# Stratanovich integral

Let $$X_t$$ be an Ito processes with
\begin{align*}
dX_t&=f_tdt + g_tdW_t
\end{align*}
and $$B_t$$ be a second (possibly correlated with $$W$$ ) Brownian
motion. We define the Stratanovich integral $$\int X_t \circ dB_t$$  by
\begin{align*}
\int_0^T X_t \circ dB_t = \int_0^T X_t dB_t + \frac12 \int_0^T \;d\langle X, B \rangle_t
\end{align*}
Recall that if $$B_t=W_t$$ then $$d\langle B, W \rangle_t =dt$$ and it is zero if they are independent. Use this definition to calculate:

1. $$\int_0^t B_t \circ dB_t$$ (Explain why this agrees with the answer you obtained here).
2. Let $$F$$ be a smooth function. Find equation satisfied by $$Y_t=F(B_t)$$ written in terms of Stratanovich integrals. (Use Ito’s formula to find the equation for $$dY_t$$ in terms of Ito integrals and then use the above definition to rewrite the Ito integrals as Stratanovich integrals“$$\circ dB_t$$”.) How does this compare to classical calculus ?
3. (Integration by parts) Let $$Z_t$$ be a second Ito process satisfying
\begin{align*}
dZ_t&=b_tdt + \sigma_tdW_t\;.
\end{align*}
Calculate $$d(X_t Z_t)$$ using Ito’s formula and then write it in terms of Stratanovich integrals. Why is this part of the problem labeled integration by parts ? (Write the integral form of the expression you derived for $$d(X_t Z_t)$$ in the two cases. What are the differences ?)