Change of measure up to a random time: theory

1991 ◽  
Vol 28 (4) ◽  
pp. 914-918 ◽  
Author(s):  
T. M. Mortimer ◽  
David Williams
Keyword(s):  
Random Time ◽  
Stopping Times ◽  
Change Of Measure ◽  
Girsanov Formula ◽  
Natural Filtration ◽  
Fixed Times ◽  
Random Times ◽  
Time Theory

Change of measure up to fixed times or stopping times is the theme of the famous Cameron–Martin–Girsanov formula. The paper studies change of measure up to random times which are not stopping times of the natural filtration. The ultimate aim is to build up a family of interesting models for physics and chemistry.

Change of measure up to a random time: theory

1991 ◽  
Vol 28 (04) ◽  
pp. 914-918 ◽  
Author(s):  
T. M. Mortimer ◽  
David Williams
Keyword(s):  
Random Time ◽  
Stopping Times ◽  
Change Of Measure ◽  
Girsanov Formula ◽  
Natural Filtration ◽  
Fixed Times ◽  
Random Times ◽  
Time Theory

Change of measure up to fixed times or stopping times is the theme of the famous Cameron–Martin–Girsanov formula. The paper studies change of measure up to random times which are not stopping times of the natural filtration. The ultimate aim is to build up a family of interesting models for physics and chemistry.

scholarly journals A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

2010 ◽  
Vol 47 (04) ◽  
pp. 1072-1083 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Prediction ◽  
Prediction Problem ◽  
Brownian Motion With Drift ◽  
Natural Filtration ◽  
Mathematical Justification ◽  
Bernoulli Random Walk

Let (B t )0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let M t := max{B s: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤T E[f(M T − B τ], where the supremum is over all stopping times τ adapted to the natural filtration of (B t ) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (B t ) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

scholarly journals Stochastic Sequences with a Regenerative Structure that May Depend Both on the Future and on the Past

2013 ◽  
Vol 45 (04) ◽  
pp. 1083-1110 ◽  
Author(s):  
Sergey Foss ◽  
Stan Zachary
Keyword(s):  
Ad Hoc ◽  
Particle System ◽  
Stopping Times ◽  
Underlying Structure ◽  
Contact Processes ◽  
The Past ◽  
Conditioning Theory ◽  
The Future ◽  
Novel Applications ◽  
Random Times

Many regenerative arguments in stochastic processes use random times which are akin to stopping times, but which are determined by the future as well as the past behaviour of the process of interest. Such arguments based on ‘conditioning on the future’ are usually developed in an ad-hoc way in the context of the application under consideration, thereby obscuring the underlying structure. In this paper we give a simple, unified, and more general treatment of such conditioning theory. We further give a number of novel applications to various particle system models, in particular to various flavours of contact processes and to infinite-bin models. We give a number of new results for existing and new models. We further make connections with the theory of Harris ergodicity.

scholarly journals A General ‘Bang-Bang’ Principle for Predicting the Maximum of a Random Walk

2010 ◽  
Vol 47 (4) ◽  
pp. 1072-1083 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Optimal Stopping ◽  
Stopping Time ◽  
Stopping Times ◽  
Optimal Prediction ◽  
Prediction Problem ◽  
Brownian Motion With Drift ◽  
Natural Filtration ◽  
Mathematical Justification ◽  
Bernoulli Random Walk

Let (Bt)0≤t≤T be either a Bernoulli random walk or a Brownian motion with drift, and let Mt := max{Bs: 0 ≤ s ≤ t}, 0 ≤ t ≤ T. In this paper we solve the general optimal prediction problem sup0≤τ≤TE[f(MT − Bτ], where the supremum is over all stopping times τ adapted to the natural filtration of (Bt) and f is a nonincreasing convex function. The optimal stopping time τ* is shown to be of ‘bang-bang’ type: τ* ≡ 0 if the drift of the underlying process (Bt) is negative and τ* ≡ T if the drift is positive. This result generalizes recent findings of Toit and Peskir (2009) and Yam, Yung and Zhou (2009), and provides additional mathematical justification for the dictum in finance that one should sell bad stocks immediately, but keep good stocks as long as possible.

