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Abstract
In this paper, we study the Carathéodory approximate solution for a class of doubly perturbed stochastic differential equations (DPSDEs). Based on the Carathéodory approximation procedure, we prove that DPSDEs have a unique solution and show that the Carathéodory approximate solution converges to the solution of DPSDEs under the global Lipschitz condition. Moreover, we extend the above results to the case of DPSDEs with non-Lipschitz coefficients.
Highlights
As the limit process from a weak polymers model, the following doubly perturbed Brownian motion xt = Bt +α sup 0≤s≤t xs β inf 0≤s≤t xs, (1.1)was discussed by Norris et al [1], and it arises as the scaling limit of some selfinteracting random walks
The Picard iterative method is a well-known procedure for approximating the solution of stochastic differential equations (SDEs)
Remark 2.2 By (2.6), we conclude that the Carathéodory approximate solution converges to the true solution of equation (2.1) in the mean square sense, i.e., for any T > 0, E sup xn(t) – x(t) 2 → 0 as n → ∞
Summary
Using the Picard iterative procedure, they showed the existence and uniqueness of the solution to equation (1.2). Let L2([a, b]; R) denote the family of Ft-measurable, R-valued processes f (t) = {f (t, ω)}, t ∈ [a, b] such that b a Consider the following doubly perturbed stochastic differential equations: t t x(t) = x(0) + f s, x(s) ds + g s, x(s) dw(s) + α sup x(s) + β inf x(s), 0≤s≤t where α, β ∈ (0, 1), the initial value x(0) = x0 ∈ R and f : [0, T] × R → R, g : [0, T] × R → R are both Borel-measurable functions.
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