Solution to an SDE

Question

This question isn't particularly interesting, but it is frustrating me. Is there a known solution to the stochastic differential equation

$$dX_t = (a + bX_t)dt + v X_t dW_t$$

where $W_t$ is standard Brownian motion?

Answer 1

You can find a list of SDE with known solutions in the book

• Kloeden, Platen: "Numerical Solution of Stochastic Differential Equations"

including the one you are asking about, for constants a, b, and v:

$$ X_t = \Phi_t( X_0 + b \int_0^{t} \Phi^{-1} _s d s) $$ with fundamental solution $$ \Phi_t = e^{ (a - \frac{1}{2} v^2) t + v W_t } $$