Stochastic differential equations with discontinuous diffusion coefficients
Authors:
Soledad Torres and Lauri Viitasaari
Journal:
Theor. Probability and Math. Statist. 109 (2023), 159-175
MSC (2020):
Primary 60H05; Secondary 60G22, 26A33
DOI:
https://doi.org/10.1090/tpms/1201
Published electronically:
October 3, 2023
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Additional Information
Abstract: We study one-dimensional stochastic differential equations of the form $dX_t = \sigma (X_t)dY_t$, where $Y$ is a suitable Hölder continuous driver such as the fractional Brownian motion $B^H$ with $H>\frac 12$. The innovative aspect of the present paper lies in the assumptions on diffusion coefficients $\sigma$ for which we assume very mild conditions. In particular, we allow $\sigma$ to have discontinuities, and as such our results can be applied to study equations with discontinuous diffusions.
References
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References
- R. Bass and Z-Q. Chen. One-dimensional stochastic differential equations with singular and degenerate coefficients. Sankhyā 67 (2005), no. 1, 19–45. MR 2203887
- H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext, Springer, New York, 2011. MR 2759829
- B. Boufoussi and Y. Ouknine. On a SDE driven by a fractional Brownian motion and with monotone drift. Electron. Comm. Probab. 8 (2003), 122–134. MR 2042751
- Z. Chen, L. Leskelä, and L. Viitasaari. Pathwise Stieltjes integrals of discontinuously evaluated stochastic processes. Stochastic Process. Appl. 129 (2019), no. 8, 2723–2757. MR 3980142
- H. J. Engelbert and W. Schmidt. On one-dimensional stochastic differential equations with generalized drift. Stochastic differential systems (Marseille–Luminy, 1984), 143–155, Lect. Notes Control Inf. Sci., 69, Springer, Berlin. MR 0798317
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- Y. Mishura. Stochastic calculus for fractional Brownian motion and related processes. Springer, Berlin, 2008. MR 2378138
- Y. Mishura and D. Nualart. Weak solutions for stochastic differential equations with additive fractional noise. Statist. Probab. Lett. 70 (2004), no. 4, 253–261. MR 2125162
- S. Nakao. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations. Osaka Math. J. 9 (1972), 513–518. MR 0326840
- D. Nualart and A. Răşcanu. Differential equations driven by fractional Brownian motion. Collect. Math. 53 (2002), no. 1, 55–81. MR 1893308
- S.G. Samko, A.A. Kilbas, and O.I. Marichev. Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Yverdon, 1993. MR 1347689
- M.S. Veillette and M.S. Taqqu. Properties and numerical evaluation of the Rosenblatt distribution. Bernoulli 19 (2013), no. 3, 982–1005. MR 3079303
- M. Zähle. Integration with respect to fractal functions and stochastic calculus. I. Probab. Theory Related Fields 111 (1998), no. 3, 333–372. MR 1640795
- M. Zähle. On the link between fractional and stochastic calculus. Stochastic dynamics, 305–325, 1999, Springer, New York. MR 1678495
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Additional Information
Soledad Torres
Affiliation:
Universidad de Valparaíso, Facultad de Ingeniería, CIMFAV, Chile
Email:
soledad.torres@uv.cl
Lauri Viitasaari
Affiliation:
Uppsala University, Department of Mathematics, Sweden
Email:
lauri.viitasaari@math.uu.se
Keywords:
Stochastic differential equation,
fractional calculus,
Hölder continuity,
discontinuity,
bounded variation
Received by editor(s):
October 27, 2022
Accepted for publication:
March 1, 2023
Published electronically:
October 3, 2023
Additional Notes:
The first author was partially supported by the grant Fondecyt Regular 1230807.
The second author wishes to thank Vilho, Yrjö, and the Kalle Väisälä foundation for financial support.
Article copyright:
© Copyright 2023
Taras Shevchenko National University of Kyiv
