Existence and uniqueness of obliquely reflecting Brownian motion in nonpolyhedral, piecewise smooth cones, with an example of application to diffusion approximation of bandwidth sharing queues

The paper considers the stochastic differential equation with reflection that defines a semimartingale Brownian motion in a nonpolyhedral, piecewise smooth cone, with radially constant, onlique direction of reflection on each face. Existence and uniqueness in distribution of the solution are proved, under suitable assumptions. The assumptions are shown to be verified by the conjectured limit, under diffusive space-time scaling, of the workload process in an example of stochastic network operating under an $\alpha$-fair bandwidth sharing policy, $\alpha\geq 2$, thus ensuring that the conjectured limit is uniquely characterized. This is a key step in proving diffusion approximation for the network.