Stochastic differential equations for Dirichlet processes
Let \(W\) denote a 1-dimensional Brownian motion. The authors study the stochastic differential equation (SDE) \(dX_t= a(X_t)dW_t+ b(X_t) dt\), where \(W\) is a 1-dimensional Brownian motion. Their goal is to generalize former results by \textit{D. W. Stroock} and \textit{S. R. S. Varadhan} [``Multidimensional diffusion processes'' (1979; Zbl 0426.60069)], \textit{A. K. Zvonkin} [Mat. Sb., n. Ser. 93(135), 129-149 (1974; Zbl 0291.60036)] and \textit{J. F. Le Gall} [in: Stochastic analysis and applications. Lect. Notes Math. 1095, 51-82 (1984; Zbl 0551.60059)] on the pathwise existence and uniqueness. They consider \(a\) as Hölder continuous of order \(1/2\) and \(b\) as a generalized function (\(b= a^2 B'\), where \(B\) is Hölder continuous), and they construct explicitly a Dirichlet process \(X\) which is proved to be the unique solution of the above SDE. Particular attention is also payed to the case \(b= aa'\) for which the infinitesimal operator associated to the SDE is of divergence form. It is shown that if \(a\in C^{1/2}\), there is a unique strong solution (which is a semimartingale if and only if \(a\) is a signed Radon measure), while for every \(\alpha\in (0,1/2)\) there is some \(a\in C^\alpha\) bounded away from zero and infinity such that the SDE does not have neither a strong solution nor pathwise uniqueness. Finally, in the last section the above SDE with \(b=0\) is interpreted in Stratonovich's sense. Existence and uniqueness of the strong solution is obtained for bounded, measurable coefficients \(a\) bounded below away from zero. For \(a\) being a signed Radon measure, this solution is characterized as semimartingale.
scientific article; zbMATH DE number 1709420