The chapter presents stochastic differential equations (SDEs) and their connections with diffusion processes and partial differential equations (PDEs). The existence and uniqueness of solutions of SDEs are proved under Lipschitz’s conditions. Two important processes geometric Brownian motion (GBM) and the Ornstein-Uhlenbeck process are constructed on this theoretical base. The difference between ordinary differential equations and SDEs is discussed. As a part of this discussion, the existence of a solution (weak solution) of any SDE with measurable bounded drift coefficient and unit diffusion is proved with the help of the miracle Girsanov theorem. Moreover, it is shown by mean of the method of monotonic approximations that such a solution will be strong if the grift coefficient is a bounded piece-wise smooth function. Diffusion processes are defined as Markov processes for which their transition densities satisfy the asymptotic properties of Kolmogorov. The backward and forward equations of Kolmogorov are derived. A connection between SDEs and PDEs are stated with the help of the Feynman-Kac theorem. Absolute continuity of distributions of diffusion processes is studied with the help of the Girsanov theorem. A special attention is paid to the class of controlled diffusion processes for which the Hamilton-Jacobi-Bellman optimality equation is derived. It is shown how the theory of diffusion processes and SDEs are helpful in mathematical finance (Bachelier and Black-Scholes models) and in statistics of random processes (see Bishwal, Parameter estimation in stochastic differential equations, 2008; Borodin, Stochastic processes, 2018; Ikeda and Watanabe, Stochastic differential equations and diffusion processes, 2nd edn., 1989; Karatzas and Shreve, Brownian motion and stochastic calculus, 1998; Krylov, Introduction to the theory of random processes, 2002; Krylov, Controlled diffusion processes, 1980; Lamberton and Lapeyre, Introduction to stochastic calculus applied to finance, 1996; Le Gall, Brownian motion, martingales, and stochastic calculus, 2016; Liptser and Shiryaev, Statistics of random processes, 2nd edn., 2001; Melnikov, Russian Math Surveys 51(5):43–136, 1996; Øksendal, Stochastic differential equations, 5th edn., 2000; Shiryaev, Essentials of stochastic finance, 1999; Skorokhod, Lectures on the theory of stochastic processes, 1996; Tikhonov et al., Differential equations, 1985, and Wentzell, A course in the theory of stochastic processes, 1981).

Stochastic Differential Equations, Diffusion Processes, and Their Applications

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发表日期：

2026年

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Alexander Melnikov

摘要

The chapter presents stochastic differential equations (SDEs) and their connections with diffusion processes and partial differential equations (PDEs). The existence and uniqueness of solutions of SDEs are proved under Lipschitz’s conditions. Two important processes geometric Brownian motion (GBM) and the Ornstein-Uhlenbeck process are constructed on this theoretical base. The difference between ordinary differential equations and SDEs is discussed. As a part of this discussion, the existence of a solution (weak solution) of any SDE with measurable bounded drift coefficient and unit diffusion is proved with the help of the miracle Girsanov theorem. Moreover, it is shown by mean of the method of monotonic approximations that such a solution will be strong if the grift coefficient is a bounded piece-wise smooth function. Diffusion processes are defined as Markov processes for which their transition densities satisfy the asymptotic properties of Kolmogorov. The backward and forward equations of Kolmogorov are derived. A connection between SDEs and PDEs are stated with the help of the Feynman-Kac theorem. Absolute continuity of distributions of diffusion processes is studied with the help of the Girsanov theorem. A special attention is paid to the class of controlled diffusion processes for which the Hamilton-Jacobi-Bellman optimality equation is derived. It is shown how the theory of diffusion processes and SDEs are helpful in mathematical finance (Bachelier and Black-Scholes models) and in statistics of random processes (see Bishwal, Parameter estimation in stochastic differential equations, 2008; Borodin, Stochastic processes, 2018; Ikeda and Watanabe, Stochastic differential equations and diffusion processes, 2nd edn., 1989; Karatzas and Shreve, Brownian motion and stochastic calculus, 1998; Krylov, Introduction to the theory of random processes, 2002; Krylov, Controlled diffusion processes, 1980; Lamberton and Lapeyre, Introduction to stochastic calculus applied to finance, 1996; Le Gall, Brownian motion, martingales, and stochastic calculus, 2016; Liptser and Shiryaev, Statistics of random processes, 2nd edn., 2001; Melnikov, Russian Math Surveys 51(5):43–136, 1996; Øksendal, Stochastic differential equations, 5th edn., 2000; Shiryaev, Essentials of stochastic finance, 1999; Skorokhod, Lectures on the theory of stochastic processes, 1996; Tikhonov et al., Differential equations, 1985, and Wentzell, A course in the theory of stochastic processes, 1981).