Conversion from Ito equations to Stratonovich

Question

I have the stochastic equations

$$ dx = pdt + \beta (x^2 + p)dV \\ dp = xdt - \gamma (x^3 + p^2)dW \\ $$

where $dV$ and $dW$ are mutually independent Wiener processes. I am asked to calculate the corresponding Stratonovich equations.

I know how to go from an Ito-integral to Stratonovich, but unsure what is meant by "calculating the corresponding Stratonovich equations". Can anybody shed light on this?

Answer 1

Hint:

Assume $n$-dimensional Ito process $\mathbf{X}$ which has the following dynamics: \begin{equation} d\textbf{X}_t = \textbf{b}(\textbf{X}_t,t)dt + \textbf{B}(\textbf{X}_t, t)d\textbf{W}_t \end{equation} where $\mathbf{b}$ : $\mathbb{R}^n\to\mathbb{R}^n$ and $\mathbf{B}: \mathbb{R}^n\to\mathbb{R}^{n\times m}$.

It exits an equivalent dynamic using the Stratonovich integral: \begin{equation} d\textbf{X}_t = \mathbf{\tilde{b}}(\textbf{X}_t,t)dt +\textbf{B}(\textbf{X}_t, t)\circ d\textbf{W}_t \end{equation} where \begin{equation} \left[\mathbf{\tilde{b}}(\textbf{X}_t,t)\right]^i= \left[\mathbf{b}(\textbf{X}_t,t)\right]^i -\frac12\sum_{k=1}^m\sum_{j=1}^n \frac{\partial{B}^{i,k}(\textbf{X}_{t},t)}{\partial x_j}{B}^{j,k}(\textbf{X}_{t},t)dt \end{equation}

In your case, we have $\mathbf{X}_t = \{x_t,p_t\}$. I will let you define $\mathbf{b}$ and $\mathbf{B}$.