The issue comes from some simulations I'm running in which a process ($X_t$) should always be positive, but using an Euler-Murayama scheme allows the process to become negative. To alleviate the problem, I define a new process ($Z_t$) such that $X_t$ is the exponential of $Z_t$, ensuring that $X_t$ is always positive. However, the new process goes to zero "to fast", that is the numerical solution is clearly incorrect. Can anyone shed some light on this issue?
Here is a more complete description:
Given an Ito process $X_t$ such that $$ dX_t = \mu^X_t dt + \sigma^X_t dW_t $$ When $dW_t$ has several negative realizations, an Euler-Murayama scheme allows $X_t$ to move into a region ($X_t < 0$) which is not allowed. This is driven by a large volatility $\sigma^X_t$.
To try to alleviate this problem, define a new process $Z_t$ such that $X_t = \exp Z_t$. Ito's lemma implies that $$ \begin{align*} dZ_t &= d(ln X_t)\\ &= \frac{dX_t}{X_t} - \frac{1}{2}\frac{(dX_t)^2}{X_t^2}\\ &= \left(\frac{\mu_t^X}{X_t} - \frac{1}{2}\frac{(\sigma^X_t)^2}{X_t^2}\right)dt + \frac{\sigma_t^X}{X_t} dW_t \end{align*} $$ Now $X_t \geq 0$, but the large volatility causes a different problem, namely a very negative drift in $Z_t$, which causes $Z_t$ to go to $-\infty$ very quickly and thus $X_t$ goes to zero very quickly, no matter the value of $W_t$.
Does anyone have suggestions on how to alleviate this issue? Is there some nice middle ground? Am I making a mistake?
Any guidance would be greatly appreciated.