Linear transformations of two independent Brownian motions and orthogonal decompositions of Brownian filtrations

Abstract

Brownian motions defined as linear transformations of two independent Brownian motions are studied, together with certain orthogonal decompositions of Brownian filtrations.

Transformations of Two Independent Brownian Motions and Orthogonal Decompositions of Brownian Filtrations

“…We notice also that the transformation T is closely related to the Hardy transform (see [20]). It is therefore interesting to explore the same questions as that in [19] from this point of view.…”

“…Recently, Wu and Yor [19] generalized this result to the case of two Brownian motions by considering the process X given by…”

“…In Section 2, in order to prepare our analysis, we introduce the definition of the Volterra operator jRT g and its adjoint K* % and provide some additional properties which will be useful in the rest of the paper. In Section 3, as in [19] we construct, with the help of the operator (/ -AX*), two independent Brownian motions X and Y in the natural filtration of В and В having a noncanonical Volterra representation (i.e., ^Х' У) С j^B' B) ). In Section 4, by using the method of iteration we get two new sequences of Brownian motions which are independent of each other that enables us to construct a new orthogonal decomposition of the filtration generated by X and Y.…”

“…On the other hand, we conclude that for fixed t ^ 0, the sequence (f* g(u) dB£> s ) N^0 is not strongly L 2 -convergent, but converges weakly to 0 in L 2 . For more details on the subject the reader can con sult [11], [12], and [19] Moreover, they satisfy the following stochastic differential equation:…”

“…Motivated by the Kyle-Back model of «insider trad ings Follmer, Wu, and Yor [8], [19] investigate some mathematical problems which appear in this context. We will give a brief outline of [19]: Wu and Yor consider certain classes of linear transformations of two independent Brow nian motions and study their canonical decompositions. More precisely, let (B t )t^o and (B t ) t^0 be two independent Brownian motions defined on a com plete probability space (fi,^",P).…”