Given a process what is the stochastic differential equation it fulfils?

Question

Given the process $X_t = (2+t+\exp(W_t))_t$ where $W_t$ is Brownian motion. How can I find the SDE that it fulfils. I am actually looking for two functions $\sigma, \tau$ such that $X_t = X_0 + \int_0^t \sigma(W_s,s)dW_s + \int_0^t\tau(W_s,s)ds$. Is there any straight forward way to get them? Cheers

Answer 1

Hint Apply Itô's formula to $f(t,x) := 2+t+e^x$. This gives you an expression of the form

$$X_t = X_0 + \int_0^t \sigma(s,e^{W_s}) \, dW_s + \int_0^t b(s,e^{W_s} ) \, ds.$$

Now use that

$$e^{W_s} = X_s-2-s.$$