On solutions of Kolmogorovʼs equations for nonhomogeneous jump Markov processes

Abstract

This paper studies three ways to construct a nonhomogeneous jump Markov process: (i) via a compensator of the random measure of a multivariate point process, (ii) as a minimal solution of the backward Kolmogorov equation, and (iii) as a minimal solution of the forward Kolmogorov equation. The main conclusion of this paper is that, for a given measurable transition intensity, commonly called a Q-function, all these constructions define the same transition function. If this transition function is regular, that is, the probability of accumulation of jumps is zero, then this transition function is the unique solution of the backward and forward Kolmogorov equations. For continuous Q-functions, Kolmogorov equations were studied in Feller's seminal paper. In particular, this paper extends Feller's results for continuous Q-functions to measurable Q-functions and provides additional results.

Average Optimality for Continuous-Time Markov Decision Processes Under Weak Continuity Conditions

“…Now, as in the argument for (8) with z being replaced by x in the beginning of this proof, we see lim sup 0<α↓0 αW α (x) ≤ W (x, π) < ∞, which together with the arbitrariness of the policy π and the fact that g = lim sup 0<α↓0 α inf x∈S W α (x) ≤ lim sup 0<α↓0 αW α (x) (recalling here the definition of g given by (11)), leads to inf π∈ W (x, π) ≥ g. Thus, we see the validity of (19) with inequalities being replaced by equalities. Since g < ∞, we see that inf π ∈ W (x, π) < ∞.…”

“…According to [11], under each Markov policy π, the process ξ t is a Markov jump process in the sense of [12] with respect to ( , F , {F t } t≥0 , P π x ) for each x ∈ S. We are also interested in policies in more specific forms. , x n , θ n+1 ).…”

“…Under each Markov policy π ∈ M , the process ξ t is a Markov jump process [11], and there exists a transition (subprobability, in general) function p π (u, x, t, dy) such that P π (ξ t ∈ dy | ξ u ) = p π (u, ξ u , t, dy) with t ≥ u ≥ 0 almost surely with respect to P π [29]. Under each Markov policy π ∈ M , the process ξ t is a Markov jump process [11], and there exists a transition (subprobability, in general) function p π (u, x, t, dy) such that P π (ξ t ∈ dy | ξ u ) = p π (u, ξ u , t, dy) with t ≥ u ≥ 0 almost surely with respect to P π [29].…”

“…Then, by the definition of the constant g (see (11)), there exists 0 < α 1 ≤ α 0 such that, for each α ∈ (0, α 1 ], g ≥ g α − ε. Let ε > 0 be arbitrarily fixed.…”

Strategies on Initial Public Offering of Company Equity at Stock Exchanges in Imperfect Highly Volatile Global Capital Markets with Induced Nonlinearities

“…The multiple evidences of the fact that the diffusion processes have the considerable influences on the various econophysical and econometrical parameters of the diffusion-type financial systems have been described in Bachelier (1900), Volterra (1906), Slutsky (1910Slutsky ( , 1912Slutsky ( , 1913Slutsky ( , 1914Slutsky ( , 1915Slutsky ( , 1922aSlutsky ( , b, 1923aSlutsky ( , b c, 1925aSlutsky ( , b, 1926Slutsky ( , 1927aSlutsky ( , b, 1929Slutsky ( , 1935Slutsky ( , 1937aSlutsky ( , b, 1942, Osborne (1959), Alexander (1961), Shiryaev (1961Shiryaev ( , 1963Shiryaev ( , 1964Shiryaev ( , 1965Shiryaev ( , 1967Shiryaev ( , 1978Shiryaev ( , 1998aShiryaev ( , b, 2002Shiryaev ( , 2008aShiryaev ( , b, 2010, Grigelionis, Shiryaev (1966), Graversen, Peskir, Shiryaev (2001), Kallsen, Shiryaev (2001, Jacod, Shiryaev (2003), Peskir, Shiryaev (2006), Feinberg, Shiryaev (2006), du Toit, Peskir, Shiryaev (2007), Eberlein, Papapantoleon, Shiryaev (2008), Shiryaev, Zryumov (2009), Shiryaev, Novikov (2009), Gapeev, Shiryaev (2010), Karatzas, Shiryaev, Shkolnikov (2011), , , Feinberg, Mandava, Shiryaev (2013), Akerlof, Stiglitz (1966), …”