Stochastic flow and noise associated with the Tanaka stochastic differential equation (Q2722148)

Consider the following one-dimensional stochastic differential equation (SDE) \( dX_t=\text{sign}(X_{t}) dw_{t}\), where \(\text{sign}(y)=1_{[0,\infty)}(y)-1_{(-\infty,0)}(y),\) \(w_{t}\) is a one-dimensional Wiener process and \(dw_{t}\) is the stochastic differential in the Itô sense, \(t\geq s \in R,\) \(X_{s}=x.\) This SDE was first introduced by \textit{H. Tanaka} [Japanese, Sem. Probab. 1964. 17. Kakurit. Sem.] as a simplest example of SDE's having law unique solutions which, however, can not possess any strong solutions. NEWLINENEWLINENEWLINEThe existence and the uniqueness in law of a family \({\mathbb X}=\{X_{s,t}(x)\}\) of solutions to the above-mentioned SDE which form a coalescing stochastic flow is shown. This stochastic flow \({\mathbb X}\) naturally generates a noise. This noise has been introduced and studied deeply by \textit{B. Tsirelson} [in: Doc. Math., J. DMV Extra Vol. ICM III, 311-320 (1998; Zbl 0907.60053); ``Unitary Brownian motions are linearizable'' (Preprint 1998); ``Scaling limit of Fourier-Walsh coefficients'' (Preprint 1999)]. The noise generated by the flow \({\mathbb X}\) of solutions to the SDE may be a simplest example of predictable, non-Gaussian or nonwhite noise. The properties of a noise (in the Tsirelson sense), which is generated by solutions of the Tanaka's equation, are studied. The approach is somewhat different from the famous one: The spectral measure for the noise is computed directly without relating it to a random walk approximation.