# 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 *L*^{2}[*a, b*]). The function space

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 *L*^{2}(Ω, , *P*)). The function space

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 *L*^{2}[*a, b*] given by

where *a* = *t*_{0} < *t*_{1} < ··· < *t*_{n} = *b*. The Wiener integral of *f* is defined as

where {*B*_{t}}_{t≥0} is the standard ...

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