Solution of SDE

Question

I would like to find the solution of the following SDE: $$dX(t)=(X(t)+a)\sigma(t)dW(t)$$ with $X(0)=X_0$ and with $X_0$, $a$ are real numbers and dW is the classical Browninan motion.

I tried using Ito with the logarithm function $ln(X+a)$, does it sound correct?

Thank you

Answer 1

Let $Y(t) = X(t) +a $. Then, per Ito,

$$ d(\ln Y) = \sigma(t)dW(t)-\frac12 \sigma^2(t)dt$$

which leads to the solution

$$X(t)= -a + (X_0+a)\exp\left(\int_0^t\sigma(s)dW(s)-\frac12\int_0^t\sigma^2(s)ds \right)$$