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**Question set up and motivation:**

We work over the interval $[0, T]$.

Let $W$ be a standard Brownian motion, and $\sigma$ an adapted almost surely Lipschitz continuous process with uniform Lipschitz constant $L$ and uniformly bounded with constant $M$.

We know we can write the Ito integral $\int_{0}^T \sigma_t \ dW_t$ as a limit in probability of Riemann sums - namely for any sequence of partitions $\mathcal P_n$ of $[0, T]$ with mesh going to $0$, we have

$$\int_{0}^T \sigma_t \ dW_t = \lim_{|\mathcal P_n| \to 0} \sum_{a_i \in \mathcal P_n}(\sigma(a_{i+1}) - \sigma(a_{i}))(W_{i+1} - W_{i})$$

where the limit is in probability.

Can we get “best/worst case” bounds on the rate at which these sums converge? More precisely:

**Definitions:**

Let $\text{Lip}(L, M)$ denote the set of adapted processes with uniform Lipschitz constant $L$ and uniform bound $M$. For a given partition $\mathcal P$, write $|\mathcal P|$ and $\|\mathcal P\|$ for $\underset{a_i \in \mathcal P}{\text{max}} \ |a_{i+1} - a_i|$ and $\underset{a_i \in \mathcal P}{\text{min}} \ |a_{i+1} - a_i|$ respectively.

Define the best case and worst case convergence rates as follows - given $\varepsilon, \delta > 0$, and $\sigma \in \text{Lip}(L)$, and a sequence of partitions $\mathcal P_n$ with mesh going to $0$, define

$$C(\{ \mathcal P_n \},\varepsilon, \delta, \sigma) := \inf \big \{N \in \mathbb N \ |\ \mathbb P\big(\sum_{a_i \in \mathcal P_n}[\sigma(a_{i+1}) - \sigma(a_{i}))(W_{i+1}$$

$$- W_{i})] - \int_{0}^T \sigma_t \ dW_t| < \delta \big) < \varepsilon \ ,\text{ for all } n \geq N\big \}$$

Further, define the best and worst case rates $\mathcal B$ and $\mathcal W$ as follows:

$$\mathcal B(\varepsilon, \delta) := \underset {\sigma \in \text{Lip}(L, M)}{\sup} \underset{s_k}{\sup} \underset {\{\mathcal P_n\}}{\sup} \ \|\mathcal P_{C(\{\mathcal P_n\}, \varepsilon, \delta, \sigma)} \ \ \|$$

$$\mathcal W(\varepsilon, \delta) := \underset {\sigma \in \text{Lip}(L, M)}{\sup} \underset {s_k}{\sup} \underset {\{\mathcal P_n\}}{\inf} \ \|\mathcal P_{C(\{\mathcal P_n\}, \varepsilon, \delta, \sigma)} \ \ \|$$

Where the second supremum is taken over all strictly decreasing sequences $s_k$ of positive numbers and the third supremum/infimum is taken over all sequences of partitions $\mathcal P_n$ with $\|\mathcal P_n\| = s_n$ and $|\mathcal P_n| \to 0$.

> **Question:** Given $\varepsilon, \delta > 0$, what are the values of $\mathcal B(\varepsilon, \delta)$ and $\mathcal W(\varepsilon, \delta)$?

*Note: By construction, these values will depend only on $T$, $M$, $L$, $\varepsilon$, and $\delta$.*

Sat, 25 Sep 2021 01:59 GMT