Help on solving a stochastic differential equation

Question

I am trying to solve the following SDE $$dX(t)=rdt+aX(t)dW(t),\ t>0$$ $$X(0)=x$$ where W() is a Wiener process and r,a and x real numbers. I have proceeded by using the integrating factor $$F(t)=exp^{-aW(t)+(1/2)a^{2}t}$$ I have calculated dF using Ito's Lemma

$$dF_{t}=(1/2)a^{2}{F}_{t}dt-a{F}_{t}dW+(1/2)a^{2}{F}_{t}dW^{2}=a^{2}F_{t}dt-aF_{t}dW_{t}$$ and then I proceeded in finding $$d(X_{t}F_{t})=X_{t}dF_{t}+F_{t}dX_{t}+dX_{t}dF_{t}=rF_{t}dt+(a-1)X_{t}F_{t}dW_{t}$$ I have 2 questions:

1. Am I correct until now?
2. How do I proceed in finally solving the SDE and finding X?

Answer 1

With your SDE for $F$, I get:

$$ dXdF = -a^2XFdt $$

$$FdX = rFdt + aXFdW $$

$$XdF = a^2XF dt -aXF dW$$

So, adding up:

$$ d(XF) = rF dt, $$

giving

$$ X_t = F^{-1}_t X_0 + rF^{-1}_t \int_0^t F_u du $$