On entrance—exit distributions of Markov processes

1978 ◽  
Vol 15 (1) ◽  
pp. 78-86
Author(s):  
Cristina Gzyl ◽  
Henryk Gzyl
Keyword(s):  
Markov Processes ◽  
Additive Functional ◽  
Stochastic Integrals ◽  
Standard Process ◽  
Integration By Parts ◽  
Infinitesimal Generators ◽  
The Right ◽  
Continuous Inverse ◽  
Exit Distributions

We use a result on integration by parts for stochastic integrals together with a technique developed by Getoor in [6], to express entrance—exit distributions for a standard process X, and a set Φ which is the support of a continuous additive functional C, in terms of the infinitesimal generators of semigroups associated with the time-changed process (Xτt), where (τt) is the right-continuous inverse of C.

On entrance—exit distributions of Markov processes

1978 ◽  
Vol 15 (01) ◽  
pp. 78-86
Author(s):  
Cristina Gzyl ◽  
Henryk Gzyl
Keyword(s):  
Markov Processes ◽  
Additive Functional ◽  
Stochastic Integrals ◽  
Standard Process ◽  
Integration By Parts ◽  
Infinitesimal Generators ◽  
The Right ◽  
Continuous Inverse ◽  
Exit Distributions

We use a result on integration by parts for stochastic integrals together with a technique developed by Getoor in [6], to express entrance—exit distributions for a standard process X, and a set Φ which is the support of a continuous additive functional C, in terms of the infinitesimal generators of semigroups associated with the time-changed process (X τ t ), where (τ t ) is the right-continuous inverse of C.

scholarly journals On a Comparison Result for Markov Processes

2008 ◽  
Vol 45 (01) ◽  
pp. 279-286
Author(s):  
Ludger Rüschendorf
Keyword(s):  
Markov Processes ◽  
Comparison Theorem ◽  
General Class ◽  
Stochastic Monotonicity ◽  
Comparison Result ◽  
State Spaces ◽  
Stochastic Orderings ◽  
Infinitesimal Generators ◽  
Real Functions

A comparison theorem is stated for Markov processes in Polish state spaces. We consider a general class of stochastic orderings induced by a cone of real functions. The main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators imply ordering of the processes. Several applications to convex type and to dependence orderings are given. In particular, Liggett's theorem on the association of Markov processes is a consequence of this comparison result.

scholarly journals Integration by Parts and Martingale Representation for a Markov Chain

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Tak Kuen Siu
Keyword(s):  
Markov Chain ◽  
Financial Market ◽  
Continuous Time ◽  
Contingent Claims ◽  
Jump Processes ◽  
Stochastic Integrals ◽  
Integration By Parts ◽  
Martingale Representation ◽  
Finite State ◽  
Change Of Measures

Integration-by-parts formulas for functions of fundamental jump processes relating to a continuous-time, finite-state Markov chain are derived using Bismut's change of measures approach to Malliavin calculus. New expressions for the integrands in stochastic integrals corresponding to representations of martingales for the fundamental jump processes are derived using the integration-by-parts formulas. These results are then applied to hedge contingent claims in a Markov chain financial market, which provides a practical motivation for the developments of the integration-by-parts formulas and the martingale representations.

scholarly journals Additive functionals and excursions of Kuznetsov processes

2005 ◽  
Vol 2005 (13) ◽  
pp. 2031-2040
Author(s):  
Hacène Boutabia
Keyword(s):  
Additive Functional ◽  
Standard Process ◽  
Additive Functionals ◽  
Isolated Points ◽  
Classical Duality

LetBbe a continuous additive functional for a standard process(Xt)t∈ℝ+and let(Yt)t∈ℝbe a stationary Kuznetsov process with the same semigroup of transition. In this paper, we give the excursion laws of(Xt)t∈ℝ+conditioned on the strict past and future without duality hypothesis. We study excursions of a general regenerative system and of a regenerative system consisting of the closure of the set of times the regular points ofBare visited. In both cases, those conditioned excursion laws depend only on two pointsXg−andXd, where]g,d[is an excursion interval of the regenerative setM. We use the(FDt)-predictable exit system to bring together the isolated points ofMand its perfect part and replace the classical optional exit system. This has been a subject in literature before (e.g., Kaspi (1988)) under the classical duality hypothesis. We define an “additive functional” for(Yt)t∈ℝwithB, we generalize the laws cited before to(Yt)t∈ℝ, and we express laws of pairs of excursions.

scholarly journals Stochastic calculus over symmetric Markov processes with time reversal

2015 ◽  
Vol 220 ◽  
pp. 91-148
Author(s):  
K. Kuwae
Keyword(s):  
Markov Processes ◽  
Time Reversal ◽  
Harmonic Functions ◽  
Stochastic Calculus ◽  
Additive Functional ◽  
Zero Energy ◽  
Additive Functionals ◽  
Stochastic Characterization ◽  
Symmetric Markov Processes

AbstractWe develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao's divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on L2-space obtained by lower-order perturbations.

scholarly journals On a Comparison Result for Markov Processes

2008 ◽  
Vol 45 (1) ◽  
pp. 279-286 ◽  
Author(s):  
Ludger Rüschendorf
Keyword(s):  
Markov Processes ◽  
Comparison Theorem ◽  
General Class ◽  
Stochastic Monotonicity ◽  
Comparison Result ◽  
State Spaces ◽  
Stochastic Orderings ◽  
Infinitesimal Generators ◽  
Real Functions

A comparison theorem is stated for Markov processes in Polish state spaces. We consider a general class of stochastic orderings induced by a cone of real functions. The main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators imply ordering of the processes. Several applications to convex type and to dependence orderings are given. In particular, Liggett's theorem on the association of Markov processes is a consequence of this comparison result.

Sign in / Sign up

Export Citation Format

Share Document