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Question about Protter's proof of the Ito's formula
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The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential Equations at page 80.
The Ito's formula is this 
And my doubt is that I have not understood yet what does the sets $A = A(epsilon, t)$ and $B = B(epsilon, t)$ mean in order to conclude that $A cup B $ equals the set of stopping times of $X$ on $(0,t]$. The definition of these two sets is the following
For me the definition of $ A= A(epsilon, t)$ (as a set that depends on $epsilon $ and $t$) should be: let $A(w) = A(epsilon, t)(w)$ be any finite set of jump times of $X_cdot(w)$ on $(0, t]$. However, this definition and the definition of $B(w) = B(epsilon, t)(w)$ as the set of jump times $s$ on $(0,t]$ such that $sum_s in B(epsilon, t)(w) (Delta X_s (w))^2 leq epsilon^2$, and $A(epsilon, t)(w) cup B(epsilon, t)(w)$ equals the stopping times of $X_cdot(w)$ on $(0,t]$ does not match. Moreover, the definition of $A = A( epsilon, t)$ does not depend on $epsilon $, and $A$ and $B$ are not necessarily disjoint according the definition.
Any comment, idea, or hint would be welcome
probability stochastic-calculus stochastic-integrals stochastic-analysis
asked Jul 25 at 16:24
Ivan
587
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The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential Equations at page 80.
The Ito's formula is this
And my doubt is that I have not understood yet what does the sets $A = A(epsilon, t)$ and $B = B(epsilon, t)$ mean in order to conclude that $A cup B $ equals the set of stopping times of $X$ on $(0,t]$. The definition of these two sets is the following
For me the definition of $ A= A(epsilon, t)$ (as a set that depends on $epsilon $ and $t$) should be: let $A(w) = A(epsilon, t)(w)$ be any finite set of jump times of $X_cdot(w)$ on $(0, t]$. However, this definition and the definition of $B(w) = B(epsilon, t)(w)$ as the set of jump times $s$ on $(0,t]$ such that $sum_s in B(epsilon, t)(w) (Delta X_s (w))^2 leq epsilon^2$, and $A(epsilon, t)(w) cup B(epsilon, t)(w)$ equals the stopping times of $X_cdot(w)$ on $(0,t]$ does not match. Moreover, the definition of $A = A( epsilon, t)$ does not depend on $epsilon $, and $A$ and $B$ are not necessarily disjoint according the definition.
Any comment, idea, or hint would be welcome
probability stochastic-calculus stochastic-integrals stochastic-analysis
asked Jul 25 at 16:24
Ivan
587
I'll give it a shot. Let $C(omega)$ be set of jump times on $[0,t)$. Note that C is countable a.s. by cadlag property (real analysis result). A is not an arbitrary (finite) set, but CB. It should be clear that B depends on $epsilon, t$ and therefore A does as well. The only nontrivial question is whether A is finite. Since C is countable and $sum_0leq s <t (Delta X_s)^2$ converges, for every $epsilon$, $exists N$ s.d. $sum_n> N (Delta X_s_n)^2 leq epsilon^2$. Choose the smallest $N$ then A contains precisely the first $s_1,dots, s_N$ and B contains the rest.
– James Yang
Jul 28 at 3:46
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential Equations at page 80.
The Ito's formula is this
And my doubt is that I have not understood yet what does the sets $A = A(epsilon, t)$ and $B = B(epsilon, t)$ mean in order to conclude that $A cup B $ equals the set of stopping times of $X$ on $(0,t]$. The definition of these two sets is the following
For me the definition of $ A= A(epsilon, t)$ (as a set that depends on $epsilon $ and $t$) should be: let $A(w) = A(epsilon, t)(w)$ be any finite set of jump times of $X_cdot(w)$ on $(0, t]$. However, this definition and the definition of $B(w) = B(epsilon, t)(w)$ as the set of jump times $s$ on $(0,t]$ such that $sum_s in B(epsilon, t)(w) (Delta X_s (w))^2 leq epsilon^2$, and $A(epsilon, t)(w) cup B(epsilon, t)(w)$ equals the stopping times of $X_cdot(w)$ on $(0,t]$ does not match. Moreover, the definition of $A = A( epsilon, t)$ does not depend on $epsilon $, and $A$ and $B$ are not necessarily disjoint according the definition.
Any comment, idea, or hint would be welcome
probability stochastic-calculus stochastic-integrals stochastic-analysis
asked Jul 25 at 16:24
Ivan
587
The following is a question about a notation that Protter uses in the proof of the Ito's formula for cadlag processes of finite variation (FV) that appears on Stochastic Integration and Differential Equations at page 80.
