Non-linear Affine Processes with Jumps

TL;DR

This paper introduces a probabilistic framework for non-linear affine jump processes, enabling the modeling of Knightian uncertainty through sublinear expectations and associated integro-differential equations.

Contribution

It provides a novel construction of non-linear affine processes with jumps using sublinear expectations, extending affine process theory to uncertain environments.

Findings

Constructs a family of sublinear expectations for affine jump processes.

Shows the process is a non-linear Markov process with a generator.

Derives a PDE for calculating expectations of Markovian functionals.

Abstract

We present a probabilistic construction of $\mathbb{R}^d$-valued non-linear affine processes with jumps. Given a set $\Theta$ of affine parameters, we define a family of sublinear expectations on the Skorokhod space under which the canonical process $X$ is a (sublinear) Markov process with a non-linear generator. This yields a tractable model for Knightian uncertainty for which the sublinear expectation of a Markovian functional can be calculated via a partial integro-differential equation.