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RESEARCH PRODUCT

Stochastic Differential Equations

Achim Klenke

subject

Stochastic partial differential equationExamples of differential equationsStochastic differential equationWeak solutionApplied mathematicsMartingale (probability theory)Malliavin calculusNumerical partial differential equationsIntegrating factorMathematics

description

Stochastic differential equations describe the time evolution of certain continuous n-dimensional Markov processes. In contrast with classical differential equations, in addition to the derivative of the function, there is a term that describes the random fluctuations that are coded as an Ito integral with respect to a Brownian motion. Depending on how seriously we take the concrete Brownian motion as the driving force of the noise, we speak of strong and weak solutions. In the first section, we develop the theory of strong solutions under Lipschitz conditions for the coefficients. In the second section, we develop the so-called (local) martingale problem as a method of establishing weak solutions. In the third section, we present some examples in which the method of duality can be used to prove weak uniqueness.

yearjournalcountryeditionlanguage
2020-01-01
https://doi.org/10.1007/978-1-4471-5361-0_26