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$begingroup$

Let $(Omega, mathcalA, P, (X_n)_n in mathbbN)$ be a discrete-time stochastic process on a state space $S$ (which we assume to be finite and discrete for simplicity). Denote by $mathcalF_n := sigma(X_0, X_1, dots, X_n)$ the filtration generated by $X_n$.

For $sigma$-algebras $mathcalF$, $mathcalG$ and $mathcalH$ denote by $mathcalF perp!!!perp_mathcalG mathcalH$ conditional independence of $mathcalF$ and $mathcalH$ given $mathcalG$.

$(X_n)$ is called a Markov chain if it satisfies the Markov property: $X_n+1 perp!!!perp_X_n mathcalF_n$ for all $n$, that is $P(X_n+1 = j mid mathcalF_n) = P(X_n+1 = j mid X_n)$ almost surely.
A Markov chain $(X_n)$ is called time-homogeneous if this conditional probability is independent of $n$.

For a process $(X_n)_n in mathbbN$ (a priori not a Markov chain) consider the following properties:

1. 
   $X_N+1 perp!!!perp_X_N mathcalF_N$ a.s. on $ N < infty $ for all $mathcalF_n$-stopping times $N$
   


2. 
   $X_N+1 perp!!!perp_N, X_N mathcalF_N$ a.s. on $ N < infty $ for all $mathcalF_n$-stopping times $N$
   

Both properties imply that $(X_n)$ is a Markov chain. Which of these two properties is the right definition of "strong Markov property"? Does any of these properties imply that $(X_n)$ is time-homogeneous?

Remark: In most of the literature one considers a time-homogeneous Markov chain $(X_n)$ equipped with a set of probability measures $P_i$ for $i in S$ and defines for such processes the strong Markov property differently, namely as $P_i(X_N+1 = j mid mathcalF_N) = P_X_N(X_1 = j)$ $P_i$-almost surely. Above, I'm interested in the cases when we have only one probability measure and without forcing time-homogeneity a priori.

stochastic-processes markov-chains markov-process

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asked Mar 21 at 12:57

yadaddyyadaddy

1,285816

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0

$begingroup$

Let $(Omega, mathcalA, P, (X_n)_n in mathbbN)$ be a discrete-time stochastic process on a state space $S$ (which we assume to be finite and discrete for simplicity). Denote by $mathcalF_n := sigma(X_0, X_1, dots, X_n)$ the filtration generated by $X_n$.

For $sigma$-algebras $mathcalF$, $mathcalG$ and $mathcalH$ denote by $mathcalF perp!!!perp_mathcalG mathcalH$ conditional independence of $mathcalF$ and $mathcalH$ given $mathcalG$.

$(X_n)$ is called a Markov chain if it satisfies the Markov property: $X_n+1 perp!!!perp_X_n mathcalF_n$ for all $n$, that is $P(X_n+1 = j mid mathcalF_n) = P(X_n+1 = j mid X_n)$ almost surely.
A Markov chain $(X_n)$ is called time-homogeneous if this conditional probability is independent of $n$.

For a process $(X_n)_n in mathbbN$ (a priori not a Markov chain) consider the following properties:

1. 
   $X_N+1 perp!!!perp_X_N mathcalF_N$ a.s. on $ N < infty $ for all $mathcalF_n$-stopping times $N$
   


2. 
   $X_N+1 perp!!!perp_N, X_N mathcalF_N$ a.s. on $ N < infty $ for all $mathcalF_n$-stopping times $N$
   

Both properties imply that $(X_n)$ is a Markov chain. Which of these two properties is the right definition of "strong Markov property"? Does any of these properties imply that $(X_n)$ is time-homogeneous?

Remark: In most of the literature one considers a time-homogeneous Markov chain $(X_n)$ equipped with a set of probability measures $P_i$ for $i in S$ and defines for such processes the strong Markov property differently, namely as $P_i(X_N+1 = j mid mathcalF_N) = P_X_N(X_1 = j)$ $P_i$-almost surely. Above, I'm interested in the cases when we have only one probability measure and without forcing time-homogeneity a priori.

stochastic-processes markov-chains markov-process

share|cite|improve this question

asked Mar 21 at 12:57

yadaddyyadaddy

1,285816

$endgroup$

add a comment | 

$begingroup$

Let $(Omega, mathcalA, P, (X_n)_n in mathbbN)$ be a discrete-time stochastic process on a state space $S$ (which we assume to be finite and discrete for simplicity). Denote by $mathcalF_n := sigma(X_0, X_1, dots, X_n)$ the filtration generated by $X_n$.

For $sigma$-algebras $mathcalF$, $mathcalG$ and $mathcalH$ denote by $mathcalF perp!!!perp_mathcalG mathcalH$ conditional independence of $mathcalF$ and $mathcalH$ given $mathcalG$.

$(X_n)$ is called a Markov chain if it satisfies the Markov property: $X_n+1 perp!!!perp_X_n mathcalF_n$ for all $n$, that is $P(X_n+1 = j mid mathcalF_n) = P(X_n+1 = j mid X_n)$ almost surely.
A Markov chain $(X_n)$ is called time-homogeneous if this conditional probability is independent of $n$.

For a process $(X_n)_n in mathbbN$ (a priori not a Markov chain) consider the following properties:

1. 
   $X_N+1 perp!!!perp_X_N mathcalF_N$ a.s. on $ N < infty $ for all $mathcalF_n$-stopping times $N$
   


2. 
   $X_N+1 perp!!!perp_N, X_N mathcalF_N$ a.s. on $ N < infty $ for all $mathcalF_n$-stopping times $N$
   

Both properties imply that $(X_n)$ is a Markov chain. Which of these two properties is the right definition of "strong Markov property"? Does any of these properties imply that $(X_n)$ is time-homogeneous?

Remark: In most of the literature one considers a time-homogeneous Markov chain $(X_n)$ equipped with a set of probability measures $P_i$ for $i in S$ and defines for such processes the strong Markov property differently, namely as $P_i(X_N+1 = j mid mathcalF_N) = P_X_N(X_1 = j)$ $P_i$-almost surely. Above, I'm interested in the cases when we have only one probability measure and without forcing time-homogeneity a priori.

stochastic-processes markov-chains markov-process

share|cite|improve this question

asked Mar 21 at 12:57

yadaddyyadaddy

1,285816

$endgroup$

share|cite|improve this question

asked Mar 21 at 12:57

yadaddyyadaddy

1,285816

share|cite|improve this question

asked Mar 21 at 12:57

yadaddyyadaddy

1,285816

asked Mar 21 at 12:57

yadaddyyadaddy

1,285816

asked Mar 21 at 12:57

yadaddyyadaddy

1,285816