Dependence ordering for Markov processes on partially ordered spaces

We compare dependence in stochastically monotone Markov processes with partially ordered Polish state spaces using the concordance and supermodular orders. We show necessary and sufficient conditions for the concordance order to hold both in terms of the one-step transition probabilities for discrete-time processes and in terms of the corresponding infinitesimal generators for continuous-time processes. We give examples showing that a stochastic monotonicity assumption is not necessary for such orderings. We indicate relations between dependence orderings and, variously, the asymptotic variance-reduction effect in Monte Carlo Markov chains, Cheeger constants, and positive dependence for Markov processes.

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Stochastic monotonicity and conditional Monte Carlo for likelihood ratios

Advances in Applied Probability ◽

10.1017/s0001867800025192 ◽

1993 ◽

Vol 25 (01) ◽

pp. 103-115 ◽

Cited By ~ 1

Author(s):

Paul Glasserman

Keyword(s):

Likelihood Ratio ◽

Variance Reduction ◽

Stochastic Monotonicity ◽

Likelihood Ratios ◽

Monotone Likelihood Ratio ◽

Positive Dependence ◽

Poisson Input ◽

Conditional Expectations ◽

Conditional Monte Carlo ◽

Simulated Time

Likelihood ratios are used in computer simulation to estimate expectations with respect to one law from simulation of another. This importance sampling technique can be implemented with either the likelihood ratio at the end of the simulated time horizon or with a sequence of likelihood ratios at intermediate times. Since a likelihood ratio process is a martingale, the intermediate values are conditional expectations of the final value and their use introduces no bias. We provide conditions under which using conditional expectations in this way brings guaranteed variance reduction. We use stochastic orderings to get positive dependence between a process and its likelihood ratio, from which variance reduction follows. Our analysis supports the following rough statement: for increasing functionals of associated processes with monotone likelihood ratio, conditioning helps. Examples are drawn from recursively defined processes, Markov chains in discrete and continuous time, and processes with Poisson input.

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Stochastic monotonicity and conditional Monte Carlo for likelihood ratios

Advances in Applied Probability ◽

10.2307/1427498 ◽

1993 ◽

Vol 25 (1) ◽

pp. 103-115 ◽

Cited By ~ 5

Author(s):

Paul Glasserman

Keyword(s):

Likelihood Ratio ◽

Variance Reduction ◽

Stochastic Monotonicity ◽

Likelihood Ratios ◽

Monotone Likelihood Ratio ◽

Positive Dependence ◽

Poisson Input ◽

Conditional Expectations ◽

Conditional Monte Carlo ◽

Simulated Time

Likelihood ratios are used in computer simulation to estimate expectations with respect to one law from simulation of another. This importance sampling technique can be implemented with either the likelihood ratio at the end of the simulated time horizon or with a sequence of likelihood ratios at intermediate times. Since a likelihood ratio process is a martingale, the intermediate values are conditional expectations of the final value and their use introduces no bias.We provide conditions under which using conditional expectations in this way brings guaranteed variance reduction. We use stochastic orderings to get positive dependence between a process and its likelihood ratio, from which variance reduction follows. Our analysis supports the following rough statement: for increasing functionals of associated processes with monotone likelihood ratio, conditioning helps. Examples are drawn from recursively defined processes, Markov chains in discrete and continuous time, and processes with Poisson input.

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CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025799000060 ◽

1999 ◽

Vol 02 (01) ◽

pp. 105-129 ◽

Cited By ~ 1

Author(s):

UWE FRANZ

Keyword(s):

Markov Process ◽

Hopf Algebra ◽

Markov Processes ◽

Lévy Processes ◽

Sufficient Conditions ◽

Markov Property ◽

Vacuum State ◽

Levy Processes ◽

Quantum Process ◽

Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

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On transient Markov processes of Ornstein-Uhlenbeck type

Nagoya Mathematical Journal ◽

10.1017/s002776300000653x ◽

1998 ◽

Vol 149 ◽

pp. 19-32 ◽

Cited By ~ 5

Author(s):

Kouji Yamamuro

Keyword(s):

Markov Processes ◽

Lévy Processes ◽

Transition Probabilities ◽

Levy Processes ◽

Exit Times ◽

Linear Drift ◽

Lévy Measures ◽

Drift Terms

Abstract.For Hunt processes on Rd, strong and weak transience is defined by finiteness and infiniteness, respectively, of the expected last exit times from closed balls. Under some condition, which is satisfied by Lévy processes and Ornstein-Uhlenbeck type processes, this definition is expressed in terms of the transition probabilities. A criterion is given for strong and weak transience of Ornstein-Uhlenbeck type processes on Rd, using their Lévy measures and coefficient matrices of linear drift terms. An example is discussed.

