scholarly journals ABSOLUTELY CONTINUOUS COMPENSATORS

2011 ◽  
Vol 14 (03) ◽  
pp. 335-351 ◽  
Author(s):  
SVANTE JANSON ◽  
SOKHNA M'BAYE ◽  
PHILIP PROTTER
Keyword(s):  
Differential Equation ◽  
Markov Process ◽  
Stochastic Differential Equation ◽  
Markov Processes ◽  
Sufficient Conditions ◽  
Random Measure ◽  
Stopping Times ◽  
Poisson Random Measure ◽  
Absolutely Continuous ◽  
Strong Markov

We give sufficient conditions on the underlying filtration such that all totally inaccessible stopping times have compensators which are absolutely continuous. If a semimartingale, strong Markov process X has a representation as a solution of a stochastic differential equation driven by a Wiener process, Lebesgue measure, and a Poisson random measure, then all compensators of totally inaccessible stopping times are absolutely continuous with respect to the minimal filtration generated by X. However Çinlar and Jacod have shown that all semimartingale strong Markov processes, up to a change of time and slightly of space, have such a representation.

scholarly journals A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tong Wang ◽  
Hao Liang
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Finite Number ◽  
Random Measure ◽  
Approximate Solutions ◽  
The Other ◽  
Poisson Random Measure ◽  
Verification Theorem ◽  
Hamilton Jacobi Bellman ◽  
Duopoly Market

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.

Two extreme value processes arising in hydrology

1976 ◽  
Vol 13 (1) ◽  
pp. 190-194 ◽  
Author(s):  
Alan F. Karr
Keyword(s):  
Markov Process ◽  
Random Measure ◽  
Extreme Value ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Flood Peak ◽  
Base Level ◽  
Strong Markov Process ◽  
Hydrological System ◽  
Strong Markov

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn, Xn)) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn, Xn)). The set-parameterized process {MA} defined by MA = sup {Xn:Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt), where Mt = sup{Xn: Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.

scholarly journals Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise

2011 ◽  
Vol 43 (3) ◽  
pp. 688-711 ◽  
Author(s):  
Anita Diana Behme
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Autocorrelation Function ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Necessary And Sufficient Conditions ◽  
Tail Behavior ◽  
Absolutely Continuous ◽  
Distributional Properties ◽  
Necessary And Sufficient

For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

scholarly journals On Multivariate Wide-sense Markov Processes

1968 ◽  
Vol 33 ◽  
pp. 7-19 ◽  
Author(s):  
V. Mandrekar
Keyword(s):  
Differential Equation ◽  
Markov Process ◽  
Stochastic Differential Equation ◽  
Markov Processes ◽  
Wide Sense ◽  
Linear Vector ◽  
Gaussian Case

The idea of multivariate wide-sense Markov processes has been recently used by F.J. Beutler [1], In his paper, he shows that the solution of a linear vector stochastic differential equation in a wide-sense Markov process. We obtain here a characterization of such processes and as its consequence obtain the conditions under which it satisfies Beutler’s equation. Furthermore, in stationary Gaussian case we show that these are precisely stationary Gaussian Markov processes studied by J. Doob [5].

Two extreme value processes arising in hydrology

1976 ◽  
Vol 13 (01) ◽  
pp. 190-194 ◽  
Author(s):  
Alan F. Karr
Keyword(s):  
Markov Process ◽  
Random Measure ◽  
Extreme Value ◽  
Poisson Random Measure ◽  
One Dimensional ◽  
Flood Peak ◽  
Base Level ◽  
Strong Markov Process ◽  
Hydrological System ◽  
Strong Markov

Let Tn be the time of occurrence of the nth flood peak in a hydrological system and Xn the amount by which the peak exceeds a base level. We assume that ((Tn , Xn )) is a Poisson random measure with mean measure μ(dx) K(x, dy). In this note we characterize two extreme value processes which are functionals of ((Tn , Xn )). The set-parameterized process {MA } defined by MA = sup {Xn :Tn ∈ A} is additive and we compute its one-dimensional distributions explicitly. The process (Mt ), where Mt = sup{Xn : Tn ≦ t}, is a non-homogeneous strong Markov process. Our results extend but computationally simplify those of previous models.

scholarly journals Distributional properties of solutions of dV t = V t-dU t + dL t with Lévy noise

2011 ◽  
Vol 43 (03) ◽  
pp. 688-711
Author(s):  
Anita Diana Behme
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Autocorrelation Function ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Necessary And Sufficient Conditions ◽  
Tail Behavior ◽  
Absolutely Continuous ◽  
Distributional Properties ◽  
Necessary And Sufficient

For a given bivariate Lévy process (U t , L t ) t≥0, distributional properties of the stationary solutions of the stochastic differential equation dV t = V t-dU t + dL t are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
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.

On Approximate Large Deviations for 1D Diffusion

2003 ◽  
Vol 10 (2) ◽  
pp. 381-399
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Approximation Scheme ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Euler Approximation ◽  
One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

scholarly journals Exponential Stability of Stochastic Differential Equation with Mixed Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Wenli Zhu ◽  
Jiexiang Huang ◽  
Xinfeng Ruan ◽  
Zhao Zhao
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Stochastic Differential Equation ◽  
Exponential Stability ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Martingale Inequality ◽  
Mixed Delay ◽  
Exponentially Stable ◽  
Theoretical Results

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

scholarly journals The stationary probability density of a class of bounded Markov processes

2010 ◽  
Vol 42 (04) ◽  
pp. 986-993 ◽  
Author(s):  
Muhamad Azfar Ramli ◽  
Gerard Leng
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Markov Process ◽  
Probability Density ◽  
Markov Processes ◽  
Transition Probability ◽  
Stationary Probability ◽  
Probability Functions ◽  
Stationary Probability Density

In this paper we generalize a bounded Markov process, described by Stoyanov and Pacheco-González for a class of transition probability functions. A recursive integral equation for the probability density of these bounded Markov processes is derived and the stationary probability density is obtained by solving an equivalent differential equation. Examples of stationary densities for different transition probability functions are given and an application for designing a robotic coverage algorithm with specific emphasis on particular regions is discussed.

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