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 \(\mathcal{F}_t\).

# Category Archives: Stratonovich Integral

## Stratonovich integral: A first example

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Tagged JCM_math545_HW2_S23

## 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:

- \(\int_0^t B_t \circ dB_t\) (Explain why this agrees with the answer you obtained here).
- 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 ?
- (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 ?)

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Posted in Ito Integrals, Stratonovich Integral

Tagged JCM_math545_HW5_S14

## 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

- \[\lim_{N\rightarrow \infty}\mathbf E I_N^\alpha(t) \ .\]
- * \[\lim_{N\rightarrow \infty}\mathbf E \big( I_N^\alpha(t)\big)^2\]
- * 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 ?

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