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Sandwiched SDEs with unbounded drift driven by Hölder noises

Part of: Stochastic processes Stochastic analysis Mathematical finance

Published online by Cambridge University Press:  08 March 2023

Giulia Di Nunno ,
Yuliya Mishura  and
Anton Yurchenko-Tytarenko
Giulia Di Nunno*
Affiliation:
University of Oslo and NHH Norwegian School of Economics
Yuliya Mishura*
Affiliation:
Taras Shevchenko National University of Kyiv
Anton Yurchenko-Tytarenko*
Affiliation:
University of Oslo
*
*Postal address: Department of Mathematics, University of Oslo, Moltke Moes vei 35, 0851 Oslo, Norway. Email address: giulian@math.uio.no
**Postal address: Department of Probability, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, Volodymyrska St. 64/13, Kyiv 01601, Ukraine. Email address: yuliyamishura@knu.ua
***Postal address:Department of Mathematics, University of Oslo, Moltke Moes vei 35, 0851 Oslo, Norway. Email address: antony@math.uio.no
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Abstract

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.

Keywords

Sandwiched processunbounded driftHölder continuous noisenumerical schemestochastic volatility

MSC classification

Primary: 60H10: Stochastic ordinary differential equations 60H35: Computational methods for stochastic equations
Secondary: 60G22: Fractional processes, including fractional Brownian motion 91G30: Interest rates (stochastic models)
Type
Original Article
Information
Advances in Applied Probability , Volume 55 , Issue 3 , September 2023 , pp. 927 - 964
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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