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 ?

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