White Noise Approach to Multiparameter Stochastic Integration.

In this dissertation we will set up the Hida theory of generalized Brownian functionals, or white noise analysis, on {\cal L}\sp{\*}(\IR\sp{\rm d}), the space of tempered distributions, and apply it to multiparameter stochastic integration. With the partial ordering on \IR\sbsp{+}{\rm d}: (s\sb1, dots,s\sb{\rm d}) $\u3c$ (t\sb1, dots,t\sb{\rm d}) if s\sb{\rm i} $\u3c$ t\sb{\rm i}, 1 $\leq$ i $\leq$ d, the Wiener process W((t\sb1, dots,t\sb{\rm d}),x) = $\langle$x,1\sb{\rm \lbrack 0,t\sb1)\times\cdots \times\lbrack 0,t\sb{d})}\rangle, x \in {\cal L}\sp{\*}(\IR\sp{\rm d}) is a generalization of a Brownian motion and there is the Wiener-Ito decomposition: L\sp2({\cal L}\sp{\*}(\IR\sp{\rm d})) = \sum\sbsp{\rm n = 0}{\infty}\timesK\sb{\rm n}, where K\sb{\rm n} is the space of n-tuple Weiner integrals. As in the one-dimensional case, there are the continuous inclusions (L\sp2)\sp+ \subset L\sp2({\cal L}\sp{\*}(\IR\sp{\rm d})) \subset (L\sp2)\sp-, and (L\sp2)\sp- is considered the space of generalized Wiener functionals. We will define the differentiation operator \partial\sb{\rm (t\sb1,\dots,t\sb{d}}) and its adjoint \partial\sp{\*}\sb{\rm (t\sb1,\dots,t\sb{d})} and give some properties. We prove that the multidimensional time Ito stochastic integral is a special case of a white noise integral and give conditions for its existence. For d = 2 the Ito integral is not sufficient for representing elements of L\sp2({\cal L}\sp{\*}(\IR\sp2)). We show that the other integral involved can also be realized in the white noise setting. For F \in {\cal L}\sp{\*}(\IR\sp{\rm d}) we will then define F(W((s,t),x) as an element of (L\sp2)\sp- and obtain a generalized Ito formula

White Noise Approach to Multiparameter Stochastic Integration

AbstractIn this paper we will set up the Hida theory of generalized Wiener functionals using S∗(Rd), the space of tempered distributions on Rd, and apply the theory to multiparameter stochastic integration. With the partial ordering on R+d: (s1, …, sd) < (t1, …, td) if si < ti, 1 ≤ i ≤ d, the Wiener process W((t1 …, td), x) = 〈x, 1[0,t1)x … x[0,td)〉, ξ ϵ I∗(Td) is a generalization of a Brownian motion and there is the Wiener-Ito decomposition: L2(S∗(Rd)) = Σn = 0∞⊕Kn, where Kn is the space of n-tuple Wiener integrals. As in the one-dimensional case, there are the continuous inclusions (L2)+ ⊂ L2(I∗(Rd)) ⊂ (L2− and (L2)− is considered the space of generalized Wiener functionals. We prove that the multidimensional Ito stochastic integral is a special case of an element of (L2)−. For d = 2 the Ito integral is not sufficient for representing elements of L2(S∗(R2)). We show that the other stochastic integral involved can also be realized in the Hida setting. For F∈S∗(R) we will define F(W(s, t), x) as an element of (L2)− and obtain a generalized Ito formula

Two-Parameter Stratonovich Integrals

In a space of generalized white noise functionals we define several integrals of the Stratonovich type. Our approach uses multiple Stratonovich integrals defined via a renormalization operator on Fock space. Once defined we relate these integrals in a Green\u27s theorem. AMS(MOS) subject classification: 60G20, 60H05

Complex White Noise Analysis

This paper describes a new space, (D[sup *, sub C]), of complex Wiener distributions for the analysis of multi-parameter generalized stochastic processes θ : R[sup N] → (D[sup *, sub C]). For a certain class of functions F : C[sup m] → C and complex Wiener integrals Φ[sub 1], …, Φ[sub m], F(Φ[sub 1], …, Φ[sub m]) is defined as an element of (D[sup *, sub C]) and its Pock space decomposition determined

White noise approach to multiparameter stochastic integration

In this paper we will set up the Hida theory of generalized Wiener functionals using *(d), the space of tempered distributions on d, and apply the theory to multiparameter stochastic integration. With the partial ordering on +d: (s1, ..., sd) , [xi] [epsilon] I*(Td) is a generalization of a Brownian motion and there is the Wiener-Ito decomposition: L2(*(d)) = [Sigma]n = 0[infinity][circle plus operator]Kn, where Kn is the space of n-tuple Wiener integrals. As in the one-dimensional case, there are the continuous inclusions (L2)+ [subset of] L2(I*(Rd)) [subset of] (L2- and (L2)- is considered the space of generalized Wiener functionals. We prove that the multidimensional Ito stochastic integral is a special case of an element of (L2)-. For d = 2 the Ito integral is not sufficient for representing elements of L2(*(2)). We show that the other stochastic integral involved can also be realized in the Hida setting. For F[set membership, variant]*() we will define F(W(s, t), x) as an element of (L2)- and obtain a generalized Ito formula.Hida theory multiparameter stochastic integral Ito formula

Programming Projects

This chapter contains a number of programming projects that can be assigned at any time during the course (but only after some initial programming skills have been developed). They can also form the basis for individual studies or a second course on computing in Maple

Graphics Programming

In this chapter we discuss programming techniques involved in producing graphic images for mathematical objects, such as curves and surfaces. We will also examine the plot structures, PLOT and PLOT3D, that Maple uses to render these images on plot devices such as the monitor screens or printers

Wiener Distributions and White Noise Analysis

The paper describes the structure of a new space of generalized Wiener functionals, (D∞)*, called the Wiener algebra, or space of Wiener distributions, and demonstrates its use in the white noise analysis. The concepts of derivatives and integrals for multi-time parameter generalized stochastic process θ:RN→(D∞)* are introduced, and a derivative version of Itô\u27s lemma is proved. The algebraic structure of (D∞)* and its lattice of subspaces is elaborated, and within this framework a generalized version of the Malliavin calculus is presented

Stochastic Integrals for Nonprevisible, Multiparameter Processes

We develop a general theory for stochastic integrals of generalized stochastic processes X(t), depending on multidimensional time, within the framework of the space of Wiener distributions (D*)