I have a process defined for a Brownian motion $W_t$: $$X(t)=X_0\exp(-\frac{1}{2}\sigma^2t+\sigma W_t)$$ This process is positive provided $X_0>0$. I would like to modify it to allow paths to go negative up to a size $-d$ but I don't want to modify the mean or the variance! Is that even possible? What I know for this process $X$ is that the mean is $X_0$ and the variance is $X_0^2(exp(\sigma^2t)-1)$.
My first attempt was to define $$Y(t)=(X_0+d)\exp(-\frac{1}{2}\sigma^2t+\sigma W_t)-d$$ that keeps mean in place but than the variance blows up. Is there to adjust it differently?