The Ito's formula is this
And my doubt is that I have not understood yet what does the sets $A = A(epsilon, t)$ and $B = B(epsilon, t)$ mean in order to conclude that $A cup B $ equals the set of stopping times of $X$ on $(0,t]$. The definition of these two sets is the following
For me the definition of $ A= A(epsilon, t)$ (as a set that depends on $epsilon $ and $t$) should be: let $A(w) = A(epsilon, t)(w)$ be any finite set of jump times of $X_cdot(w)$ on $(0, t]$. However, this definition and the definition of $B(w) = B(epsilon, t)(w)$ as the set of jump times $s$ on $(0,t]$ such that $sum_s in B(epsilon, t)(w) (Delta X_s (w))^2 leq epsilon^2$, and $A(epsilon, t)(w) cup B(epsilon, t)(w)$ equals the stopping times of $X_cdot(w)$ on $(0,t]$ does not match. Moreover, the definition of $A = A( epsilon, t)$ does not depend on $epsilon $, and $A$ and $B$ are not necessarily disjoint according the definition.
Any comment, idea, or hint would be welcome
probability stochastic-calculus stochastic-integrals stochastic-analysis
asked Jul 25 at 16:24
Ivan
587
edited Jul 25 at 16:36
asked Jul 25 at 16:24
Ivan
587
asked Jul 25 at 16:24
Ivan
587
asked Jul 25 at 16:24
Ivan
587
587
I'll give it a shot. Let $C(omega)$ be set of jump times on $[0,t)$. Note that C is countable a.s. by cadlag property (real analysis result). A is not an arbitrary (finite) set, but CB. It should be clear that B depends on $epsilon, t$ and therefore A does as well. The only nontrivial question is whether A is finite. Since C is countable and $sum_0leq s <t (Delta X_s)^2$ converges, for every $epsilon$, $exists N$ s.d. $sum_n> N (Delta X_s_n)^2 leq epsilon^2$. Choose the smallest $N$ then A contains precisely the first $s_1,dots, s_N$ and B contains the rest.
– James Yang
Jul 28 at 3:46
add a comment |Â
I'll give it a shot. Let $C(omega)$ be set of jump times on $[0,t)$. Note that C is countable a.s. by cadlag property (real analysis result). A is not an arbitrary (finite) set, but CB. It should be clear that B depends on $epsilon, t$ and therefore A does as well. The only nontrivial question is whether A is finite. Since C is countable and $sum_0leq s <t (Delta X_s)^2$ converges, for every $epsilon$, $exists N$ s.d. $sum_n> N (Delta X_s_n)^2 leq epsilon^2$. Choose the smallest $N$ then A contains precisely the first $s_1,dots, s_N$ and B contains the rest.
– James Yang
Jul 28 at 3:46
I'll give it a shot. Let $C(omega)$ be set of jump times on $[0,t)$. Note that C is countable a.s. by cadlag property (real analysis result). A is not an arbitrary (finite) set, but CB. It should be clear that B depends on $epsilon, t$ and therefore A does as well. The only nontrivial question is whether A is finite. Since C is countable and $sum_0leq s <t (Delta X_s)^2$ converges, for every $epsilon$, $exists N$ s.d. $sum_n> N (Delta X_s_n)^2 leq epsilon^2$. Choose the smallest $N$ then A contains precisely the first $s_1,dots, s_N$ and B contains the rest.
– James Yang
Jul 28 at 3:46
I'll give it a shot. Let $C(omega)$ be set of jump times on $[0,t)$. Note that C is countable a.s. by cadlag property (real analysis result). A is not an arbitrary (finite) set, but CB. It should be clear that B depends on $epsilon, t$ and therefore A does as well. The only nontrivial question is whether A is finite. Since C is countable and $sum_0leq s <t (Delta X_s)^2$ converges, for every $epsilon$, $exists N$ s.d. $sum_n> N (Delta X_s_n)^2 leq epsilon^2$. Choose the smallest $N$ then A contains precisely the first $s_1,dots, s_N$ and B contains the rest.
– James Yang
Jul 28 at 3:46
add a comment |Â
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I'll give it a shot. Let $C(omega)$ be set of jump times on $[0,t)$. Note that C is countable a.s. by cadlag property (real analysis result). A is not an arbitrary (finite) set, but CB. It should be clear that B depends on $epsilon, t$ and therefore A does as well. The only nontrivial question is whether A is finite. Since C is countable and $sum_0leq s <t (Delta X_s)^2$ converges, for every $epsilon$, $exists N$ s.d. $sum_n> N (Delta X_s_n)^2 leq epsilon^2$. Choose the smallest $N$ then A contains precisely the first $s_1,dots, s_N$ and B contains the rest.
– James Yang
Jul 28 at 3:46