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 \(\mathcal{F}_t\).

# Stratonovich integral: A first example

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