The Non-uniform Riemann Approach to Stochastic Integration

The following sections are included:

• Preface
• Contents

In this chapter, we provide the background and the fundamentals of stochastic integrals using the generalized Riemann approach and classical stochastic integrals.

In this chapter, we shall use the non-uniform (generalized) Riemann approach to define the Itô integral. The Riemann sum is given by belated δ-fine partial division. The belated δ-fine full division may not exist, therefore the Vitali cover needed in the definition. The necessary and sufficient condition is given for an adapted Stochastic process to be Itô-integrable. Itô’s isometry and Itô’s Lemma are proved using Henstock’s Lemma and absolute continuity. An alternative definition of the Itô integral using a variation approach is given. The Vitali cover is not needed in the variation approach. Finally, the Itô integral and the classical Itô integral are proved to be equivalent.

This chapter consists of two parts. In the first part, we will define the belated derivative of a stochastic process and thereby characterize the class of all Itô integrable processes on [a, b] by its primitive process.

The notion of differential has been used extensively in stochastic analysis and mathematical finance. However, a rigorous definition of the differential is not given.

In the second part, we provide a definition of the differential of a stochastic process and a rigorous computation for the differential of the function of a stochastic process.

In this chapter, first we give an alternative definition of the Itô integral using a variational approach. In the variational approach, Vitali’s cover is not needed. We also use the variational approach to define other stochastic integrals. The variational integral is equivalent to the classical stochastic integral if the integrator is a semimartingale.

The theory of Multiple Stochastic Integral was first introduced by N. Wiener in 1938, who used the term polynomial chaos. This study was followed up by K. Itô in 1951, see [Itô (1951)]. Itô’s approach of studying the Multiple Stochastic Integral was a generalization of his approach to the one-dimensional Wiener integral, in which case the integrator is a canonical Brownian motion and the integrand is a real-valued deterministic function. His one-dimensional integral is what we call the Itô–Wiener Integral while his generalization to m-dimension is the Multiple Itô–Wiener Integral…

In this chapter, first we shall define Multiple Stochastic integrals. The integrators may not be Brownian Motions in one-dimensional space. We shall derive the Henstock–Fubini’s Theorem for Multiple Stochastic Integrals based on the Henstock approach and show that the Iterated Multiple Integral Formula is a direct consequence of Henstock–Fubini’s Theorem.

In Chapter 5, Section 5.1, we saw that the Itô–McShane integral (see Definition 5.2) does not take into account the values of the integrand f on the diagonal (and replace the integrand f by f₀, which vanishes on the diagonal). As we have seen in the preceding chapter, it turns out that the Multiple Itô–McShane integral is identical to the classical Multiple Itô–Wiener Integral. In this chapter we shall study the integral on the diagonal and offer a proof of the classical Hu–Meyer Theorem.

The following sections are included:

• Bibliography
• Index of Symbols
• Index