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
\int_0^t W(s)\circ dW(t)\;.
we then define the integral buy
\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)
where \(t_k^n=k\frac{t}n\). Prove that with probability one
X_t= \int_0^t W(s)\circ dW(s)= \frac12 W(t)^2\;.
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
dX_t&=f_tdt + g_tdW_t
and \(B_t\) be a second (possibly correlated with \(W\) ) Brownian
motion. We define the Stratanovich integral \(\int X_t \circ dB_t\)  by
\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
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
    dZ_t&=b_tdt + \sigma_tdW_t\;.
    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

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

  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 ?