How can I show that the stochastic integral of a jump process w.r.t. Brownian motion is a local martingale by...
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Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then
begin{equation}
M_t=int_0^t Y_s,dW_s=sum_{i=0}^{N_t-1} Y_{T_i}(W_{T_{i+1}}-W_{T_i})+Y_t(W_t-W_{T_{N_t}}).
end{equation}
The process $M$ is continuous. I want to show that $M$ is a local martingale by using the sequence $tau_n=inf{t: |M_t|>n}$. That is, I have to show that ${M_{tau_nwedge t}}$ is a martingale. I have read that one can use the above sequence of stopping times for continuous local martingales. But why or how?
brownian-motion martingales stochastic-integrals stochastic-analysis local-martingales
asked Nov 13 at 16:48
Sofia
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Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then
begin{equation}
M_t=int_0^t Y_s,dW_s=sum_{i=0}^{N_t-1} Y_{T_i}(W_{T_{i+1}}-W_{T_i})+Y_t(W_t-W_{T_{N_t}}).
end{equation}
The process $M$ is continuous. I want to show that $M$ is a local martingale by using the sequence $tau_n=inf{t: |M_t|>n}$. That is, I have to show that ${M_{tau_nwedge t}}$ is a martingale. I have read that one can use the above sequence of stopping times for continuous local martingales. But why or how?
brownian-motion martingales stochastic-integrals stochastic-analysis local-martingales
asked Nov 13 at 16:48
Sofia
62
New contributor
Sofia is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
1
down vote
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up vote
1
down vote
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Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then
begin{equation}
M_t=int_0^t Y_s,dW_s=sum_{i=0}^{N_t-1} Y_{T_i}(W_{T_{i+1}}-W_{T_i})+Y_t(W_t-W_{T_{N_t}}).
end{equation}
The process $M$ is continuous. I want to show that $M$ is a local martingale by using the sequence $tau_n=inf{t: |M_t|>n}$. That is, I have to show that ${M_{tau_nwedge t}}$ is a martingale. I have read that one can use the above sequence of stopping times for continuous local martingales. But why or how?
brownian-motion martingales stochastic-integrals stochastic-analysis local-martingales
asked Nov 13 at 16:48
Sofia
62
New contributor
Sofia is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Suppose that $Y$ is a pure jump process with $N_t$ jumps in $(0,t]$ and $E[N_t]<infty$. Denote the jump times by $T_i$. Let $W$ be a Brownian motion. If $T_0=0$, then
begin{equation}
M_t=int_0^t Y_s,dW_s=sum_{i=0}^{N_t-1} Y_{T_i}(W_{T_{i+1}}-W_{T_i})+Y_t(W_t-W_{T_{N_t}}).
end{equation}
The process $M$ is continuous. I want to show that $M$ is a local martingale by using the sequence $tau_n=inf{t: |M_t|>n}$. That is, I have to show that ${M_{tau_nwedge t}}$ is a martingale. I have read that one can use the above sequence of stopping times for continuous local martingales. But why or how?
brownian-motion martingales stochastic-integrals stochastic-analysis local-martingales
brownian-motion martingales stochastic-integrals stochastic-analysis local-martingales
asked Nov 13 at 16:48
Sofia
62
New contributor
Sofia is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked Nov 13 at 16:48
Sofia
62
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edited Nov 14 at 10:10
asked Nov 13 at 16:48
Sofia
62
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asked Nov 13 at 16:48
Sofia
62
asked Nov 13 at 16:48
Sofia
62
62
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Sofia is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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1 Answer
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I guess you could use the sequence $tau_n=inf{t,:, N_t=n}$ and then Corollary 6.14 (or 7.14 depending on the edition, it is called "martingale transforms") from Kallenberg's Foundations of Modern Probability yields the result.
answered Nov 14 at 11:10
TimBraun
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1 Answer
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1 Answer
1
active
oldest
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active
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active
oldest
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up vote
0
down vote
I guess you could use the sequence $tau_n=inf{t,:, N_t=n}$ and then Corollary 6.14 (or 7.14 depending on the edition, it is called "martingale transforms") from Kallenberg's Foundations of Modern Probability yields the result.
answered Nov 14 at 11:10
TimBraun
111
add a comment |
up vote
0
down vote
I guess you could use the sequence $tau_n=inf{t,:, N_t=n}$ and then Corollary 6.14 (or 7.14 depending on the edition, it is called "martingale transforms") from Kallenberg's Foundations of Modern Probability yields the result.
answered Nov 14 at 11:10
TimBraun
111
add a comment |
up vote
0
down vote
up vote
0
down vote
I guess you could use the sequence $tau_n=inf{t,:, N_t=n}$ and then Corollary 6.14 (or 7.14 depending on the edition, it is called "martingale transforms") from Kallenberg's Foundations of Modern Probability yields the result.
answered Nov 14 at 11:10
TimBraun
111
I guess you could use the sequence $tau_n=inf{t,:, N_t=n}$ and then Corollary 6.14 (or 7.14 depending on the edition, it is called "martingale transforms") from Kallenberg's Foundations of Modern Probability yields the result.
answered Nov 14 at 11:10
TimBraun
111
answered Nov 14 at 11:10
TimBraun
111
answered Nov 14 at 11:10
TimBraun
111
answered Nov 14 at 11:10
TimBraun
111
111
add a comment |
add a comment |
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