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]:
Define Δti = ti − ti−1, i = 1, 2, . . . , n.
- Introduce an intermediate partition Qn for the partition Pn:
- Define the Riemann (nth partial) sum as a weighted average of the values of f:
- 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 ...