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Construction of semimartingales
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Let $(B,C,nu)$ be the characteristics of a semimartingale $X_t_t$ on $(Omega,mathcalF,mathcalF_t,mathbbP)$.
If
beginequation
B_t(omega)=bt, C_t(omega)=ct, nu(omega,dt,dx)=dt times
F(dx),
endequation
for suitable $b,c,F$, we know that $X_t_t$ is a Lévy process.
On the other hand, by each $(B,C,nu)$ with the above representation a Lévy process $Z_t_t$ is defined.
Now, similar to the statement for Lévy processes, I would like to construct a semimartingale from a given triplet $(B,C,nu)$ (which is not deterministic).
To be more specific, suppose
beginequation
tildeB_t=int_0^t b_s ds, tildeC_t=int_0^t c_s ds, tildenu(dt,dx)=dtF_t(dx),
endequation
for $b_t_t$ a predictable processes, $c_t_t$ a positive predictable processes, and $F_t=F_t(omega,dx)$ for each $(omega,t)$ a measure on $mathbbR$.
Then, motivated by the Lévy–Itô decomposition for Lévy processes, I thought about a process $X_t_t$ defined by
beginequation
X_t=int_0^t b_s ds+int_0^t sqrtc_sdW_s+int_0^tint_leq 1x(tildemu-tildenu)(ds,dx)+int_0^tint_>1xtildemu(ds,dx),
endequation
where $tildemu$ is a random measure with compensator $tildenu$.
My question is: Does this construction define a semimartingale with characteristics $(tildeB,tildeC,tildenu)$?
In particular, I am not sure if the last integral is a process of finite variation.
probability-theory stochastic-processes stochastic-integrals levy-processes
edited Aug 1 at 14:46
TheBridge
3,60611324
asked Jul 31 at 15:04
Daniel Lingohr
163
add a comment |Â
up vote
2
down vote
favorite
Let $(B,C,nu)$ be the characteristics of a semimartingale $X_t_t$ on $(Omega,mathcalF,mathcalF_t,mathbbP)$.
If
beginequation
B_t(omega)=bt, C_t(omega)=ct, nu(omega,dt,dx)=dt times
F(dx),
endequation
for suitable $b,c,F$, we know that $X_t_t$ is a Lévy process.
On the other hand, by each $(B,C,nu)$ with the above representation a Lévy process $Z_t_t$ is defined.
Now, similar to the statement for Lévy processes, I would like to construct a semimartingale from a given triplet $(B,C,nu)$ (which is not deterministic).
To be more specific, suppose
beginequation
tildeB_t=int_0^t b_s ds, tildeC_t=int_0^t c_s ds, tildenu(dt,dx)=dtF_t(dx),
endequation
for $b_t_t$ a predictable processes, $c_t_t$ a positive predictable processes, and $F_t=F_t(omega,dx)$ for each $(omega,t)$ a measure on $mathbbR$.
Then, motivated by the Lévy–Itô decomposition for Lévy processes, I thought about a process $X_t_t$ defined by
beginequation
X_t=int_0^t b_s ds+int_0^t sqrtc_sdW_s+int_0^tint_leq 1x(tildemu-tildenu)(ds,dx)+int_0^tint_>1xtildemu(ds,dx),
endequation
where $tildemu$ is a random measure with compensator $tildenu$.
My question is: Does this construction define a semimartingale with characteristics $(tildeB,tildeC,tildenu)$?
In particular, I am not sure if the last integral is a process of finite variation.
probability-theory stochastic-processes stochastic-integrals levy-processes
edited Aug 1 at 14:46
TheBridge
3,60611324
asked Jul 31 at 15:04
Daniel Lingohr
163
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $(B,C,nu)$ be the characteristics of a semimartingale $X_t_t$ on $(Omega,mathcalF,mathcalF_t,mathbbP)$.
If
beginequation
B_t(omega)=bt, C_t(omega)=ct, nu(omega,dt,dx)=dt times
F(dx),
endequation
for suitable $b,c,F$, we know that $X_t_t$ is a Lévy process.
On the other hand, by each $(B,C,nu)$ with the above representation a Lévy process $Z_t_t$ is defined.
Now, similar to the statement for Lévy processes, I would like to construct a semimartingale from a given triplet $(B,C,nu)$ (which is not deterministic).
To be more specific, suppose
beginequation
tildeB_t=int_0^t b_s ds, tildeC_t=int_0^t c_s ds, tildenu(dt,dx)=dtF_t(dx),
endequation
for $b_t_t$ a predictable processes, $c_t_t$ a positive predictable processes, and $F_t=F_t(omega,dx)$ for each $(omega,t)$ a measure on $mathbbR$.
Then, motivated by the Lévy–Itô decomposition for Lévy processes, I thought about a process $X_t_t$ defined by
beginequation
X_t=int_0^t b_s ds+int_0^t sqrtc_sdW_s+int_0^tint_leq 1x(tildemu-tildenu)(ds,dx)+int_0^tint_>1xtildemu(ds,dx),
endequation
where $tildemu$ is a random measure with compensator $tildenu$.
My question is: Does this construction define a semimartingale with characteristics $(tildeB,tildeC,tildenu)$?
In particular, I am not sure if the last integral is a process of finite variation.
probability-theory stochastic-processes stochastic-integrals levy-processes
edited Aug 1 at 14:46
TheBridge
3,60611324
asked Jul 31 at 15:04
Daniel Lingohr
163
Let $(B,C,nu)$ be the characteristics of a semimartingale $X_t_t$ on $(Omega,mathcalF,mathcalF_t,mathbbP)$.
If
beginequation
B_t(omega)=bt, C_t(omega)=ct, nu(omega,dt,dx)=dt times
F(dx),
endequation
for suitable $b,c,F$, we know that $X_t_t$ is a Lévy process.
On the other hand, by each $(B,C,nu)$ with the above representation a Lévy process $Z_t_t$ is defined.
Now, similar to the statement for Lévy processes, I would like to construct a semimartingale from a given triplet $(B,C,nu)$ (which is not deterministic).
To be more specific, suppose
beginequation
tildeB_t=int_0^t b_s ds, tildeC_t=int_0^t c_s ds, tildenu(dt,dx)=dtF_t(dx),
endequation
for $b_t_t$ a predictable processes, $c_t_t$ a positive predictable processes, and $F_t=F_t(omega,dx)$ for each $(omega,t)$ a measure on $mathbbR$.
Then, motivated by the Lévy–Itô decomposition for Lévy processes, I thought about a process $X_t_t$ defined by
beginequation
X_t=int_0^t b_s ds+int_0^t sqrtc_sdW_s+int_0^tint_leq 1x(tildemu-tildenu)(ds,dx)+int_0^tint_>1xtildemu(ds,dx),
endequation
where $tildemu$ is a random measure with compensator $tildenu$.
My question is: Does this construction define a semimartingale with characteristics $(tildeB,tildeC,tildenu)$?
In particular, I am not sure if the last integral is a process of finite variation.
probability-theory stochastic-processes stochastic-integrals levy-processes
edited Aug 1 at 14:46
TheBridge
3,60611324
asked Jul 31 at 15:04
Daniel Lingohr
163
edited Aug 1 at 14:46
TheBridge
3,60611324
edited Aug 1 at 14:46
TheBridge
3,60611324
edited Aug 1 at 14:46
TheBridge
3,60611324
3,60611324
asked Jul 31 at 15:04
Daniel Lingohr
163
asked Jul 31 at 15:04
Daniel Lingohr
163
asked Jul 31 at 15:04
Daniel Lingohr
163
163
add a comment |Â
add a comment |Â
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