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Stochastic differential equation with piecewise continuous arguments: Markov property, invariant measure and numerical approximation

  • * Corresponding author: Yulan Lu

    * Corresponding author: Yulan Lu

This work is funded by National Natural Science Foundation of China (No. 11971470, No. 11871068, No. 12031020, No. 12022118, No. 12026428), by Youth Innovation Promotion Association CAS, and by National Postdoctoral Program for Innovative Talents (No. BX20180347)

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  • For the stochastic differential equation with piecewise continuous arguments, multiplicative noises and dissipative drift coefficients, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to numerically preserve the invariant measure, we apply the backward Euler method to the equation, and prove that the numerical solution at integer time is also a Markov chain and possesses a unique numerical invariant measure. By establishing several a priori estimations, we present the time-independent weak error analysis for the method via Malliavin calculus, which implies that the numerical invariant measure converges to the original one with weak order 1. Numerical experiments verify the theoretical analysis.

    Mathematics Subject Classification: Primary: 60H35, 37M25; Secondary: 65C30.


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  • Figure 1.  The expectations and variances of $ X(t) $ and $ X(k) $

    Figure 2.  The expectations (left column) and variances (right column) of $ X(t) $ with different $ \theta_2 $

    Figure 3.  Order of weak convergence of BE method

    Figure 4.  The evolution of $ \mathbb{E}\phi(Y_k) $ started from different initial data

    Figure 5.  Order of weak convergence of the BE method

    Figure 6.  The evolution of $ \mathbb{E}\phi(Y_k) $ started from different initial data

    Figure 7.  Order of weak convergence of the BE method

    Figure 8.  The evolution of $ \mathbb{E}\phi(Y_k) $ started from different initial data

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