Chapter 11

# Introduction to Continuous-Time Stochastic Calculus

## 11.1 The Riemann Integral of Brownian Motion

### 11.1.1 The Riemann Integral

Let f be a real-valued function defined on [0, T]. We now recall the precise definition of the Riemann integral of f on [0, T] as follows.

- For n ∈ ℕ, consider a partition Pn of the interval [0, T]:
${P}_{n}=\left\{{t}_{0},{t}_{1},\mathrm{...},{t}_{n}\right\},0={t}_{0}<{t}_{1}<\cdots <{t}_{n}=T.$

Define Δti = ti − ti−1, i = 1, 2, . . . , n.

- Introduce an intermediate partition Qn for the partition Pn:
${Q}_{n}=\left\{{s}_{1},{s}_{2},\mathrm{...},{s}_{n}\right\},{t}_{i-1}\le {s}_{i}\le {t}_{i},i=1,2,\mathrm{...},n.$

- Define the Riemann (nth partial) sum as a weighted average of the values of f:
${S}_{n}={S}_{n}\left(f,{P}_{n},{Q}_{n}\right)={\displaystyle \sum _{i=1}^{n}f\left({s}_{i}\right)\Delta {t}_{i}.}$

- Suppose that the mesh size δ(Pn) ≔ max1≤i≤n Δti goes to zero as n → ∞. If the limit limn→∞ Sn exists and does not depend on the ...