Characteristic Function of an Ornstein Uhlenbeck process

Question

Consider the Ornstein Uhlenbeck process, defined by the SDE: $$ dX_t = \alpha(\mu - X_t) dt + \sigma dW_t $$

I want to compute the characteristic function of this process. My approach is simply to apply Ito's lemma and then taking the expectation. Therefore, since the characteristic function is defined by $$ \varphi(X_t)=\mathbb{E}[e^{itX_t}] $$ Therefore, we apply Ito's lemma to the original SDE, with the function $g(X_t,t)=e^{itX_t}$. This yields $$ dg(X_t) = (iX_t e^{itX_t} + ite^{itX_t}(\alpha(\mu -X_t)) - \frac{1}{2}t^2e^{itX_t} \sigma^2) dt + ite^{itX_t} \sigma dW_t $$ Hence, taking the expectation should yield $$ \mathbb{E}[g(X_t)] = e^{itX_0} + (iX_t e^{i t X_t} + i t e^{iX_t}(\alpha(\mu - X_t)) - \frac{1}{2} t^2 e^{itX_t}\sigma^2)t $$

Does the above reasoning make sense? I hope that somebody can help! :)

Answer 1

See 21.1 in Lecture #33, 34: The Characteristic Function for a Diffusion.

It refers to Lecture #31, 32: The Ornstein-Uhlenbeck Process as a Model of Volatility