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?



Euler Schemes in Stochastic Differential Equations
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**](http://dx.doi.org/10.1090/memo/1112), *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. 


