Does solution exit for the stochastic differential equation (SDE) $d X_t = -\frac{1}{2} e^{-2 X_t} dt + e^{-X_t} dW_t$

Question

Let $W_t$ be a standard Brownian Motion. Solve the stochastic differential equation: $$d X_t = -\frac{1}{2} e^{-2 X_t} dt + e^{-X_t} dW_t$$ Does the solution exist for all times t?

Answer 1

Set $Y=e^X-W$, then by the Ito theorem $$ dY=e^XdX+\frac12e^Xd[X]-dW=0\implies Y=e^{X_0} $$ and the equation $e^{X_t}=e^{X_0}+W_t$ can only be true as long as the right side is positive. As there is a non-zero probability that it becomes non-negative, the solution will not exist for all times.

---

How to get to this idea with a little hand-waving

On an atomic level one has $(dW)^2=dt$. Then the expression $e^{-X}dW-\frac12e^{-2X}(dW)^2$ is the start of the Taylor series of $\ln(1+e^{-X}dW)$. Taking the exponential gives thus $$ e^{X_{t+dt}-X_t}=1+e^{-X_t}dW_t\implies e^{X_{t+dt}}=e^{X_t}+dW_t\implies e^{X_t}=e^{X_0}+W_t $$ To confirm this heuristic computation, apply as above the Ito theorem to $Y=e^X-W$ to confirm that it is a constant process.