2013/07/16 by Mykhaylo Shkolnikov, Ioannis Karatzas, Shkolnikov, Mykhaylo +1
Mathematics · #60F17 #60G10 #60H10 #60J27 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F17 #msc:60G10 #msc:60H10 #msc:60J27
paper · pdf · doi:10.48550/arxiv.1307.4422
13 pages
arxiv created 2013/07/16 · arxiv updated 2013/07/18
We determine the processes obtained from a large class of reflected Brownian motions (RBMs) in the nonnegative orthant by means of time reversal. The class of RBMs we deal with includes, but is not limited to, RBMs in the so-called Harrison-Reiman class [4] having diagonal covariance matrices. For such RBMs our main result resolves the long-standing open problem of determining the time reversal of RBMs beyond the skew-symmetric case treated by R.J. Williams in [16]. In general, the time-reversed process itself is no longer a RBM, but its distribution is absolutely continuous with respect to a certain auxiliary RBM. In the course of the proof we introduce a novel discrete approximation scheme for the class of RBMs described above, and use it to determine the semigroups dual to the semigroups of such RBMs.