A transformation function to be used with Ito's lemma for a specific SDE

Question

So I am trying to find a closed-form solution for the following SDE.

$$dX_t=a(e^{-t}-X_t)dt+be^{-t}dB \quad X_0=0\quad (I)$$ where $B$ represents a Brownian motion.

My approach, so far, is to use the so-called variation of constant method, i.e. find a suitable $Z=F(X_t,t)$ to be substituted in the original SDE.

Using such an $F$ and Ito's lemma, I can convert the original SDE into $$dZ=\mu_Z(t,Z)dt+\sigma_z(t,z)dB\quad (II)$$

where

$$\mu_Z=\frac{\partial F}{\partial t}+a(e^{-t}-X_t) \frac{\partial F}{\partial X}+\frac{1}{2}\Big(be^{-t}\Big)^2\frac{\partial^2 F}{\partial X^2}\quad $$

and

$$\sigma_Z=be^{-t}\frac{\partial F}{\partial X}.$$

Now this is the problem: a "suitable" $F$ must be such that the second SDE (II) turns into another form whose closed-form solution is already known, for example, a GBM.

I have been thinking of several functional forms for $F$ to no avail. Could anyone provide a hint? (Assuming that I am on the right track at all.) Thank you.

Answer 1

Hint:

Let $Y_t := F(X_t, t) := e^{at}(X_t - \frac{a}{1-a}(e^{-at} - e^{-t}))$, if $a \neq 1$, and $e^{t}X_t - t$, if $a = 1$, where $X_t$ is a solution to the above SDE.

1. Find SDE of $Y_t$ (should be quite nice).
2. Substitute the integral form of the SDE for $Y_t$ into its definition and rearrange for $X_t$.

The choice of $F$ is not arbitrary. One can derive this using ODE and considering $b=0$, i.e. when there is zero volatility. After solving the ODE, you may notice that the chosen $F(X_t, t)$ (note that $X_t$ is not random under zero volatility assumption) will be constant. The details for this method are outlined here.