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Abstract
We study a class of stochastic differential equations driven by semimartingale with non-Lipschitz coefficients. New sufficient conditions on the strong uniqueness and the nonexplosion are derived ford-dimensional stochastic differential equations onRd(d>2)with non-Lipschitz coefficients, which extend and improve Fei’s results.
Highlights
Consider the following stochastic differential equations driven by a nonlinear integrator of the form t dXt = x0 + ∫ F (Xs, ds) (1) t t= x0 + ∫ b (Xs, s) ds + ∫ M (Xs, ds), equivalently dXt = b (Xt, t) dt + M (Xt, dt), X0 = x0, (2)where (Ω, F, P) is a probability space, (Ft)t⩾0 is σ fields, b(t, ⋅, ⋅) is a d-dimensional process of finite variation and F × B(Wd)/B(Rd) adapted function, M(x, t) is a d-dimensional continuous local martingale, and the pair (a(x, y, t), b(x, t)) is the local characteristic of F(x, t).Recently, many studies have focused on the strong uniqueness and the nonexplosion of stochastic differential equations with the coefficients satisfying the local Lipchitz condition
We study a class of stochastic differential equations driven by semimartingale with non-Lipschitz coefficients
Many studies have focused on the strong uniqueness and the nonexplosion of stochastic differential equations with the coefficients satisfying the local Lipchitz condition
Summary
Fang and Zhang discussed the pathwise uniqueness and the nonexplosion for a class of stochastic differential equations driven by Brownian motion with non-Lipschitz coefficients (see [8]). When the diffusion coefficient is uniformly nondegenerate and non-Lipschitz and drift coefficient is locally integrable, Zhang proved the existence of a unique strong solution up to the explosion time for a stochastic differential equation. For the stochastic differential equations with non-Lipschitz coefficients driven by Brown motion, Lan proved the pathwise uniqueness and nonexplosion, deriving the new sufficient condition (see [14]). We derive the new sufficient conditions on the strong uniqueness and the nonexplosion for ddimensional stochastic differential equations (1) driven by semimartingale with non-Lipschitz coefficients on Rd (d > 2), and the new conditions are sharp in a sense.
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