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?