Financial Mathematics

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 ...