Convergence of delay differential equations driven by fractional Brownian motion
Tóm tắt
In this note, we prove an existence and uniqueness result of solution for stochastic differential delay equations with hereditary drift driven by a fractional Brownian motion with Hurst parameter H > 1/2. Then, we show that, when the delay goes to zero, the solutions to these equations converge, almost surely and in L
p
, to the solution for the equation without delay. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann–Stieltjes integral.
Tài liệu tham khảo
X. Fernique. Regularité des trajectoires des fonctions aléatoires gaussienes. In: École d’Été de Probabilités de Saint Flour, IV-1974. Lecture Notes in Math 480, 1–96 (1975).
Ferrante M., Rovira C.: Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H > 1/2. Bernoulli 12, 85–100 (2006)
J. León and S. Tindel: Malliavin calculus for fractional delay equations. ArXiv:0912.2180.
Lyons T.: Differential equations driven by rough signals (I): An extension of an inequality of L. C. Young. Mathematical Research Letters 1, 451–464 (1994)
S.-E. A. Mohammed: Stochastic differential systems with memory: theory, examples and applications. In: Stochastic Analysis and Related Topics VI (L. Decreusefond, J. Gjerde, B. Oksendal and A.S. Üstünel, eds), Birkhäuser, Boston, 1–77 (1998).
Neuenkirch A., Nourdin I., Tindel S.: Delay equations driven by rough paths. Electron. J. Probab. 13, 2031–2068 (2008)
Nualart D., Rascanu A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53, 55–81 (2002)
S.G. Samko, A.A. Kilbas and O. Marichev: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach (1993).
Young L.C.: An inequality of the Hölder type connected with Stieltjes integration. Acta Math. 67, 251–282 (1936)
Zähle M.: Integration with respect to fractal functions and stochastic calculus. I. Prob. Theory Relat. Fields 111, 333–374 (1998)