Continuous-Space Markov Processes with Jumps
From now on, Markov processes with continuous state space (\(\mathbb{R}^{d}\) for some or one of its closed subsets) are considered. Their rigorous study requires advanced measure-theoretic tools, but we limit ourselves to developing the reader’s intuitio
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Continuous-Space Markov Processes with Jumps
Abstract From now on, Markov processes with continuous state space (Rd for some or one of its closed subsets) are considered. Their rigorous study requires advanced measure-theoretic tools, but we limit ourselves to developing the reader’s intuition, notably by pathwise constructions leading to simulations. We first emphasize the strong similarity between such Markov processes with constant trajectories between isolated jumps and discrete space ones. We then introduce Markov processes with sample paths following an ordinary differential equation between isolated jumps. In both cases, the Kolmogorov equations and Feynman–Kac formula are established. This is applied to kinetic equations coming from statistical Mechanics. These describe the time evolution of the instantaneous distribution of particles in phase space (position-velocity), when the particle velocity jumps at random instants in function of the particle position and velocity.
6.1 Preliminaries The construction of continuous-space Markov processes requires the use of measure and integration theory, and it is always assumed implicitly that functions and subsets are measurable w.r.t. the Borel σ -field B(V ) of V . Intrinsic notations will be used for integrals and adjoint operators, as well as more intuitive notations which establish links with the discrete space theory and allow to quickly write formulas; there will be some redundancy.
6.1.1 Measures, Functions, and Transition Kernels Positive and Signed Measures, Integrals of Functions A positive measure μ on V is a σ -additive function from B(V ) to [0, ∞]. It is finite if its total mass μ(V ) is finite, and a probability measure, or a law, if μ(V ) = 1. The space of probability measures is denoted by P = P(V ). A signed measure μ is a σ -additive function from B(V ) to R. It can be written as the difference μ = μ+ − μ− of two positive finite measures which are mutually C. Graham, D. Talay, Stochastic Simulation and Monte Carlo Methods, Stochastic Modelling and Applied Probability 68, DOI 10.1007/978-3-642-39363-1_6, © Springer-Verlag Berlin Heidelberg 2013
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Continuous-Space Markov Processes with Jumps
singular, i.e., such that μ+ (A)μ− (A) = 0 for every A ∈ B(V ). The vector space of signed measures is denoted by M = M (V ). A positive measure belongs to M if and only if it is finite. The Lebesgue integral by a measure μ of a function f is defined in [0, ∞] if μ and f are positive (wide-sense), and in R if μ belongs to M and f to the space of bounded functions L∞ = L∞ (V ). It provides a natural duality bracket (μ, f ) → μ, f := f dμ = f (x)μ(dx) = μ(dx)f (x) (6.1) and all these notations will be used. The notations μ(dx)f (x) and μ, f are close to the discrete state space matrix notation, and allow to recover some formulas in an easier way. Transition Kernels and Their Actions A transition or Markovian kernel P is a measurable mapping x ∈ V → P (x, ·) = P (x, dy) ∈ P(V ). A kernel of positive measures or of signed measures is
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