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Functional Solutions of Stochastic Differential Equations.
- Source :
-
Mathematics (2227-7390) . Apr2024, Vol. 12 Issue 8, p1258. 17p. - Publication Year :
- 2024
-
Abstract
- We present an integration condition ensuring that a stochastic differential equation d X t = μ (t , X t) d t + σ (t , X t) d B t , where μ and σ are sufficiently regular, has a solution of the form X t = Z (t , B t) . By generalizing the integration condition we obtain a class of stochastic differential equations that again have a functional solution, now of the form X t = Z (t , Y t) , with Y t an Ito process. These integration conditions, which seem to be new, provide an a priori test for the existence of functional solutions. Then path-independence holds for the trajectories of the process. By Green's Theorem, it holds also when integrating along any piece-wise differentiable path in the plane. To determine Z at any point (t , x) , we may start at the initial condition and follow a path that is first horizontal and then vertical. Then the value of Z can be determined by successively solving two ordinary differential equations. Due to a Lipschitz condition, this value is unique. The differential equations relate to an earlier path-dependent approach by H. Doss, which enables the expression of a stochastic integral in terms of a differential process. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 22277390
- Volume :
- 12
- Issue :
- 8
- Database :
- Academic Search Index
- Journal :
- Mathematics (2227-7390)
- Publication Type :
- Academic Journal
- Accession number :
- 176878993
- Full Text :
- https://doi.org/10.3390/math12081258