Euler Schemes in Stochastic Differential Equations

Question

So i am trying to understand what happens in Implicit (backward) and Explicit (forward) Euler in Stochastic Differential Equations

I'll start with explicit. Say I have the following SDE known as Geometric Brownian Motion $dX(t)=aX(t)dt + bX(t)dW(t)$, for $a,b$ constants

The explicit scheme goes as follows

$X_{n+1} = X_n + aX_n\Delta t+bX_n\Delta W_n$ then I can rewrite this as

$X_n=\prod^{n-1}_{j=0} (1 + a\Delta t + b\Delta W_j)X_0$.

Now you can prove that you have a finite second moment by computing $E[X_n^2]=\prod_{j=0}^{n-1}((1+a\Delta t)^2 + b^2 \Delta t)X_0^2$. So now, if we further assume that $a<0$, then $E[X_n^2]\rightarrow0$ as $n\rightarrow\infty$ iff $|(1+a\Delta t)^2 + b^2 \Delta t)|=1 + 2\Delta t (a+\frac{b^2}{2}+\frac{a^2}{2}\Delta t )<1$ so the timestep $\Delta t$ must satisfy $0<\Delta t< \frac{-2(a+\frac{b^2}{2})}{a^2}$.

Now the fully Implicit Euler scheme goes as $X_{n+1}=X_n + aX_{n+1}\Delta t + bX_{n+1}\Delta W_n$, which can be rewritten as $X_n=X_0\prod_{j=0}^{n-1}\frac{1}{1-a\Delta t - b\Delta W_j}$ but it does not have finite moments.

If my research and calculations are correct, what could someone do to go around this problem and use implicit euler on a stiff stochastic differential equation?

Answer 1

The current state of the art is to tame the forward Euler-Maruyama scheme in an explicit way, and completely avoid tricky convergence questions that may come up when using drift-implicit Euler-Maruyama. The tamed Euler-Maruyama scheme is easy to implement and provably works if the drift coefficient is globally one-sided Lipschitz continuous and the noise coefficient is globally Lipschitz continuous. There are several variants of this scheme. For the drift-tamed and increment-tamed Euler-Maruyama schemes check out (1.5) and (3.122):

Martin Hutzenthaler and Arnulf Jentzen, MR 3364862 Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc. 236 (2015), no. 1112, v+99.

However, due to the presence of noise these schemes may not satisfy positivity ($X_t > 0$), which is a fundamental property of geometric Brownian motion.

Answer 2

The answer seems easier than expected. Before answering let me also note that $bX_{n+1}\Delta W_n$ will not converge to Ito as the discretization grid becomes finer.

So the answer is that the discretisation should become $X_{n+1}=X_n +aX_{n+1}\Delta t +bX_n \Delta W_n$.

Answer 3

To follow-up on your answer, if you look at the implicit schemes in Kloden's Numerical Solution of Stochastic Differential Equations you will notice that all of the implicit schemes are "semi-implicit" where the only implicit part is on the drift (or deterministic) term. This is because of the following:

We also saw in Section 8 of Chapter 9 that difficulties can arise in applying fully implicit schemes to obtain strong approximations of solutions of stochastic differential equations, because they usually involve reciprocals of Gaussian random variables which do not have finite absolute moments. Consequently, finite absolute moments generally will not exist for fully implicit strong approximations and a strong convergence analysis would not make sense. For mainly this reason we shall restrict our attention her to "semi-implicit" strong approximations, which we shall call implicit

So this is actually a more general idea that the proper implicit form of a method for SDEs is for it to be semi-implicit by only making the drift term implicit. The diffusion term should be kept the same to make sure there are finite moments.