Existence and pathwise uniqueness of solutions for stochastic differential equations involving the local time at point zero
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Cited by 4 publications
References 18 publications
In this paper, we study the Carathéodory approximate solution for a class of stochastic differential equations involving the local time at point zero. Based on the Carathéodory approximation procedure, we prove that stochastic differential equations involving the local time at point zero have a unique solution, and we show that the Carathéodory approximate solution converges to the solution of stochastic differential equations involving the local time at point zero with one-sided Lipschitz drift coefficient.
Existence and uniqueness of solutions for perturbed stochastic differential equations with reflected boundary
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.
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