Calculating the average of $\sin^2$ of a stochastic process

Question

I have a random process $\phi_t$ which evolves according to the SDE $$d \phi_t = \mu dt+ \sigma \sin \phi_t \,dW_t$$ with $\mu$ and $\sigma$ constants and $W_t$ a Wiener process. The initial condition is $\phi_0 =0$. I would like to know $$\langle \sin^2 \phi_t \rangle$$ which is a function of $t$, $\mu$, and $\sigma$. I do not know how to approach this problem. Is the evaluation of $\langle \sin^2 \phi_t \rangle$ turns out to be difficult, I would be at least interested in the stationary value for $t\to \infty$.

Answer 1

You are intrested on the quadratic variation of the process $(X_t)_{t\geq0}$ defined by $X_t=u(\phi_t) $where $u(x) = \sin^2(x)$.

By Itô's lemma follows that \begin{align} X_t = u(\phi_t)& = u(\phi_0) + \int _0^t u'(\phi_u) ~d\phi_u +\frac{1}{2}\int _0^t u''(\phi_u) ~d\langle\phi\rangle_u \\&= \int _0^t 2 \sin(\phi_u)\cos(\phi_u) ~(\mu ~du +\sigma \sin(\phi_u) ~dW_u) \\ & \quad +\frac{\sigma^2}{2}\int _0^t 2(\cos^2(\phi_u) -\sin^2(\phi_u))\sin^2(\phi_u) ~du \\&= \int _0^t (2\mu \sin(\phi_u)\cos(\phi_u) +\sigma^2(\cos^2(\phi_u) -\sin^2(\phi_u))\sin^2(\phi_u)~du \\ & \quad+2\sigma\int _0^t \sin^2(\phi_u) ~dW_u\end{align}

then $$ \langle X\rangle_t = 4 \sigma^2\int _0^t \sin^4(\phi_u) ~du$$

Edit

I've just saw that for you $ \langle sin^2(\phi_t)\rangle$ means $\mathbb E \left \{sin^2(\phi_t) \right\}$ and not the quadratic variation. So, always by the result obtained from application of Itô's lemma, the fact that the mean of stochastic integral is zero and by Fubinni's theorem, we have

\begin{align} \mathbb E \left \{sin^2(\phi_t) \right\} &= \mathbb E \left \{ \int _0^t (2\mu \sin(\phi_u)\cos(\phi_u) +\sigma^2(\cos^2(\phi_u) -\sin^2(\phi_u))\sin^2(\phi_u)~du \right\} \\ & \quad+ \mathbb E \left \{ 2\sigma\int _0^t \sin^2(\phi_u) ~dW_u \right\} \\&= \int _0^t \mathbb E \left \{(2\mu \sin(\phi_u)\cos(\phi_u) +\sigma^2(\cos^2(\phi_u) -\sin^2(\phi_u))\sin^2(\phi_u) \right\}~du \end{align}

As you can see it doesn't seem to solve the problem. But, by the same aproach applying Itô's lemma to $ \sin( \phi_t )\cos( \phi_t) $ and $\cos (2\phi_t) \sin^2 (\phi_t)$ and exploating trygonometric relations, I guess you can find a simpler ODE envolvin $\mathbb E \left \{sin^2(\phi_t) \right\}$ that you must solve to finally find the solution.