On the Partial Ordering of One-Dimensional Markov Processes

In this article a class of Markov processes on the ring of finite adeles of the rational numbers are introduced. A class of non-Archimedean metrics on \(\mathbb{A}_{f}\) are chosen in order to describe this ring as a general polyadic ring and to introduce a family of pseudodifferential operators and parabolic-type equations on \({L^2}(\mathbb{A}_{f})\). The fundamental solutions of these parabolic equations determine transition functions of time and space homogeneous Markov processes on \(\mathbb{A}_{f}\) which are invariant under multiplication by units. Considering the infinite place ℝ, we extend these results to the complete ring of adeles.

In a probabi!istic treatment of semi-linear parabolic equations key steps were to construct a certain class of Markov processes on a big state space and to prove the branching property of semi-groups of the processes (cf. ikeda-NagasawaWatanabe [1]). In this paper branching property will be characterized in terms of multiDlicativity (or decomposability) of semigroup of a Markov process on a multipliqat~ve state space (cf. § I for definitions). This characterization will then be applied to branching Markov processes with age and sign in § 2~ and to a branching Markov process treated recently by Sirao [7] in § 3.

On the possibility of extinction in a class of Markov processes in economics

A class of Markov processes in continuous time, with local transition rules, acting on colourings of a lattice, is defined. An algorithm is described for dynamic simulation of such processes. The computation time for the next state is O(logb), where b is the number of possible next states. This technique is used to give some evidence that the limiting shape for a random growth process in the plane with exponential distribution is approximately a circle.

Introduction.Let xt{t), Xt(0)=0, 0< (k = l, , n) be elements of n independent Wiener spaces.Let x(t) be an element of the product space of the n independent Wiener spaces.Throughout the paper we shall assume V(t, x) to be a Borel measurable function of t and the n-dimensional vector x=(xi, , xn) that is bounded in every finite Euclidean sphere of the (t, x) space, 0=< oo.These assumptions will not be restated in the rest of the paper.We shall often assume that V(t, x) satisfies additional regularity conditions specified later on in the paper. Let(1.1) y(t) = f V(r,x(r))dr. J oWe wish to relate the study of the Markov process(1.2) (x(t),y(t))to the study of certain differential and integral equations.Such a study is of interest for the following reasons.We obtain information about Markov processes of type (1.2).For certain choices of the function V(t, x), the study yields interesting information about the problem of the absorbing barrier in diffusion theory.Moreover, if we look at the problem from a different point of view, we obtain a general method of getting limit theorems of a certain type [2] (2).Consider the distribution function (1.3) a(a;t) = Pr {y(t) ^ a}.Let 5m=(G2), , G), where G?\ G?\ (k = l, , n) are independent, normally distributed random variables with mean 0 and variance 1.The distribution function of 1 z r(l, _+ m k/mt \m m112is the same as that of

Stochastic processes of temporary equilibria

Let {λn,μn} be a set of birth-death parameters, A the associated generator (1.4.2) and {X(t): 0 ≤ t < ∞} a birth-death process with generator A. In this and the following chapters we shall be concerned with natural birth-death processes only, i.e., A is assumed to satisfy the conditions C(A) and D(A). The state -1 will be disregarded and the term transition matrix will be used for the matrix P(⋅) = (pij(⋅)), where i, j = 0, 1,....... Since the properties to be discussed in this chapter are independent of the initial distribution of the process, we shall often identify the birth-death process {X(t)} with its transition matrix P(.). KARLIN and McGREGOR (1957a, 1959b) have proved the following important feature of the transition matrix P(⋅) of a natural birth-death process: (2.1.1) P(t) is strictly totally positive (STP) for t > 0, which means that every subdeterminant of P(t) is strictly positive for t > 0. KARLIN and McGREGOR (1959a) showed that property (2.1.1) is characteristic for birth-death processes in the class of Markov processes as defined in section 1.1. An immediate conclusion from (2.1.1) is (2.1.2) pij(t) > 0 i, j = 0, 1,...; t > 0.KeywordsTransition MatrixSpectral FunctionMoment ProblemConjugate PointClassical PolynomialThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We consider an open quantum system, with dissipation applied only to a part of its degrees of freedom, evolving via a quantum Markov dynamics. We demonstrate that, in the Zeno regime of large dissipation, the relaxation of the quantum system towards a pure quantum state is linked to the evolution of a classical Markov process towards a single absorbing state. The rates of the associated classical Markov process are determined by the original quantum dynamics. Extension of this correspondence to absorbing states with internal structure allows us to establish a general criterion for having a Zeno-limit nonequilibrium stationary state of arbitrary finite rank. An application of this criterion is illustrated in the case of an open XXZ spin-1/2 chain dissipatively coupled at its edges to baths with fixed and different polarizations. For this system, we find exact nonequilibrium steady-state solutions of ranks 1 and 2.

