Diffusions, Jump-Diffusions and Heat Equations
We study diffusions and jump-diffusions on a Euclidean space determined by SDE studied in Chap. 3. We select topics which are related to the stochastic flow generated by the SDE; topics are concerned with heat equations and backward heat equations.
In Sects. 4.1, 4.2, 4.3, and 4.4, we consider diffusion processes on a Euclidean space. In Sect. 4.1, we show that a stochastic flow generated by a continuous symmetric SDE defines a diffusion process. Its generator A(t) is represented explicitly as a second order linear partial differential operator with time parameter t, using coefficients of the SDE. Kolmogorov’s forward and backward equation associated with the operator A(t) will be derived. In Sect. 4.2, we discuss exponential transformation of the diffusion process by potentials. It will be shown that solutions of various types of backward heat equations will be obtained by exponential transformations by potentials. In Sect. 4.3, we study backward SDE and backward diffusions. We will see that Kolmogorov equations for backward diffusion will give the solution of heat equations. In Sect. 4.4, we present a new method of constructing the dual (adjoint) semigroup, making use of the geometric property of diffeomorphic maps Φs,t of stochastic flows. The method will present a clear geometric explanation of the adjoint operator A(t)∗ and the dual semigroup. We will see that the dual semigroup will be obtained by a certain exponential transformation (Feynman–Kac–Girsanov transformation) of a backward diffusion.
In Sects. 4.5 and 4.6, we consider jump-diffusions on a Euclidean space. We will extend results for diffusion studied in Sects. 4.2, 4.3, and 4.4 to those for jump-diffusions. We show that the dual semigroup of jump-diffusion is well defined if the jump coefficients are diffeomorphic. In Sect. 4.7, we return to a problem of the stochastic flow. We discuss the volume-preserving property of stochastic flows by applying properties of dual jump-diffusions. In Sect. 4.8, we consider processes killed outside of a subdomain of an Euclidean space and its dual processes.