Strong Markov Process Research Articles

Diffusion spiders: Green kernel, excessive functions and optimal stopping

A diffusion spider is a strong Markov process with continuous paths taking values on a graph with one vertex and a finite number of edges (of infinite length). An example is Walsh’s Brownian spider where the process on each edge behaves as a Brownian motion. In this paper we calculate firstly the density of the resolvent kernel in terms of the characteristics of the underlying diffusion. Excessive functions are studied via the Martin boundary theory. A crucial result is an expression for the representing measure of a given excessive function. These results are used to solve optimal stopping problems for diffusion spiders.

Some further classes of pseudo-differential operators in the p-adic context and their applications.

The purpose of this article is to study new non-Archimedean pseudo-differential operators whose symbols are determined from the behavior of two functions defined on the p-adic numbers. Thanks to the characteristics of our symbols, we can find connections between these operators and new types of non-homogeneous differential equations, Feller semigroups, contraction semigroups and strong Markov processes.

New classes of p-adic evolution equations and their applications

In this article, we study new classes of evolution equations in the p-adic context. We establish rigorously that the fundamental solutions of the homogeneous Cauchy problem, naturally associated to these equations, are transition density functions of some strong Markov processes {mathfrak {X}} with state space the n-dimensional p-adic unit ball ({mathbb {Z}}_{p}^{n}). We introduce a family of operators {T_{t}}_{tge 0} (obtained explicitly) that determine a Feller semigroup on C_{0}({mathbb {Z}}_{p}^{n}). Also, we study the asymptotic behavior of the survival probability of a strong Markov processes {mathfrak {X}} on a ball B_{-m}^{n}subset {mathbb {Z}}_{p}^{n}, min {mathbb {N}}. Moreover, we study the inhomogeneous Cauchy problem and we will show that its mild solution is associated with the mentioned above Feller semigroup.

Maximal inequalities and some applications

A maximal inequality is an inequality which involves the (absolute) supremum sups⩽t|Xs| or the running maximum sups⩽tXs of a stochastic process (Xt)t⩾0. We discuss maximal inequalities for several classes of stochastic processes with values in an Euclidean space: Martingales, Lévy processes, Lévy-type – including Feller processes, (compound) pseudo Poisson processes, stable-like processes and solutions to SDEs driven by a Lévy process –, strong Markov processes and Gaussian processes. Using the Burkholder–Davis–Gundy inequalities we also discuss some relations between maximal estimates in probability and the Hardy–Littlewood maximal functions from analysis.

On the Feller–Dynkin and the martingale property of one-dimensional diffusions

We show that a one-dimensional regular continuous strong Markov process \(X\) with scale function \(s\) is a Feller-Dynkin process precisely if the space transformed process \(s (X)\) is a martingale when stopped at the boundaries of its state space. As a consequence, the Feller-Dynkin and the martingale property are equivalent for regular diffusions on natural scale with open state space. Furthermore, for Ito diffusions we discuss relations to existence and uniqueness properties of Cauchy problems, and we identify the infinitesimal generator.

The Brownian Castle

AbstractWe introduce a 1+1‐dimensional temperature‐dependent model such that the classical ballistic deposition model is recovered as its zero‐temperature limit. Its ∞‐temperature version, which we refer to as the 0‐Ballistic Deposition (0‐BD) model, is a randomly evolving interface which, surprisingly enough, does not belong to either the Edwards–Wilkinson (EW) or the Kardar–Parisi–Zhang (KPZ) universality class. We show that 0‐BD has a scaling limit, a new stochastic process that we call Brownian Castle (BC) which, although it is “free”, is distinct from EW and, like any other renormalisation fixed point, is scale‐invariant, in this case under the 1:1:2 scaling (as opposed to 1:2:3 for KPZ and 1:2:4 for EW). In the present article, we not only derive its finite‐dimensional distributions, but also provide a “global” construction of the Brownian Castle which has the advantage of highlighting the fact that it admits backward characteristics given by the (backward) Brownian Web (see [37, 16]). Among others, this characterisation enables us to establish fine pathwise properties of BC and to relate these to special points of the Web. We prove that the Brownian Castle is a (strong) Markov and Feller process on a suitable space of càdlàg functions and determine its long‐time behaviour. Finally, we give a glimpse to its universality by proving the convergence of 0‐BD to BC in a rather strong sense. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

