Itô and Stratonovich interpretation of stochastic integrals
This follows mostly chapter 3.2 of Pavlotis.
For a stochastic integral of the form
$$I(t) = \int_0^t f(s) dW(s)$$
we are interested in integrating some process $f(t)$ up to some intermediate time $t \in [0, T]$ that depends only on the past of the Brownian motion $W(t)$.
We do a Riemann sum approximation of the integral:
The time is partitioned into small intervals of size $\delta t$, i.e.
$$t_k = k * \delta t \qquad \text{for}\quad k=0,…,K-1$$
and we define with the parameter $\lambda \in [0,1]$
$$\tau_k = (1 - \lambda) t_k + \lambda t_{k+1}$$
the stochastic integral as the $L^2$ limit of the Riemann sum approximation
$$I(t) := lim_{K \rightarrow \infty} \sum_k f(\tau_k) ( W(t_{k+1}) - W(t_k) )$$
The result depends on the choice of $\lambda$: Where do we evaluate $f(t)$ for the “rectangles”? (Compare with the midpoint for a standard Riemann integral)
The most common choices are
- $\lambda = 0$ (Itô): evaluate at left endpoint (advantage: no correlation to the current increment)
- $\lambda = 1/2$ (Stratonovich): evaluate at the midpoint (advantage: standard chain rule)
Choice depends on interpretation, seldomly others than these two are used.