Some motivation:
Let $W$ be a standard Brownian motion, and $X$ an integrable process with respect to $W$, i.e. progressively measurable with respect to the natural filtration of $W$ and square integrable on compacts almost surely.
It is known that if $X$ is continuous, then for any sequence of partitions $\mathcal P_n$ of $[0, T]$ with mesh going to $0$, the Riemann sums
$$\sum_{i = 0}^{K_n - 1} X_{t^n_i} (W_{t^n_{i+1}} - W_{t^n_i})$$
converge in probability to the Itô integral
$$\int_0^T X_t \, dW_t.$$
Here we have written $\mathcal P_n := \{t^n_0, \dots, t^n_{K_n}\}$.
Now it follows that if we take a suitable subsequence of partitions, the Riemann sums converge almost surely to the Itô integral. But I am wondering how almost sure convergence fails to begin with! It is unclear to me how a certain choice of partitions could lead to the limit fluctuating instead of existing.
Questions:
- Can anyone provide an explicit example of an almost surely continuous process $X$, integrable with respect to $W$, and a sequence of partitions $\mathcal P_n$ with mesh going to $0$ such that
$$\lim_{n \to \infty} \sum_{i = 0}^{K_n - 1} X_{t^n_i} (W_{t^n_{i+1}} - W_{t^n_i})$$
fails to exist with some positive probability, or even full probability?
- For a given $X$, are there some growth conditions on $\mathcal P_n$ that ensure the integral would converges a.s.?