A class of multidimensional nonlinear diffusions with the Feller property

TL;DR

This paper studies a family of multidimensional nonlinear diffusions with uncertain drift, establishing the Feller property and linking the value function to a semilinear Kolmogorov equation, highlighting a smoothing effect due to randomness.

Contribution

It introduces a new class of nonlinear diffusions with set-valued drift depending on time and path, proving the Feller property and connecting to semilinear PDEs.

Findings

Feller property established for the nonlinear diffusion semigroup

Smoothing effect observed due to randomness in the model

Connection made between the value function and semilinear Kolmogorov equation

Abstract

In this note we consider a family of nonlinear (conditional) expectations that can be understood as a multidimensional diffusion with uncertain drift and certain volatility. Here, the drift is prescribed by a set-valued function that depends on time and path in a Markovian way. We establish the Feller property for the associated sublinear Markovian semigroup and we observe a smoothing effect as our framework carries enough randomness. Furthermore, we link the corresponding value function to a semilinear Kolmogorov equation.