Properties of Hida processes on \({\mathbb{R}}^ 2\). I: N-Hida processes
This paper deals with representations of a square-integrable centered process \(Y_ t\) \((Y_ t=\int^{t}F(t,u)B(du)\) with \(B(u)=Brownian\) motion, \(F(t,u)=\sum^{N}_{1}f_ n(t)g_ n(u)\) where \(f_ n,g_ n\) are some deterministic functions) in the spirit of N-Hida processes by allowing \(t\in {\mathbb{R}}^ 2\) and \(\dim (sp\{E(Y_ t| F_ s)\), \(t\geq s'\})\leq d_ s\) for s'\(\geq s\) as opposed to Hida's \(t\in {\mathbb{R}}\) and \(d_ s=N\). First two chapters have basically preparatory character. It is shown in chapter 3 that for \(t\in {\mathbb{R}}\), \(Y_ t=\int^{t}F(t,u)B(du)\) has an N-Goursat representation \(Y_ t=\sum^{N}_{1}f_ n(t)M_ n(t)\) where \(f_ n(t)\) are deterministic and \((M_ 1(t),...,M_ N(t))\) is a martingale iff \(Y_ t\) is N-Hida. As an application one obtains an explicit form of solution Y(t) of the stochastic differential equation \[ a(t)^{-1}\quad [Y^{(N- 1)}_{(dt)}-(\sum^{N-1}_{1}a_ j(t)Y^{(j)}_{(t)})dt]=B(dt)\quad with\quad Y_ 0=Y_ 0'=...=Y_ 0^{(N-1)}=0 \] i.e. \(Y(t)=\int^{t}_{0}\sum^{N}_{1}f_ n(t)g_ n(u)B(du)\) where \(f_ n,g_ n\) solve certain deterministic equations of order N. Chapter 4 extends some results to \(t\in {\mathbb{R}}^ 2\).
scientific article; zbMATH DE number 3879835