Existence and uniqueness of solutions for perturbed stochastic differential equations with reflected boundary

Abstract

Abstract In this paper, under some suitable conditions, we prove existence of a strong solution and uniqueness for the perturbed stochastic differential equations with reflected boundary (PSDERB), that is, { x ⁢ ( t ) = x ⁢ ( 0 ) + ∫ 0 t σ ⁢ ( s , x ⁢ ( s ) ) ⁢ d B ⁢ ( s ) + ∫ 0 t b ⁢ ( s , x ⁢ ( s ) ) ⁢ d s + α ⁢ ( t ) ⁢ H ⁢ ( max 0 ≤ u ≤ t ⁡ x ⁢ ( u ) ) + β ⁢ ( t ) ⁢ L t 0 ⁢ ( x ) , x ⁢ ( t ) ≥ 0 for all ⁢ t ≥ 0 , \left\{\begin{aligned} {}x(t)&=x(0)+\int_{0}^{t}\sigma(s,x(s))\,dB(s)+\int_{0}^{t}b(s,x(s))\,ds+\alpha(t)H\bigl{(}\max_{0\leq u\leq t}x(u)\bigr{)}+\beta(t)L_{t}^{0}(x),\\ x(t)&\geq 0\quad\text{for all}\ t\geq 0,\end{aligned}\right. where 𝐻 is a continuous R-valued function, σ , b , α \sigma,b,\alpha and 𝛽 are measurable functions, L t 0 L_{t}^{0} denotes a local time at point zero for the time of the semi-martingale 𝑥.