On a Lévy process pinned at random time

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mohamed Erraoui ◽  
Astrid Hilbert ◽  
Mohammed Louriki
Keyword(s):  
Lévy Process ◽  
Markov Property ◽  
Levy Process ◽  
Random Time ◽  
Natural Filtration ◽  
The Right

AbstractIn this paper, our first goal is to rigorously define a Lévy process pinned at random time. Our second task is to establish the Markov property with respect to its completed natural filtration and thus with respect to the usual augmentation of the latter. The resulting conclusion is the right-continuity of completed natural filtration. Certain examples of such process are considered.

scholarly journals Random Times for Markov Processes with Killing

2021 ◽  
Vol 5 (4) ◽  
pp. 254
Author(s):  
Yuri G. Kondratiev ◽  
José Luís da Silva
Keyword(s):  
Markov Processes ◽  
Schrödinger Operators ◽  
Random Time ◽  
Schrodinger Operators ◽  
Time Behavior ◽  
Non Local ◽  
Time Changes ◽  
Random Times

We consider random time changes in Markov processes with killing potentials. We study how random time changes may be introduced in these Markov processes with killing potential and how these changes may influence their time behavior. As applications, we study the parabolic Anderson problem, the non-local Schrödinger operators as well as the generalized Anderson problem.

scholarly journals Subgeometric Ergodicity under Random-Time State-Dependent Drift Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Mokaedi V. Lekgari
Keyword(s):  
Markov Chains ◽  
Stochastic Networks ◽  
Random Time ◽  
Stopping Times ◽  
Lyapunov Techniques ◽  
State Dependent ◽  
Drift Conditions ◽  
The Stability

Motivated by possible applications of Lyapunov techniques in the stability of stochastic networks, subgeometric ergodicity of Markov chains is investigated. In a nutshell, in this study we take a look atf-ergodic general Markov chains, subgeometrically ergodic at rater, when the random-time Foster-Lyapunov drift conditions on a set of stopping times are satisfied.

scholarly journals Random time change and related evolution equations. Time asymptotic behavior

2019 ◽  
Vol 20 (05) ◽  
pp. 2050034
Author(s):  
Anatoly N. Kochubei ◽  
Yuri G. Kondratiev ◽  
José L. da Silva
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Analysis ◽  
Evolution Equations ◽  
Random Time ◽  
Asymptotic Behavior Of Solutions ◽  
Kolmogorov Equations ◽  
Random Time Change ◽  
Time Changes ◽  
Random Times ◽  
Time Asymptotic Behavior

In this paper, we investigate the time asymptotic behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination principle for the solutions to forward Kolmogorov equations. The classes of subordinators for which asymptotic analysis may be realized are described.

scholarly journals Stochastic Sequences with a Regenerative Structure that May Depend Both on the Future and on the Past

2013 ◽  
Vol 45 (4) ◽  
pp. 1083-1110 ◽  
Author(s):  
Sergey Foss ◽  
Stan Zachary
Keyword(s):  
Ad Hoc ◽  
Particle System ◽  
Stopping Times ◽  
Underlying Structure ◽  
Contact Processes ◽  
The Past ◽  
Conditioning Theory ◽  
The Future ◽  
Novel Applications ◽  
Random Times

Many regenerative arguments in stochastic processes use random times which are akin to stopping times, but which are determined by the future as well as the past behaviour of the process of interest. Such arguments based on ‘conditioning on the future’ are usually developed in an ad-hoc way in the context of the application under consideration, thereby obscuring the underlying structure. In this paper we give a simple, unified, and more general treatment of such conditioning theory. We further give a number of novel applications to various particle system models, in particular to various flavours of contact processes and to infinite-bin models. We give a number of new results for existing and new models. We further make connections with the theory of Harris ergodicity.

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