On a skew stable Lévy process

The skew Brownian motion is a strong Markov process which behaves like a Brownian motion until hitting zero and exhibits an asymmetry at zero. We address the following question: what is a natural counterpart of the skew Brownian motion in the situation that an underlying Brownian motion is replaced with a stable Lévy process with finite mean and infinite variance. We define a skew stable Lévy process X as a limit of a sequence of stable Lévy processes which are perturbed at zero. We derive a formula for the resolvent of X and show that X is a solution to a stochastic differential equation with a local time. Also, we provide a representation of X in terms of Itô‘s excursion theory.

On transition density functions of skew Brownian motions with two-valued drift

In this article, based on the results in Gairat and Shcherbakov (2017), we derive two-sided bounds for the full family of transition density functions of skew Brownian motion (SBM in abbreviation) with two-valued drift for all t>0. As an important step of this result, it is also shown in this paper that SBM with two-valued drift is a strong Markov process by finding its symmetrizing measure and canonical scale function, from which one can tell what values of the drift make such a process transient or recurrent.

On Diffusion Processes with Drift in Ld+ 1

This paper is a natural continuation of Krylov (2020 and 2021) where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(\mathbb {R}^{d+1})$ and some properties of their Green’s functions and probability of passing through narrow tubes are investigated. On the basis of this here we study some further properties of these processes such as Harnack inequality, Hölder continuity of potentials, Fanghua Lin estimates and so on.

Operators and Boundary Problems in Finance, Economics and Insurance: Peculiarities, Efficient Methods and Outstanding Problems

The price V of a contingent claim in finance, insurance and economics is defined as an expectation of a stochastic expression. If the underlying uncertainty is modeled as a strong Markov process X, the Feynman–Kac theorem suggests that V is the unique solution of a boundary problem for a parabolic equation. In the case of PDO with constant symbols, simple probabilistic tools explained in this paper can be used to explicitly calculate expectations under very weak conditions on the process and study the regularity of the solution. Assuming that the Feynman–Kac theorem holds, and a more general boundary problem can be localized, the local results can be used to study the existence and regularity of solutions, and derive efficient numerical methods. In the paper, difficulties for the realization of this program are analyzed, several outstanding problems are listed, and several closely efficient methods are outlined.

On some generalizations of non-archimedean pseudo-differential operators and their applications

In this article we define and study various generalizations non-archimedean pseudo-differential operators and its connections with probability theory. We provide explicit and sufficient conditions to ensure that these operators can satisfy the positive maximum principle, the heat Kernel or fundamental solution of the Cauchy problem (or heat equation) naturally associated with these operators determine a Feller semigroup and a transition function of some strong Markov processes X with state space the p-adic numbers. Also, we will study the property of Conservation of mass, the Comparison principle and some aspects of these processes including the first passage time problem and the probability of survival. These equations are proposed as mathematical models (connected with energy landscapes) to study the spread of an infectious or contagious disease that takes into account social clusters in a situation of extreme social isolation.

Some properties of solutions of Itô equations with drift in [formula omitted

This paper is a natural continuation of Krylov (2020), where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in Ld+1(Rd+1). Here we study some properties of these processes such as higher summability of Green’s functions, a maximum principle for related transport equations with irregular coefficients, boundedness of resolvent operators in Lebesgue spaces, establish Itô’s formula, and so on.

