Probability Theory pp 567-590 | Cite as

# Stochastic Differential Equations

Stochastic differential equations describe the time evolution of certain continuous Markov processes with values in ℝ^{ n }. 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 Itô 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.

## Keywords

Brownian Motion Weak Solution Stochastic Differential Equation Strong Solution Local Martingale## Preview

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