Abstract We give an overview of the concept random time changes in evolution processes. First all, we discuss times Markov Secondly, propose to use for dynamical systems. In both cases did appear fractional equations. case processes arrive Kolmogorov For systems it leads Liouville

Conditional expected values in Markov chains are solutions to a set of associated backward differential equations, which may be ordinary or partial depending on the number of relevant state variables. This paper investigates the validity of these differential equations by locating the points of non-smoothness of the state-wise conditional expected values, and it presents a numerical method for ...

We present a sufficient condition for robust permanence of ecological (or Kolmogorov) differential equations based on average Liapunov functions. Via the minimax theorem we rederive Schreiber’s sufficient condition [S. Schreiber, J. Differential Equations, 162 (2000), pp. 400–426] in terms of Liapunov exponents and give various generalizations. Then we study robustness of permanence criteria ag...

We adapt the Levi’s parametrix method to prove existence, estimates and qualitative properties of a global fundamental solution to ultraparabolic partial differential equations of Kolmogorov type. Existence and uniqueness results for the Cauchy problem are also proved.

Recent results about linear partial differential equations of Kolmogorov type are reviewed. They are examined in the context of financial mathematics: specifically, applications to arbitrage valuation, model calibration and estimation of stochastic processes are discussed.

High order splitting schemes with complex timesteps are applied to Kolmogorov backward equations stemming from stochastic differential equations in Stratonovich form. In the setting of weighted spaces, the necessary analyticity of the split semigroups can be easily proved. A numerical example from interest rate theory, the CIR2 model, is considered. The numerical results are robust for drift-do...

We obtain Calderón-Zygmund estimates for some degenerate equations of Kolmogorov type with inhomogeneous nonlinear coefficients. We then derive the well-posedness of the martingale problem associated with related degenerate operators, and therefore uniqueness in law for the corresponding stochastic differential equations. Some density estimates are established as well.

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size (N) and can be written as u̇N = ANuN . Our results rely on the convergence of the transition matrices AN to an operator A...