Fokker–Planck equation for Feynman–Kac transform of anomalous processes

In this paper, we develop a novel and rigorous approach to the Fokker–Planck equation, or Kolmogorov forward equation, for the Feynman–Kac transform of non-Markov anomalous processes. The equation describes the evolution of the density of the anomalous process Yt=XEt under the influence of potentials, where X is a strong Markov process on a Lusin space X that is in weak duality with another strong Markov process X̂ on X and {Et,t≥0} is the inverse of a driftless subordinator S that is independent of X and has infinite Lévy measure. We derive a probabilistic representation of the density of the anomalous process under the Feynman–Kac transform by the dual Feynman–Kac transform in terms of the weak dual process X̂t and the inverse subordinator {Et;t≥0}. We then establish the regularity of the density function, and show that it is the unique mild solution as well as the unique weak solution of a non-local Fokker–Planck equation that involves the dual generator of X and the potential measure of the subordinator S. During the course of the study, we are naturally led to extend the notation of Riemann–Liouville integral to measures that are locally finite on [0,∞).

On potentials of Itô’s Processes with Drift in Ld+ 1

This paper is a natural continuation of Krylov (2020), where strong Markov processes are constructed in the time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in \(L_{d+1}(\mathbb {R}^{d+1})\). Here we study some properties of these processes such as the probability to pass through narrow tubes, higher summability of Green’s functions, and so on. The results seem to be new even if the diffusion is constant.

Properties of the EMCEL scheme for approximating irregular diffusions

We prove several properties of the EMCEL scheme, which is capable of approximating one-dimensional continuous strong Markov processes in distribution on the path space (the scheme is briefly recalled). Special cases include irregular stochastic differential equations and processes with sticky features. In particular, we highlight differences from the Euler scheme in the case of stochastic differential equations and discuss a certain “stabilizing” behavior of the EMCEL scheme like “smoothing and tempered growth behavior”.

Probability density functions and the dynamics of complex systems associated to some classes of non-archimedean pseudo-differential operators

In this article, we study certain p-adic master equations which describe the dynamics of a large class of complex systems such as glasses, macromolecules and proteins. These equations are naturally associated to certain non-archimedean pseudo-differential operators whose symbols are connected via Fourier transform with radial probability density functions defined on the p-adic numbers. We show that the fundamental solutions of these equations are probability measures and determine a convolution semigroup on the p-adic numbers. Also, we show that the classical solution of this equations preserves the mass and satisfies the comparison principle. Moreover, we study some strong Markov processes corresponding to radial probability density functions of linear and logarithmic type.

Feynman--Kac Transform for Anomalous Processes

We develop a new approach to the study of the Feynman--Kac transform for non-Markov anomalous process $Y_t=X_{E_t}$ using methods from stochastic analysis, where $X$ is a strong Markov process on a...

Optimal Equilibria for Multidimensional Time-Inconsistent Stopping Problems

We study an optimal stopping problem under non-exponential discounting, where the state process is a multi-dimensional continuous strong Markov process. The discount function is taken to be log sub-additive, capturing decreasing impatience in behavioral economics. On strength of probabilistic potential theory, we establish the existence of an optimal equilibrium among a sufficiently large collection of equilibria, consisting of finely closed equilibria satisfying a boundary condition. This generalizes the existence of optimal equilibria for one-dimensional stopping problems in prior literature.

Wasserstein convergence rates for random bit approximations of continuous Markov processes

We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of certain Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of 1/4 with respect to every p-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than 1/4. Our results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points.

Playing with ghosts in a Dynkin game

We study a class of two-player optimal stopping games (Dynkin games) of preemption type, with uncertainty about the existence of competitors. The set-up is well-suited to model, for example, real options in the context of investors who do not want to publicly reveal their interest in a certain business opportunity. We show that if the underlying process is a Rd-valued, continuous, strong Markov process, and the stopping payoff is a continuous function (with mild integrability properties) there exists a Nash equilibrium in randomised stopping times for the game. Moreover, the equilibrium strategies and the expected payoffs of the two players are computed explicitly in terms of the corresponding one-player game. To the best of our knowledge this is the first paper to address this version of Dynkin games.