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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 $t>s$

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    $\begingroup$ 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 $s$ $\endgroup$ – Makina Dec 23 '18 at 1:36

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