On near-martingales and a class of anticipating linear stochastic differential equations. (English) Zbl 07923760
Summary: 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.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60F10 | Large deviations |
| 60G48 | Generalizations of martingales |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
| 60H05 | Stochastic integrals |
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 60H20 | Stochastic integral equations |
Keywords:
anticipating integral; stochastic integral; stochastic differential equation; near-martingale; optional stopping theorem; large deviation principlesReferences:
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