Ito vs Stratonovich SDE with irregular time-dependence in coefficients

Question

Suppose I am interested in the Stratonovich SDE $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) \circ dB_t $$ If the coefficients are smooth enough in time and space, I can show this is equivalent to the Ito SDE $$ dX_t = \left[ b(t,X_t) + \frac{1}{2} \sigma(t,X_t) \sigma'(t,X_t) \right] dt + \sigma(t,X_t) dB_t $$ where $'$ denotes derivative in the space variable

I show this by using the definition of Stratonovich integral for two continuous semimartingales, given in terms of the Ito integral $$ \int_0^T Y_s \circ dZ_s = \int_0^T Y_s dZ_s + \frac{1}{2} [Y,Z]_T $$ To make this conversion in the SDE example, I have to calculate to be able to compute the covariation $[\sigma(\cdot, X_{\cdot}), B]_t$ which is easy because I can expand $\sigma(t, X_{t})$ using Ito's formula.

In the case where $\sigma(t,x)$ is smooth in $x$ but say only Holder continuous in $t$, then I cannot expand $\sigma(t, X_{t})$ using Ito's formula. But, I think the equivalence of the Ito and Stratonovich formulations should still hold true. Does it?

Answer 1

Yes, it does.

Let $dX_t = b(t,X_t) \, dt + \sigma(t,X_t) \, dB_t$ be an Itô process. Let $\Phi=\Phi(t,x)$ be such that $\Phi(\cdot,x)$ is continuous, $\Phi(t,\cdot)$ continuously differentiable, $$\mathbb{E}\int_0^T |\Phi(t,X_t)^2 | \, dt < \infty.$$ Then $$\int_0^T \Phi(t,X_t) \, \circ dB_t = \int_0^T \Phi(t,X_t) \, dB_t + \frac{1}{2} \int_0^T \frac{\partial}{\partial x} \Phi(t,X_t) \cdot \sigma(t,X_t) \, dt$$

cf. R.L. Stratonovich: A new representation for stochastic integrals and equations.