We describe a symmetric monoidal category whose arrows are automata in which the actions have probabilities. The endomorphisms of the identity for the tensor are classical finite Markov processes. The operations of the category permit the compositional description Markov processes. We illustrate by describing a Markov process with 12 n states, which represents a model of the classical Dining Philosopher problem with n dining philosophers, showing how to calculate the probability of reaching deadlock in k steps. A straightforward application of the Perron-Frobenius Theorem yields that this probability tends to 1 as k tends to infinity.

In this paper we consider a general class of Markov processes with discrete time parameter {X n : n = 0, 1, 2, …}, where the transitions from one state to another are determined by: The functions h j (·) are strictly increasing, and in general the Markov processes have dependent increments. In section 1 the model is described more precisely. Considering the sequence of distribution functions F n(x) = Pr[X n ≤ x] (n = 0, 1, 2, ….) a numerical method is given by which upper and lower bounds can be determined for F n(x) (n ≥ n o). Numerical results are given for some special cases, and the determination of an alarm level in a flame failure control system is described as an application. The results show that the upper and lower bounds can lie very close together.

We give a necessary and sucient condition for a one-dimensional Markov (not necessarily strong Markov) processes with continuous paths in a class to be equivalent to a one-dimensional bi-generalized diusion processes in the sense of (7).

A general framework is proposed for pricing both continuously and discretely monitored Asian options under one-dimensional Markov processes. For each type (continuously monitored or discretely monitored), we derive the double transform of the Asian option price in terms of the unique bounded solution to a related functional equation. In the special case of continuous-time Markov chain (CTMC), the functional equation reduces to a linear system that can be solved analytically via matrix inversion. Thus the Asian option prices under a one-dimensional Markov process can be obtained by first constructing a CTMC to approximate the targeted Markov process model, and then computing the Asian option prices under the approximate CTMC by numerically inverting the double transforms. Numerical experiments indicate that our pricing method is accurate and fast under popular Markov process models, including the CIR model, the CEV model, Merton’s jump diffusion model, the double-exponential jump diffusion model, the variance gamma model, and the CGMY model.

On 'A General Framework for Pricing Asian Options Under Markov Processes'

Mathematical formulations of applied inverse problems often involve operator equations in normed functional spaces. In many cases, these spaces can, in addition, be endowed with a partial order relation, which turns them into Banach lattices. The fact that two tools, such as a partial order relation and a monotone (with respect to this partial order) norm, are available to the researchers, gives them a clear advantage of having more freedom in the problem formulations. For instance, errors in the approximate data are sometimes easier to describe in terms of pointwise bounds. Inverse problems in partially ordered normed spaces (Banach lattices) have been studied before in the case when a compact set containing the unknown exact solution is available a priori. It turned out that under this assumption it is possible, even in the ill-posed case, to compute ‘pointwise’ bounds for the unknown exact solution (or rather, bounds by means of the appropriate partial order), thus providing an error estimate by means of the partial order. Also, a useful property of this approach was that one was able to quantify the uncertainty in the operator by means of linear inequalities that were included in the corresponding optimization problems as (linear) constraints and made the computations easier. However, the compactness assumption might sometimes be too demanding in practice. This paper aims at revealing the possibilities and advantages of using partial order in solving inverse problems in the case when no compact set of prior restrictions is available, concentrating on linear inverse (possibly ill-posed) problems.

In this paper, we construct the two-parameter Feller semigroup associated with a certain one-dimensional inhomogeneous Markov process. This process may be described as follows. At the interior points of the finite number of intervals $(-\infty,r_1(s)),\,(r_1(s),r_2(s)),\ldots,\,(r_{n}(s),\infty)$ separated by points $r_i(s)\,(i=1,\ldots,n)$, the positions of which depend on the time variable, this process coincides with the ordinary diffusions given there by their generating differential operators, and its behavior on the common boundaries of these intervals is determined by the Feller-Wentzell conjugation conditions of the integral type, each of which corresponds to the inward jump phenomenon from the boundary. The study of the problem is done using analytical methods. With such an approach, the problem of existence of the desired semigroup leads to the corresponding nonlocal conjugation problem for a second order linear parabolic equation of Kolmogorov’s type with discontinuous coefficients. The main part of the paper consists in the investigation of this parabolic conjugation problem, the peculiarity of which is that the domains on the plane, where the equations are given, are curvilinear and have non-smooth boundaries: the functions $r_i(s)\,(i=1,\ldots,n)$, which determine the boundaries of these domains satisfy only the Hölder condition with exponent greater than $\frac{1}{2}$. Its classical solvability in the space of continuous functions is established by the boundary integral equations method with the use of the fundamental solutions of the uniformly parabolic equations and the associated potentials. It is also proved that the solution of this problem has a semigroup property. The availability of the integral representation for the constructed semigroup allows us to prove relatively easily that this semigroup yields the Markov process.