On near-martingales and a class of anticipating linear stochastic differential equations

The goals of this paper are to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. For a class of anticipating linear stochastic differential equations, we prove the existence and uniqueness of solutions using two approaches: (1) Ayed-Kuo differential formula using an ansatz, and (2) a braiding technique by interpreting the integral in the Skorokhod sense. We establish a Freidlin-Wentzell type large deviations result for the solution of such equations. In addition, we prove large deviation results for small noise where the initial conditions are random.

Automated summary

The paper aims to prove a near-martingale optional stopping theorem and establish solvability and large deviations for a class of anticipating linear stochastic differential equations. Solutions' existence and uniqueness are demonstrated using two approaches, and a Freidlin-Wentzell type large deviations result is established for the equation's solution. Large deviation results are also proven for small noise with random initial conditions.