Martingale problem with time-homogeneous Markov chain and a Poisson process
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Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(mathcal F_t)_{ninmathbb N_0}$ be a filtration on $(Omega,mathcal A)$
$(E,mathcal E)$ be a measurable space
$(Y_n)_{ninmathbb N_0}$ be a $(E,mathcal E)$-valued time-homogeneous $mathcal F$-Markov chain on $(Omega,mathcal A,operatorname P)$ with transition kernel $pi$
$(N_t)_{tge0}$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with intensity $lambdage0$
$B(E,mathcal E)$ denote the space of bounded $mathcal E$-measurable $f:Etomathbb R$
Assume $Y$ and $N$ are independent. Let $$X_t:=Y_{N_t};;;text{for }tge0$$ and $$(mathcal Lf)(x):=int q(x,{rm d}y)(f(y)-f(x));;;text{for }xin Etext{ and }fin B(E,mathcal E)$$ with $q:=lambdapi$.
Let $fin B(E,mathcal E)$. How can we show that $$M_t:=f(X_t)-int_0^t(mathcal Lf)(X_s):{rm d}s;;;text{for }tge0$$ is an $mathcal F$-martingale?
I guess that this is not too complicated, but I don't see the trick we need to show it.
stochastic-processes markov-chains martingales stochastic-analysis poisson-process
asked Oct 12 at 18:16
0xbadf00d
1,86041428
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up vote
0
down vote
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Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(mathcal F_t)_{ninmathbb N_0}$ be a filtration on $(Omega,mathcal A)$
$(E,mathcal E)$ be a measurable space
$(Y_n)_{ninmathbb N_0}$ be a $(E,mathcal E)$-valued time-homogeneous $mathcal F$-Markov chain on $(Omega,mathcal A,operatorname P)$ with transition kernel $pi$
$(N_t)_{tge0}$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with intensity $lambdage0$
$B(E,mathcal E)$ denote the space of bounded $mathcal E$-measurable $f:Etomathbb R$
Assume $Y$ and $N$ are independent. Let $$X_t:=Y_{N_t};;;text{for }tge0$$ and $$(mathcal Lf)(x):=int q(x,{rm d}y)(f(y)-f(x));;;text{for }xin Etext{ and }fin B(E,mathcal E)$$ with $q:=lambdapi$.
Let $fin B(E,mathcal E)$. How can we show that $$M_t:=f(X_t)-int_0^t(mathcal Lf)(X_s):{rm d}s;;;text{for }tge0$$ is an $mathcal F$-martingale?
I guess that this is not too complicated, but I don't see the trick we need to show it.
stochastic-processes markov-chains martingales stochastic-analysis poisson-process
asked Oct 12 at 18:16
0xbadf00d
1,86041428
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(mathcal F_t)_{ninmathbb N_0}$ be a filtration on $(Omega,mathcal A)$
$(E,mathcal E)$ be a measurable space
$(Y_n)_{ninmathbb N_0}$ be a $(E,mathcal E)$-valued time-homogeneous $mathcal F$-Markov chain on $(Omega,mathcal A,operatorname P)$ with transition kernel $pi$
$(N_t)_{tge0}$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with intensity $lambdage0$
$B(E,mathcal E)$ denote the space of bounded $mathcal E$-measurable $f:Etomathbb R$
Assume $Y$ and $N$ are independent. Let $$X_t:=Y_{N_t};;;text{for }tge0$$ and $$(mathcal Lf)(x):=int q(x,{rm d}y)(f(y)-f(x));;;text{for }xin Etext{ and }fin B(E,mathcal E)$$ with $q:=lambdapi$.
Let $fin B(E,mathcal E)$. How can we show that $$M_t:=f(X_t)-int_0^t(mathcal Lf)(X_s):{rm d}s;;;text{for }tge0$$ is an $mathcal F$-martingale?
I guess that this is not too complicated, but I don't see the trick we need to show it.
stochastic-processes markov-chains martingales stochastic-analysis poisson-process
asked Oct 12 at 18:16
0xbadf00d
1,86041428
Let
$(Omega,mathcal A,operatorname P)$ be a probability space
$(mathcal F_t)_{ninmathbb N_0}$ be a filtration on $(Omega,mathcal A)$
$(E,mathcal E)$ be a measurable space
$(Y_n)_{ninmathbb N_0}$ be a $(E,mathcal E)$-valued time-homogeneous $mathcal F$-Markov chain on $(Omega,mathcal A,operatorname P)$ with transition kernel $pi$
$(N_t)_{tge0}$ be a Poisson process on $(Omega,mathcal A,operatorname P)$ with intensity $lambdage0$
$B(E,mathcal E)$ denote the space of bounded $mathcal E$-measurable $f:Etomathbb R$
Assume $Y$ and $N$ are independent. Let $$X_t:=Y_{N_t};;;text{for }tge0$$ and $$(mathcal Lf)(x):=int q(x,{rm d}y)(f(y)-f(x));;;text{for }xin Etext{ and }fin B(E,mathcal E)$$ with $q:=lambdapi$.
Let $fin B(E,mathcal E)$. How can we show that $$M_t:=f(X_t)-int_0^t(mathcal Lf)(X_s):{rm d}s;;;text{for }tge0$$ is an $mathcal F$-martingale?
I guess that this is not too complicated, but I don't see the trick we need to show it.
stochastic-processes markov-chains martingales stochastic-analysis poisson-process
stochastic-processes markov-chains martingales stochastic-analysis poisson-process
asked Oct 12 at 18:16
0xbadf00d
1,86041428
asked Oct 12 at 18:16
0xbadf00d
1,86041428
edited Nov 17 at 16:29
asked Oct 12 at 18:16
0xbadf00d
1,86041428
asked Oct 12 at 18:16
0xbadf00d
1,86041428
asked Oct 12 at 18:16
0xbadf00d
1,86041428
1,86041428
add a comment |
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