# Category Archives: Stratonovich Integral

## Stratonovich integral: A first example

Let us denote the Stratonovich integral of a standard Brownian motion $$W(t)$$ with respect to itself by
\begin{align*}
\int_0^t W(s)\circ dW(t)\;.
\end{align*}
we then define the integral buy
\begin{align*}
\int_0^t W(s)\circ dW(t) = \lim_{n \rightarrow \infty}
\sum_k\frac12\big(W(t_{k+1}^n)+W(t_{k}^n)\big)\big(W(t_{k+1}^n) -W(t_{k}^n)\big)
\end{align*}
where $$t_k^n=k\frac{t}n$$. Prove that with probability one
\begin{align*}
X_t= \int_0^t W(s)\circ dW(s)= \frac12 W(t)^2\;.
\end{align*}
Observe that this is what one would have if one used standard (as opposed to Ito) calculus. Calculate $$\mathbf E [ X_t | \mathcal{F}_s]$$ for $$s < t$$ where $$\mathcal{F}_t$$ is the $$\sigma$$-algebra generated by the Brownian motion. Is $$X_t$$ a martingale with respect to $$\mathcal{F}_t$$.

## 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 ?)

## Ito to Stratonovich

Let’s think about different ways to make sense of $\int_0^t W(s)dW(s)$ were $$W(t)$$ is a standard Brownian motion. Fix any $$\alpha \in [0,1]$$define

\begin{equation*}
I_N^\alpha(t)=\sum_{j=0}^{N-1} W(t_j^\alpha)[W(t_{j+1})-W(t_j)]
\end{equation*}
were $$t_j=\frac{j t}N$$ and $$t_j^\alpha=\alpha t_j + (1-\alpha)t_{j+1}$$.
Calculate

1. $\lim_{N\rightarrow \infty}\mathbf E I_N^\alpha(t) \ .$
2. * $\lim_{N\rightarrow \infty}\mathbf E \big( I_N^\alpha(t)\big)^2$
3. * For which choice of $$\alpha$$ is $$I_N^\alpha(t)$$ a martingale ?

What choice of $$\alpha$$ is the standard It\^o integral ? What choice is the Stratonovich integral ?