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On Strong Solutions of Stochastic Differential Equations and Their Trajectory Analogs

Abstract

We find new conditions for existence of strong solutions of ordinary differential equations with random right-hand side, stochastic differential equations with measurable random drift, and their trajectory analogs with symmetric integrals. We show that solutions of Itô equations satisfy a parabolic equation along trajectories of a Wiener process.

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REFERENCES

  1. 1

    M. A. Abdullin, N. S. Ismagilov, and F. S. Nasyrov, “One dimensional stochastic differential equations: pathwise approach,” Ufimsk. Mat. Zh. 5, no. 4, 3 (2013) [Ufa Math. J. 5:4, 3 (2013)].

    MathSciNet  Article  Google Scholar 

  2. 2

    S. V. Anulova, A. Yu. Veretennikov, N. V. Krylov, R. Sh. Liptser, and A. N. Shiryaev, “Stochastic calculus,” Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 45, 5 (1989) [Probability Theory III. Encycl. Math. Sci. 45, 1 (1998)].

    MATH  Google Scholar 

  3. 3

    Yu N. Bibikov, A Course in Ordinary Differential Equations (Vysshaya Shkola, Moscow, 1991) [in Russian].

    Google Scholar 

  4. 4

    A. V. Bulinskiĭ and A. N. Shiryaev, Theory of Stochastic Processes (Fizmatlit, Moscow, 2003) [in Russian].

    Google Scholar 

  5. 5

    M. Davis and G. Burstein, “A deterministic approach in stochastic optimal control with application to anticipative control,” Stochastics Stochastics Rep. 40, 203 (1992).

    MathSciNet  Article  Google Scholar 

  6. 6

    A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides (Nauka, Moscow, 1985; Kluwer Academic Publishers, Dordrecht, 1988).

    Google Scholar 

  7. 7

    M. L. Kleptsina, “On strong solutions of stochastic equations with degenerate coefficients,” Teor. Veroyatn. Primen. 29, 392 (1984) [Theory Probab. Appl. 29, 403 (1985)].

    Article  Google Scholar 

  8. 8

    M. A. Krasnosel’skiĭ and A. V. Pokrovskiĭ, Systems with Hysteresis (Nauka, Moscow, 1983; Springer-Verlag, Berlin, 1989).

    MATH  Google Scholar 

  9. 9

    F. S. Nasyrov, “Symmetric integrals and stochastic analysis,” Teor. Veroyatn. Primen. 51, 496 (2007) [Theory Probab. Appl. 51, 486 (2007)].

    MathSciNet  Article  Google Scholar 

  10. 10

    F. S. Nasyrov, Local Times, Symmetric Integrals, and Stochastic Analysis (Fizmatlit, Moscow, 2011) [in Russian].

    Google Scholar 

  11. 11

    A. N. Shiryaev, Fundamentals of the Stochastic Financial Mathematics (Fazis, Moscow, 1998) [Essentials of Stochastic Finance (World Scientific, Singapore, 1999)].

  12. 12

    A. Yu. Veretennikov, “On the strong solutions of stochastic differential equations,” Teor. Veroyatn. Primen. 24, 348 (1979). [Theory Probab. Appl. 24, 354 (1980)].

    Article  Google Scholar 

  13. 13

    A. K. Zvonkin and N. V Krylov, “On strong solutions of stochastic differential equations,” in: Tr. Shk.-Semin. Teor. Sluch. Prots., Druskininkaj, Pt. 2, 9 (Inst. Fiz. Mat. Akad. nauk LitSSR, 1975) [Selecta Math. Sov. 1, 19 (1981)].

    MATH  Google Scholar 

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ACKNOWLEDGMENTS

The author is grateful to Professor Albert N. Shiryaev who drew author’s attention to problems connected with strong solutions of stochastic ODEs and SDEs.

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Correspondence to F. S. Nasyrov.

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Nasyrov, F.S. On Strong Solutions of Stochastic Differential Equations and Their Trajectory Analogs. Sib. Adv. Math. 31, 147–153 (2021). https://doi.org/10.1134/S1055134421020048

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Keywords

  • stochastic differential equation
  • strong solution
  • equation with symmetric integrals
  • ordinary differential equation with random right-hand side
  • Carathéodory solution
  • Filippov solution