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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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CHAPTER 31

THE WIENER INTEGRAL

The Wiener integral is a simple stochastic integral where the integrand is a deterministic function. In this chapter, we present a definition and some results of the Wiener integral.

31.1 Basic Concepts and Facts

Definition 31.1 (Function Space L2[a, b]). The function space

equation

denotes the space of all real-valued square Lebesgue integrable functions on [a, b], where μ is the Lebesgue measure.

Definition 31.2 (Function Space L2(Ω, , P)). The function space

equation

denotes the space of all square integrable real-valued random variables with inner product X, Y = E(XY).

Definition 31.3 (Wiener Integral of Step Functions). Let f be a step function in L2[a, b] given by

equation

where a = t0 < t1 < ··· < tn = b. The Wiener integral of f is defined as

(31.1) equation

where {Bt}t≥0 is the standard ...

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