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Financial Mathematics by Roman N. Makarov, Giuseppe Campolieti

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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]:

    Pn={t0,t1,...,tn},0=t0<t1<<tn=T.

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

  • Introduce an intermediate partition Qn for the partition Pn:

    Qn={s1,s2,...,sn},ti1siti,i=1,2,...,n.

  • Define the Riemann (nth partial) sum as a weighted average of the values of f:

    Sn=Sn(f,Pn,Qn)=i=1nf(si)Δti.

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

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