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Comparing stochastic systems using regenerative simulation with common random numbers

Advances in Applied Probability ◽

10.1017/s0001867800033048 ◽

1979 ◽

Vol 11 (04) ◽

pp. 804-819 ◽

Cited By ~ 3

Author(s):

Philip Heidelberger ◽

Donald L. Iglehart

Keyword(s):

Stochastic Models ◽

Stochastic System ◽

Variance Reduction ◽

Stochastic Systems ◽

Sufficient Conditions ◽

Random Numbers ◽

Numerical Examples ◽

Common Random Numbers ◽

Regenerative Simulation ◽

Alternative Designs

Suppose two alternative designs for a stochastic system are to be compared. These two systems can be simulated independently or dependently. This paper presents a method for comparing two regenerative stochastic processes in a dependent fashion using common random numbers. A set of sufficient conditions is given that guarantees that the dependent simulations will produce a variance reduction over independent simulations. Numerical examples for a variety of simple stochastic models are included which illustrate the variance reduction achieved.

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Insensitivity and reversed Markov processes

Advances in Applied Probability ◽

10.2307/1427322 ◽

1983 ◽

Vol 15 (4) ◽

pp. 752-768 ◽

Cited By ~ 18

Author(s):

W. Henderson

Keyword(s):

Markov Processes ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient ◽

The Relationship

This paper is concerned with the relationship between insensitivity in a certain class of Markov processes and properties of that process when time is reversed. Necessary and sufficient conditions for insensitivity are established and linked to already proved results. A number of examples of insensitive systems are given.

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Stability and Stabilization of Continuous-Time Markovian Jump Singular Systems with Partly Known Transition Probabilities

Mathematical Problems in Engineering ◽

10.1155/2014/752626 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Jumei Wei ◽

Rui Ma

Keyword(s):

Continuous Time ◽

Controller Design ◽

Partial Information ◽

Transition Probabilities ◽

Singular Systems ◽

Sufficient Conditions ◽

Linear Matrix ◽

Markovian Jump ◽

Stability And Stabilization ◽

The Stability

This paper investigates the problem of the stability and stabilization of continuous-time Markovian jump singular systems with partial information on transition probabilities. A new stability criterion which is necessary and sufficient is obtained for these systems. Furthermore, sufficient conditions for the state feedback controller design are derived in terms of linear matrix inequalities. Finally, numerical examples are given to illustrate the effectiveness of the proposed methods.

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On Minimal Copulas under the Concordance Order

Journal of Optimization Theory and Applications ◽

10.1007/s10957-019-01618-4 ◽

2019 ◽

Vol 184 (3) ◽

pp. 762-780 ◽

Cited By ~ 1

Author(s):

Jae Youn Ahn ◽

Sebastian Fuchs

Keyword(s):

Optimization Problems ◽

Sufficient Conditions ◽

Necessary Condition ◽

Negative Dependence ◽

Kendall’S Tau ◽

Kendall's Tau ◽

Minimal Elements ◽

Measures Of Concordance ◽

Concordance Order

AbstractIn the present paper, we study extreme negative dependence focussing on the concordance order for copulas. With the absence of a least element for dimensions $$d\ge 3$$d≥3, the set of all minimal elements in the collection of all copulas turns out to be a natural and quite important extreme negative dependence concept. We investigate several sufficient conditions, and we provide a necessary condition for a copula to be minimal. The sufficient conditions are related to the extreme negative dependence concept of d-countermonotonicity and the necessary condition is related to the collection of all copulas minimizing multivariate Kendall’s tau. The concept of minimal copulas has already been proved to be useful in various continuous and concordance order preserving optimization problems including variance minimization and the detection of lower bounds for certain measures of concordance. We substantiate this key role of minimal copulas by showing that every continuous and concordance order preserving functional on copulas is minimized by some minimal copula, and, in the case the continuous functional is even strictly concordance order preserving, it is minimized by minimal copulas only. Applying the above results, we may conclude that every minimizer of Spearman’s rho is also a minimizer of Kendall’s tau.

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Stochastic Monotonicity and Duality ofkth Order with Application to Put-Call Symmetry of Powered Options

Journal of Applied Probability ◽

10.1239/jap/1429282608 ◽

2015 ◽

Vol 52 (1) ◽

pp. 82-101 ◽

Cited By ~ 1

Author(s):

Vassili N. Kolokoltsov

Keyword(s):

Option Pricing ◽

Markov Processes ◽

Ruin Probability ◽

Analytic Approach ◽

Stochastic Monotonicity ◽

Full Characterization

We introduce a notion ofkth order stochastic monotonicity and duality that allows us to unify the notion used in insurance mathematics (sometimes refereed to as Siegmund's duality) for the study of ruin probability and the duality responsible for the so-called put-call symmetries in option pricing. Our generalkth order duality can be interpreted financially as put-call symmetry for powered options. The main objective of this paper is to develop an effective analytic approach to the analysis of duality that will lead to the full characterization ofkth order duality of Markov processes in terms of their generators, which is new even for the well-studied case of put-call symmetries.

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