Reflected anticipated backward stochastic differential equations with nonlinear resistance

TL;DR

This paper studies reflected anticipated backward stochastic differential equations with nonlinear resistance, establishing existence, uniqueness, comparison, and minimal solutions without small time horizon restrictions.

Contribution

It introduces a new proof method that relaxes Lipschitz conditions on generators, extending the theory of RABSDEs with nonlinear resistance.

Findings

Proved existence and uniqueness without small horizon constraint.

Established a comparison theorem for RABSDEs.

Derived the minimal solution under linear growth conditions.

Abstract

In this paper, we consider reflected anticipated backward stochastic differential equations (RABSDEs, for short) with an additional resistance in the generators. Firstly, we study the existence and uniqueness results. In Luo (2020), the condition of a small time horizon is needed. Compared with the proving method in Luo (2020), we use a different proving method to avoid requiring the Lipschitz coefficients of generators $f(t,y,z,\theta,\vartheta,m,\bar{m})$ for $y,z,\theta,\vartheta$ to be small enough. We only require the Lipschitz coefficient for resistance in generator is small enough. Moreover, a probabilistic structure for solution is specified. Secondly, we give a comparison theorem for this type of equation. At last, under the linear growth condition and some other conditions on resistance , we derive the minimal solution.