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Stochastic differential equations with fractal noise
Author(s) -
Zähle M.
Publication year - 2005
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200310295
Subject(s) - mathematics , sobolev space , stochastic calculus , stochastic differential equation , fractional calculus , malliavin calculus , quantum stochastic calculus , quadratic equation , quadratic variation , fractal , stochastic partial differential equation , stochastic process , norm (philosophy) , mathematical analysis , differential equation , calculus (dental) , brownian motion , geometry , quantum process , dentistry , quantum dynamics , quantum , political science , law , medicine , physics , statistics , quantum mechanics
Stochastic differential equations in ℝ n with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W β 2 for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in W α 2 for 0 < α < 1. Linear equations are considered as a special case. This approach leads to fast computer algorithms basing on Picard's iteration method. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)