On the construction of Wiener integrals with respect to certain pseudo-Bessel processes
Let \((B_t)\) be a standard real Brownian motion, \((R_t)\) be a \((B_t)\)-driven Bessel process, and \(B_t^{(s)}:=\frac{1}{\sqrt s} B_{st}\). In order to define stochastic integrals such as \(\int^1_0h(s)d [\sqrt sf(\frac{Bs}{\sqrt s})]\) or \(\int^1_0h(s)d[\sqrt sf(\frac{Rs}{\sqrt s})]\), for \(h\in L^2[0,1]\), the authors are led to consider the existence of integrals \(\int^1_0h(s)_xF[B^{(s)}]\frac {ds}{\sqrt s}\), where \(F\) is some Brownian functional. They find that this existence is in turn guaranteed by the finiteness of \(C^*_F:=\int^1_0|\mathbb{E} [F(B)_xF (B^{(s)})]|\frac{ds}{s}\). They study some sufficient conditions ensuring \(C^*_F<\infty\), and apply them in particular to the examples: \(F(B)= g(B_1)\) and \(F(B)=g(R_1)\).
scientific article; zbMATH DE number 5083375
scientific article