A new Markov selection procedure for degenerate diffusions
The authors consider the \(d\)-dimensional stochastic differential equation \[ dX(t) = m(X(t))dt + \sigma(X(t))dW(t),\;\;X(0)=x, \] and the continuous viscosity solutions to the corresponding backward Kolmogorov equation for various time intervals and terminal conditions. Attention is paid to the limits as the noise decreases to zero, and a unique limit is found for certain conditions on \(m\) and \(\sigma\). A unique choice of limiting transition laws satisfying the Chapman-Kolmogorov equations results, and the canonical construction of a Markov process consistent with the given s.d.e.\ is hence obtained.
Markov selection: backward Kolmogorov equation
scientific article; zbMATH DE number 5799411
scientific article