Sandwiched SDEs with unbounded drift driven by Hölder noises

• Published in: Advances in applied probability Vol. 55; no. 3; pp. 927 - 964
• Main Authors: Di Nunno, Giulia, Mishura, Yuliya, Yurchenko-Tytarenko, Anton
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.09.2023; Applied Probability Trust
• Subjects: Brownian motion; Continuous noise; Differential equations; Drift; Lower bounds; Noise; Original Article; Probability; Random variables; Stochastic models; 60H10; 60G22; 91G30; unbounded drift; numerical scheme; Sandwiched process; Hölder continuous noise; stochastic volatility; 60H35
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2022.56

We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of order $\lambda \in (0,1)$ . 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 mild assumptions on the noise, we prove that the solution has moments of all orders. In addition, we provide its connection to the solution of some Skorokhod reflection problem. As an illustration of our results and motivation for applications, we also suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes. We complete the study by providing a numerical scheme for the solution.