| Title | NEW STOCHASTIC INTEGRALS, OSCILLATION THEOREMS AND ENERGY IDENTITIES |
| Publication Type | Journal Article |
| Year of Publication | 2009 |
| Authors | SCHURZ, HENRI |
| Secondary Title | Communications in Applied Analysis |
| Volume | 13 |
| Issue | 2 |
| Start Page | 181 |
| Pagination | 194 |
| Date Published | 04/2009 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 34F05, 37H10, 60H10, 65C30. |
| Abstract | This paper is divided into three parts on diverse aspects of stochastic analysis, namely
1) newly defined stochastic integrals such as the stochastic Simpson and stochastic quadrature integrals and their relation to the recently introduced stochastic α-integral by the author (DSA, Vol. 15 (2), 2006), 2) oscillation theorems for second order stochastic differential equations (SDEs) which show the almost sure oscillation property of all linear undamped oscillators perturbed by additive, non-degenerate martingale-type noise for all measurable random initial data (this generalizes results from X. Mao (1997), Markus and Weerasinghe (1988)), 3) expected energy formulas for linear stochastic oscillators with additive noise under adequate discretization by midpoint-type methods (the latter generalizes independent results from Hong, Scherer and Wang (NPSC, Vol. 14 (1), 2006) and the author (2004 for the beam problem, 2005 for stochastic wave equation)). Energy-exact stochastic-numerical methods (called improved midpoint methods) for linear second order SDEs are constructed and verified along non-equidistant partitions. These results can be applied to quadrature methods such as Newton-Cotes formulas for stochastic integrals, to analysis of the oscillatory and energy behavior of stochastically perturbed Schr ̈odinger equations, stochastic oscillators, beam models and stochastic wave equations for randomly vibrating strings. |
| URL | http://www.acadsol.eu/en/articles/13/2/3.pdf |
| Short Title | NEW STOCHASTIC INTEGRALS, OSCILLATIONS AND ENERGY |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] E. Allen, Modeling with Ito stochastic differential equations, Springer, New York, 2007.
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