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Review 54 Kent, Some probabilistic properties of Bessel functions, Ann. Review 58 T. Kurtz, Martingale problems for conditional distributions of Markov processes, Elect. Review 1 Review 50 Matsumoto and M. Paris, Serie I , McKean, Jr. Review 40 Pitman and M. Yor, Bessel processes and infinitely divisible laws, Stochastic Integrals, ed.

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Finally, we introduce continuous semimartingales and their quadratic variation processes. This chapter is at the core of the present book. We start by defining the stochastic integral with respect to a square-integrable continuous martingale, considering first the integral of elementary processes which play a role analogous to step functions in the theory of the Riemann integral and then using an isometry between Hilbert spaces to deal with the general case.

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It is easy to extend the definition of stochastic integrals to continuous local martingales and semimartingales. Our goal in this chapter is to give a concise introduction to the main ideas of the theory of continuous time Markov processes. Markov processes form a fundamental class of stochastic processes, with many applications in real life problems outside mathematics.

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The reason why Markov processes are so important comes from the so-called Markov property, which enables many explicit calculations that would be intractable for more general random processes. Although the theory of Markov processes is by no means the central topic of this book, it will play a major role in the next chapters, in particular in our discussion of stochastic differential equations.

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The present chapter is mostly independent of the previous ones, even though Brownian motion is used as a basic example, and martingale theory developed in Chap. After a section dealing with the general definitions and the problem of existence, we focus on the particular case of Feller processes, and in that framework we introduce the key notion of the generator. We establish regularity properties of Feller processes as consequences of the analogous results for supermartingales. We then discuss the strong Markov property, and we conclude the chapter by presenting three important classes of Markov processes.

In this chapter, we use the results of the preceding two chapters to discuss connections between Brownian motion and partial differential equations.

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In particular, we give the probabilistic solution of the classical Dirichlet problem in a bounded domain whose boundary satisfies the exterior cone condition. In the case where the domain is a ball, the solution is made explicit by the Poisson kernel, which corresponds to the density of the exit distribution of the ball for Brownian motion. We then discuss recurrence and transience of d -dimensional Brownian motion, and we establish the conformal invariance of planar Brownian motion as a simple corollary of the results of Chap. 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.