# Integral representation of a stochastic differential equation

If I have a stochastic differential equation gives as $$dX_t = aW_t^2dt +bW_t^3dW_t$$ where $$w_t$$ is a wiener process and $$a,b$$ are real numbers. How can I reach the integral representation of $$X_t$$? In other words how do I compute $$f,g$$ such that:

$$X_t=X_s+\int_s^tf(t,W_t)dt+\int_s^tg(t,W_t)dW_t$$ when $$t>s$$

• You just ... integrate, from $s$ to $t$...? I mean, its not like you will solve those integrals, but the representation is pretty much this. As a result, your $f(t, W_t) = aW_t^2$ and $g(t, W_t) = bW_t^3$ Also, keep in mind, the $t$ should actually be something else, like $s$, since $t$ is in your limits of integration, so replace all $t$ with $s$ – Makina Dec 23 '18 at 1:36