Abstract
This chapter is devoted to stochastic differential equations, which motivated Itô’s construction of stochastic integrals. After giving the general definitions, we provide a detailed treatment of the Lipschitz case, where strong existence and uniqueness statements hold. Still in the Lipschitz case, we show that the solution of a stochastic differential equation is a Markov process with a Feller semigroup, whose generator is a second-order differential operator. By results of Chap. 6, the Feller property immediately gives the strong Markov property of solutions of stochastic differential equations. The last section presents a few important examples.
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Le Gall, JF. (2016). Stochastic Differential Equations. In: Brownian Motion, Martingales, and Stochastic Calculus . Graduate Texts in Mathematics, vol 274. Springer, Cham. https://doi.org/10.1007/978-3-319-31089-3_8
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DOI: https://doi.org/10.1007/978-3-319-31089-3_8
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