Sandwiched SDEs with unbounded drift driven by H\"older noises

TL;DR

This paper investigates stochastic differential equations with unbounded drifts driven by H"older continuous noises, establishing existence, bounds, moments, and numerical schemes, with applications to stochastic volatility models.

Contribution

It introduces a novel analysis of SDEs with unbounded drifts and H"older noise, including solution properties and numerical methods, extending classical models like CIR and CEV.

Findings

Unique solutions exist under broad conditions.

Solutions can be bounded or stay above certain functions.

Solutions possess moments of all orders under specific assumptions.

Abstract

We study a stochastic differential equation with an unbounded drift and general H\"older continuous noise of an arbitrary order. The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some additional assumptions on the noise, we prove that the solution has moments of all orders. We complete the study providing a numerical scheme for the solution. As an illustration of our results and motivation for applications, we suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes.