**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:**


1. 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?


2. For a given $X$, are there some growth conditions on $\mathcal P\_n$ that ensure the integral would converges a.s.?



Divergence of Riemann sums in the Itô integral
Here is an example of an integrand $X$ and a deterministic sequence of partitions such that almost surely, the limit fails to exist.


Indeed, consider $X = W$. The associated Riemann sums are


$$\lim\_{n \to \infty} \sum\_{[u, v] \in \mathcal P\_n} W\_u (W\_{v} - W\_{u}).$$


We may rewrite this as


$$\frac{1}{2} W^2\_T - \frac{1}{2} \sum\_{[u, v] \in \mathcal P\_n} (W\_u - W\_v)^2.$$


Now, Exercise 1.15 in Peres and Mortens' *Brownian Motion* yields a sequence of deterministic partitions $\mathcal P\_n$ such that $\limsup\_{n \to \infty} \frac{1}{2} \sum\_{[u, v] \in \mathcal P\_n} (W\_u - W\_v)^2 = \infty$, almost surely, whence the result follows.


