×

Stochastic differential equations with fractal noise. (English) Zbl 1075.60075

A stochastic differential equation in \(\mathbb{R}^n\), \[ dX(t)=\sum_{j=0}^m a_j(X(t),t)dZ^j(t)+b(X(t),t)\,dt,\quad X(t_0)=X_0, \] is considered where \(Z^0\) is a continuous process with generalised bracket and \(Z^1,\dots, Z^m\) are processes with sample paths in a fractional Sobolev space \(W_2^\beta\) for some \(\beta>1/2\). Stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed. It combines the stochastic Itô calculus with fractional norm estimates of associated integral operators in \(W^\alpha_2\) for \(\alpha\in (0,1)\). Linear equations are considered as a special case. This approach leads to fast computer algorithms based on Picard’s iteration method.
See the author [Math. Nachr. 225, 145–183 (2001; Zbl 0983.60054)] for the case \(m=0\) and L. Coutin and L. Decreusefond [Ann. Appl. Probab. 9, No. 4, 1058–1090 (1999; Zbl 0956.60058)] for the case of Gaussian driving processes.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
34F05 Ordinary differential equations and systems with randomness
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Bertoin, Sur une intégrale pour les processus à {\(\alpha\)} variation bornée, Ann. Probab. 17 pp 1521– (1989)
[2] Coutin, Abstract non linear filtering theory in the presence of fractional Brownian motion, Ann. Appl. Probab. 9 pp 1058– (2000)
[3] R. M. Dudley R. Norvaiša An introduction to p -variation and Young integrals, Techn. Report 1 (MaPhySto, University of Aarhus, 1998).
[4] Feyel, Fractional integrals and Brownian processes, Potential Anal. 10 pp 273– (1989) · Zbl 0944.60045
[5] H. Föllmer Calcul d’Itô sans Probabilité, Séminaire de Probabilités XV, Lecture Notes in Mathematics Vol. 850 (Springer-Verlag, Berlin - Heidelberg - New York, 1989), pp. 143-150.
[6] Russo, Forward, backward and symmetric stochastic integration, Probab. Theory Related Fields 97 pp 403– (1993) · Zbl 0792.60046
[7] Russo, The generalized covariation process and Itô formula, Stoch. Process. Appl. 59 pp 81– (1995) · Zbl 0840.60052
[8] Russo, Stochastic calculus with respect to continuous finite quadratic variation processes, Stoch. Stoch. Rep. 70 pp 1– (2000) · Zbl 0981.60053
[9] S. G. Samko A. A. Kilbas O. I. Marichev Fractional Integrals and Derivatives. Theory and Applications (Gordon and Breach, 1993).
[10] A. N. Shiryaev Essentials of Stochastic Finance (World Scientific 1998).
[11] Young, An inequality of Hölder type, connected with Stieltjes integration, Acta Math. 67 pp 251– (1936)
[12] Zähle, Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 pp 333– (1998) · Zbl 0918.60037
[13] Zähle, Integration with respect to fractal functions and stochastic calculus II, Math. Nachr. 225 pp 145– (2001) · Zbl 0983.60054
[14] M. Zähle Forward integrals and stochastic differential equations, in: Seminar on Stochastic Analysis, Random Fields and Applications, edited by R. C. Dalang, M. Dozzi, and F. Russo, Progress in Probability Vol. 52 (Birkhäuser Verlag, 2002), pp. 293-302.
[15] Zähle, Long range dependence, no arbitrage and the Black-Scholes formula, Stoch. Dyn. 2 pp 265– (2002) · Zbl 1016.91053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.