On explicit tamed Milstein-type scheme for stochastic differential equation with Markovian switching

Abstract

We propose a new tamed Milstein-type scheme for stochastic differential equation with Markovian switching when drift coefficient is assumed to grow super-linearly. The strong rate of convergence is shown to be equal to 1.0 under mild regularity (e.g. once differentiability) requirements on drift and diffusion coefficients. Novel techniques are developed to tackle two-fold difficulties arising due to jumps of the Markov chain and the reduction of regularity requirements on the coefficients.

New Simplified High-Order Schemes for Solving SDEs with Markovian Switching Driven by Pure Jumps

“…Then, the existence and uniqueness of solutions for neutral SDEwMs were proven in [10] under non-Lipschitz conditions, and Euler approximate solutions were provided for solving SDEwMs. Common numerical schemes for solving SDEs with jumps or SDEwMs include Euler-Maruyama scheme [7,9,11,12], Milstein scheme [13,14], and jump-adapted scheme [15,16]. The authors of [17] studied the balanced implicit numerical methods for solving SDEs driven by Poisson jumps.…”

Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments

“…It is well known that the Milstein method for stochastic differential equations (SDEs) or stochastic delay differential equations (SDDEs) has strong convergence order one ( [4,6,7,11,16,23,26]). In addition, it can be seen from [18] that the Milstein-type method is much more computationally efficient than the Euler-type method for commutative SDEs.…”

Convergence and stability of the Milstein scheme for stochastic differential equations with piecewise continuous arguments

“…It is well known that the Milstein method for stochastic differential equations (SDEs) or stochastic delay differential equations (SDDEs) has strong convergence order one ( [4,6,7,11,16,23,26]). In addition, it can be seen from [18] that the Milstein-type method is much more computationally efficient than the Euler-type method for commutative SDEs.…”