Asymptotic properties and finite time convergence of classical and modified methods for stochastic differential equations

As few stochastic differential equations have explicit solutions, the numerical schemes are studied to approximate the underlying solution. The fast development in computer science in recent years has made large scale simulations available, then the numerical analysis for stochastic differential equations has been blooming in past decades. However, the study on numerical solutions is still far behind the study on the underlying solutions. This thesis is devoted to mathematically rigorous investigation on the numerical solutions. Among all those attractive mysteries in the numerical analysis of stochastic differential equations, one of the popular problems is that if the numerical solutions can reproduce different properties of the underlying solutions. In thesis, we present some interesting results on this topic, which includes the asymptotic moment boundedness, the stationary distribution and the almost sure stability. The methods considered in this part are two classical methods, the explicit Euler-Maruyama method and the backward Euler-Maruyama method, and one modified method, the Euler-Maruyama method with random variable step size, which is first introduced in this thesis. Another main focus of numerical analysis is the finite time convergence. Our work on this topic is to modify the explicit Euler-Maruyama method and investigate the strong convergence (in the L2 sense) of it. Our investigation first goes to reproduce the asymptotic boundedness in small moment of the underlying solutions. The explicit Euler-Maruyama method is shown to be able to achieve this goal if both the drift coefficient and the diffusion coefficient are global Lipschitz. But with the global Lipschitz condition on the drift coefficient violated, a counter example indicates the failure of the explicit Euler-Maruyama method. A natural replacement, the backward Euler-Maruyama method, then is considered and successfully reproduce the asymptotic boundedness. In the case of small moment, we are only able to reproduce the boundedness property qualitatively so far. To answer another close related question that if we could reproduce the upper bound quantitatively, we strengthen the conditions and show that for the case of second moment the upper bound of the underlying solution can be reproduced as well. As the moment boundedness is key to the existence and uniqueness of the stationary distribution, we next study this property for the numerical solution. Since the backward Euler-Maruyama method has better performance than the explicit Euler-Maruyama method, in this part we only discuss the backward Euler-Maruyama method. The coefficient related sufficient conditions are given for the existence and uniqueness of the stationary distribution of the backward Euler-Maruyama method. Then the numerical stationary distribution is proved to converge to the stationary distribution of the underlying solution as step size vanishes. These results largely extend the existing works to cover wider range of stochastic differential equations. The almost sure stability is one of the hottest topics and many papers have studied the reproduction of this property by different kinds of classical methods. Therefore, we seek to study this property by one modified method, the Euler-Maruyama method with random variable step size. To our best knowledge, this is the first work to apply the random variable step size to the analysis of the almost sure stability of the explicit Euler-Maruyama method. One of our key contributions is that we show that the time variable is a stopping time, which were ignored by many researchers, and only under this circumstance the rest results hold. Compare with those fixed step size or nonrandom variable step size methods, the Euler-Maruyama method with random variable step size is shown to be able to reproduce the almost sure stability with much weaker conditions. As the strong convergence of the classical methods has already been widely studied and the recent works have shown the good performance of the modified classical methods, we present our findings in this area by introducing the stopped Euler method and show the strong convergence of it to the underlying solution with the rate a half. Briefly, the stopped Euler method is the classical Euler-Maruyama method equipped with the stopping time technique. The stopping time is originally employed to preserve the non-negativity of the numerical solution, and it turns out that the non-negativity in return enables the strong convergence of the method with the rate arbitrarily close to a half. Compare with the explicit Euler-Maruyama method, the stopped Euler method can cover some highly non-linear stochastic differential equations.As few stochastic differential equations have explicit solutions, the numerical schemes are studied to approximate the underlying solution. The fast development in computer science in recent years has made large scale simulations available, then the numerical analysis for stochastic differential equations has been blooming in past decades. However, the study on numerical solutions is still far behind the study on the underlying solutions. This thesis is devoted to mathematically rigorous investigation on the numerical solutions. Among all those attractive mysteries in the numerical analysis of stochastic differential equations, one of the popular problems is that if the numerical solutions can reproduce different properties of the underlying solutions. In thesis, we present some interesting results on this topic, which includes the asymptotic moment boundedness, the stationary distribution and the almost sure stability. The methods considered in this part are two classical methods, the explicit Euler-Maruyama method and the backward Euler-Maruyama method, and one modified method, the Euler-Maruyama method with random variable step size, which is first introduced in this thesis. Another main focus of numerical analysis is the finite time convergence. Our work on this topic is to modify the explicit Euler-Maruyama method and investigate the strong convergence (in the L2 sense) of it. Our investigation first goes to reproduce the asymptotic boundedness in small moment of the underlying solutions. The explicit Euler-Maruyama method is shown to be able to achieve this goal if both the drift coefficient and the diffusion coefficient are global Lipschitz. But with the global Lipschitz condition on the drift coefficient violated, a counter example indicates the failure of the explicit Euler-Maruyama method. A natural replacement, the backward Euler-Maruyama method, then is considered and successfully reproduce the asymptotic boundedness. In the case of small moment, we are only able to reproduce the boundedness property qualitatively so far. To answer another close related question that if we could reproduce the upper bound quantitatively, we strengthen the conditions and show that for the case of second moment the upper bound of the underlying solution can be reproduced as well. As the moment boundedness is key to the existence and uniqueness of the stationary distribution, we next study this property for the numerical solution. Since the backward Euler-Maruyama method has better performance than the explicit Euler-Maruyama method, in this part we only discuss the backward Euler-Maruyama method. The coefficient related sufficient conditions are given for the existence and uniqueness of the stationary distribution of the backward Euler-Maruyama method. Then the numerical stationary distribution is proved to converge to the stationary distribution of the underlying solution as step size vanishes. These results largely extend the existing works to cover wider range of stochastic differential equations. The almost sure stability is one of the hottest topics and many papers have studied the reproduction of this property by different kinds of classical methods. Therefore, we seek to study this property by one modified method, the Euler-Maruyama method with random variable step size. To our best knowledge, this is the first work to apply the random variable step size to the analysis of the almost sure stability of the explicit Euler-Maruyama method. One of our key contributions is that we show that the time variable is a stopping time, which were ignored by many researchers, and only under this circumstance the rest results hold. Compare with those fixed step size or nonrandom variable step size methods, the Euler-Maruyama method with random variable step size is shown to be able to reproduce the almost sure stability with much weaker conditions. As the strong convergence of the classical methods has already been widely studied and the recent works have shown the good performance of the modified classical methods, we present our findings in this area by introducing the stopped Euler method and show the strong convergence of it to the underlying solution with the rate a half. Briefly, the stopped Euler method is the classical Euler-Maruyama method equipped with the stopping time technique. The stopping time is originally employed to preserve the non-negativity of the numerical solution, and it turns out that the non-negativity in return enables the strong convergence of the method with the rate arbitrarily close to a half. Compare with the explicit Euler-Maruyama method, the stopped Euler method can cover some highly non-linear stochastic differential equations