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

MathSciNet review:
4652998

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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.

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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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*On solutions of one-dimensional stochastic differential equations without drift.* Z. Wahrsch. Verw. Gebiete **68** (1985), no. 3, 287–314. MR **0771468**
- J. Garzon, J.A. Leon, and S. Torres.
*Fractional stochastic differential equation with discontinuous diffusion.* Stoch. Anal. Appl. **35** (2017), no. 6, 1113–1123. MR **3740753**
- M. Hinz, J. Tölle, and L. Viitasaari.
*Sobolev regularity of occupation measures and paths, variability and compositions.* Electron. J. Probab. **27** (2022), Paper No. 73, 29 pp. MR **4440066**
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*Variability of paths and differential equations with BV-coefficients*. Accepted for publication in Annales de l’Institut Henri Poincaré, 2022. MR **4440066**
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*Composing functions of bounded variation.* Proc. Amer. Math. Soc. **83** (1981), no. 2, 354–356. MR **0624930**
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*Local time applications to one-dimensional stochastic differential equations.* Seminar on probability, XVII, Lecture Notes in Math., vol. 986, Springer, Berlin, 1983, pp. 15–31. MR **0770393**
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*Young differential equations with power type nonlinearities.* Stochastic Process. Appl. **127** (2017), no. 9, 3042–3067. MR **3682123**
- 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