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I'm reading an introduction to Stochastic Calculus. I'm at the point where Ito integrals are developed and constrasted with the Stratonovich integral.

Below is a calculation of $\int_0^T W_t d W_t$.

enter image description here

How do I get from the first equation to the second? I presume ms-lim means $L^2$-limit, but even so, I can't seem to make sense of what the author's doing. I have other references for this integral, but I would like to understand this particular proof.

$W_t$ is a Wiener process and equation (4.29) just says:

$$ \mathbb{E}[(W_s-W_t)^2] = |s - t| $$

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  • $\begingroup$ Do you know that $$\sum_{k=1}^K (W_{t_k}-W_{t_{k-1}})^2$$ converges in $L^2$ to $T$....? (I agree with you that it takes much more effort to prove the statement than the author (or the answer below) are indicating; if I have enough time, I'll write a more detailed answer.) $\endgroup$ – saz Mar 6 '16 at 7:43
  • $\begingroup$ @saz: yeah, that's one of my main issues $\endgroup$ – user40167 Mar 7 '16 at 14:30
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To use intuitive terms: basically, equation 4.29 is saying that, using a much simpler notation, the variance of the process, since the mean is 0, becomes $ \mathbb{E}[(W_{t+\Delta t}-W_{t})^2] =\Delta t$, and since there is independence, $ \mathbb{E}[(W_{t+\Delta t}-W_{t}) (W_{t+2 \Delta t}-W_{t+\Delta t})] =0$. He is taking $K$ to $\infty$ means making the increments $\Delta t$ smaller and smaller since the overall interval $[0,T]$ does not change.

Now the difference between Ito and Stratonovich lies in (once again, maximally simplifying notation): When you do a sum of a function of $W$, in one case you have "nonanticipation": $\sum_i^{\frac{n}{\Delta t}} f(W_{t+i \Delta t}) ( W_{t+(i+1) \Delta t}-W_{t+i \Delta t}),$ in the other $\sum_i^{\frac{n}{\Delta t}} f(W_{t+(i+1) \Delta t})( W_{t+(i+1) \Delta t}-W_{t+i \Delta t})$. You make as with the Riemann integral $\Delta t$ smaller and smaller and increase the number of steps.

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The meaning is that if you take $\tau_k$ equal to $t_{k-1} $ you get the usual Ito integral, whereas if you take it in between $t_{k-1}$ and $t_k$ you don't. In particular if you always take exactly the midpoint then you get an additional term T/2 (check yourself). This might come as a surprise since for example for Riemann integrals (of continuous functions ) it doesn't matter which point you choose on each interval

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  • $\begingroup$ Sorry, what I really meant to ask was: how do we get from the first equation to the second $\endgroup$ – user40167 Mar 5 '16 at 13:07
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    $\begingroup$ That's what I tried to explain, skipping the Stratonovich part, staying within Ito as the question was about simple Ito identities. $\endgroup$ – Nero Mar 5 '16 at 13:32

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