $0$. When $\varepsilon \rightarrow 0$, we establish large deviation principle. The proof of the results relies on the weak convergence approach. As an application, we establish the large deviation for functionals of path-dependent SDEs in small ti...

In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) with a new kind of non-Lipschitz coefficients. We establish an existence and uniqueness result of solutions in $L^p\ (p>1)$, which includes some known results as its particular cases.

We study the uniqueness in the path-by-path sense (i.e. $\omega$-by-$\omega$) of solutions to stochastic differential equations with additive noise and non-Lipschitz autonomous drift. The notion of path-by-path solution involves considering a collection of ordinary differential equations and is, in principle, weaker than that of a strong solution, since no adaptability condition is required. We use results and ideas from the classical theory of ode's, together with probabilis...

We show two Freidlin-Wentzell type Large Deviations Principles (LDP) in path space topologies (uniform and H\"older) for the solution process of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) using techniques which directly address the presence of the law in the coefficients and altogether avoiding decoupling arguments or limits of particle systems. We provide existence and uniqueness results along with several properties for a class of MV-SDEs having random coeffi...

This paper focuses on systems of nonlinear second-order stochastic differential equations with multi-scales. The motivation stems from mathematical physics and statistical mechanics, for example, stochastic acceleration in random environment. Our main effort is to obtain averaging and large deviations principles. When the fast-varying process is a diffusion, neither Lipschitz continuity nor linear growth is needed. Our approach is based on combinations of the intuition from S...

In this paper, we are interested in solving multidimensional backward stochastic differential equations (BSDEs) in $L^p\ (p>1)$ under weaker assumptions on the coefficients, considering both a finite and an infinite time interval. We establish a general existence and uniqueness result of solutions in $L^p\ (p>1)$ to finite and infinite time interval BSDEs with non-Lipschitz coefficients, which includes the corresponding results in \citet{Par90}, \citet{Mao95}, \citet{Chen97},...

In this paper, we establish the Freidlin-Wentzell's large deviations for quasilinear parabolic stochastic partial differential equations with multiplicative noise, which are neither monotone nor locally monotone. The proof is based on the weak convergence approach.

In this paper, we establish large deviation principle for the strong solution of a doubly nonlinear PDE driven by small multiplicative Brownian noise. Motononicity arguments and the weak convergence approach have been exploited in the proof. Moreover, by using certain a-priori estimates and sequentially weakly Feller property of the associated Markov semigroup, we show existence of invariant probability measure for the strong solution of the underlying problem.