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Abstrakty
We study the existence, uniqueness and approximation of solutions of stochastic differential equations with constraints driven by processes with bounded p-variation. Our main tool are new estimates showing Lipschitz continuity of the deterministic Skorokhod problem in p-variation norm. Applications to fractional SDEs with constraints are given.
Czasopismo
Rocznik
Tom
Strony
343--365
Opis fizyczny
Bibliogr. 34 poz.
Twórcy
autor
- Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
autor
- Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
Bibliografia
- [1] S. Asmussen, Queueing simulation in heavy traffic, Math. Oper. Res. 17 (1992), pp. 84-111.
- [2] M. Besalu and C. Rovira, Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion, Bernoulli 18 (1) (2012), pp. 24-45.
- [3] R. J. Chitashvili and N. L. Lazrieva, Strong solutions of stochastic differential equations with boundary conditions, Stochastics 5 (1981), pp. 225-309.
- [4] F. Coquet, J. Mémin, and L. Słomiński, On non-continuous Dirichlet processes, J. Theoret. Probab. 16 (2003), pp. 197-216.
- [5] F. Coquet and L. Słomiński, On the convergence of Dirichlet processes, Bernoulli 5 (1999), pp. 615-639.
- [6] R. M. Dudley, Picard iteration and p-variation: The work of Lyons (1994), in: Miniproceedings: Workshop on Product Integrals and Pathwise Integration, MaPhySto 1999.
- [7] R. M. Dudley and R. Norvaiša, An Introduction to p-variation and Young Integrals, Lecture Note No. 1, Aarhus University, 1999.
- [8] R. M. Dudley and R. Norvaiša, Concrete Functional Calculus, Springer Science+Business Media 2011.
- [9] P. Dupuis and H. Ishii, On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications, Stoch. Stoch. Rep. 35 (1991), pp. 31-62.
- [10] P. Dupuis and K. Ramanan, A multiclass feedback queueing network with a regular Skorokhod problem, Queueing Syst. 36 (2000), pp. 327-349.
- [11] S. Ethier and T. Kurtz, Markov Processes. Characterization and Convergence, Wiley, New York 1986.
- [12] A. Falkowski, L. Słomiński, and B. Ziemkiewicz, On weak and strong discrete-time approximations of fractional SDEs, Lithuanian Math. J. 54 (2014), pp. 409-428.
- [13] M. Ferrante and C. Rovira, Stochastic differential equations with non-negativity constraints driven by fractional Brownian motion, J. Evol. Equ. 13 (2013), pp. 617-632.
- [14] J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, Springer, Berlin 1987.
- [15] A. Jakubowski, J. Mémin, and G. Pagès, Convergence en loi des suites d’intégrales stochastiques sur l’espace D1 de Skorokhod, Probab. Theory Related Fields 81 (1989), pp. 111-137.
- [16] P. Krée and C. Soize, Mathematics and Random Phenomena: Random Vibrations of Mechanical Structures, Reidel, Dordrecht 1986.
- [17] K. Kubilius, The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type, Stochastic Process. Appl. 98 (2002), pp. 289-315.
- [18] K. Kubilius, On weak and strong solutions of an integral equation driven by a continuous p-semimartingale, Lithuanian Math. J. 43 (2003), pp. 34-50.
- [19] P. L. Lions and A. S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1983), pp. 511-537.
- [20] T. J. Lyons, Differential equations driven by rough signals (I): An extension of an inequality of L. C. Young, Math. Res. Lett. 1 (1994), pp. 451-464.
- [21] J. Mémin and L. Słomiński, Condition UT et stabilité en loi des solutions d’équations différentielles stochastiques, in: Séminaire de Probabilités XXV, Lecture Notes in Math., Vol. 1485, Springer, 1991, pp. 162-177.
- [22] D. Nualart and A. Răşcanu, Differential equations driven by fractional Brownian motion, Collect. Math. 53 (2002), pp. 55-81.
- [23] A. A. Ruzmaikina, Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion, J. Statist. Phys. 100 (2000), pp. 1049-1069.
- [24] M. A. Shashiashvili, On the variation of the difference of singular components in the Skorokhod problem and on stochastic differential systems in a half-space, Stochastics 24 (1988), pp. 151-169.
- [25] L. A. Shepp and A. N. Shiryaev, A new look at pricing of the “Russian option”, Theory Probab. Appl. 39 (1994), pp. 103-119.
- [26] A. V. Skorokhod, Stochastic equations for diffusion processes in a bounded region 1,2, Theory Probab. Appl. 6 (1961), pp. 264-274; 7 (1962), pp. 3-23.
- [27] L. Słomiński, Stability of strong solutions of stochastic differential equations, Stochastic Process. Appl. 31 (1989), pp. 173-202.
- [28] L. Słomiński, On existence, uniqueness and stability of solutions of multidimensional SDE’s with reflecting boundary conditions, Ann. Inst. H. Poincaré 29 (1993), pp. 163-198.
- [29] L. Słomiński, Stability of stochastic differential equations driven by general semimartingales, Dissertationes Math. 349 (1996), pp. 1-113.
- [30] L. Słomiński, Euler’s approximations of solutions of SDEs with reflecting boundary, Stochastic Process. Appl. 94 (2001), pp. 317-337.
- [31] L. Słomiński and T. Wojciechowski, Stochastic differential equations with time-dependent reflecting barriers, Stochastics 85 (2013), pp. 27-47.
- [32] L. Słomiński and B. Ziemkiewicz, On weak approximations of integrals with respect to fractional Brownian motion, Statist. Probab. Lett. 79 (2009), pp. 543-552.
- [33] T. Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979), pp. 163-177.
- [34] L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936), pp. 251-282.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
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Bibliografia
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