Definition of a “bounded” stochastic process (Bass, Protter)

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I am currently reading through Protter's 'stochastic integration and differential equations' and Bass' 'stochastic processes' books.



Both seem to make use of "bounded" stochastic processes. I am unsure of whether this means



1) there exists $K > 0$ such that for each $(t, omega) in [0,infty) times Omega$, $|X(t,omega)| < K$



or



2) for each $omega in Omega$, there exists $K > 0$ such that for each $t in [0,infty)$, $|X(t, omega)| <K$.



I could not find the definition in either of these books, so I could be missing it, or it could be assumed to be known? If you know a page number then that would be great.



I suspect that definition 1) is the case, but I just wanted to make sure because I want to show that "a bounded increasing cadlag process is a submartingale" (which I think should be adapted as well?) (Bass pg 130)
Bass then goes on to use the Doob Meyer Decomposition on this process, but I think that it should be of class D. I think it would be easier to show it is of class D if the definition of "bounded" was that of 1).




stochastic-processes stochastic-analysis




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  • 1




    There is no common definition for "bounded process", I think; the definition varies from book to book. In this particular case you are mentioning I agree that 1) is more reasonable. Note that 2) does not ensure integrability of $X_t$ which is clearly a necessary ingredient for $X_t$ to be a submartingale.
    – saz
    Aug 19 at 9:07











  • Thanks saz! I will emphasize this in the notes that I am writing up so that it is explicit for me in the future.
    – Ceeerson
    Aug 19 at 9:12














up vote
1
down vote

favorite












I am currently reading through Protter's 'stochastic integration and differential equations' and Bass' 'stochastic processes' books.



Both seem to make use of "bounded" stochastic processes. I am unsure of whether this means



1) there exists $K > 0$ such that for each $(t, omega) in [0,infty) times Omega$, $|X(t,omega)| < K$



or



2) for each $omega in Omega$, there exists $K > 0$ such that for each $t in [0,infty)$, $|X(t, omega)| <K$.



I could not find the definition in either of these books, so I could be missing it, or it could be assumed to be known? If you know a page number then that would be great.



I suspect that definition 1) is the case, but I just wanted to make sure because I want to show that "a bounded increasing cadlag process is a submartingale" (which I think should be adapted as well?) (Bass pg 130)
Bass then goes on to use the Doob Meyer Decomposition on this process, but I think that it should be of class D. I think it would be easier to show it is of class D if the definition of "bounded" was that of 1).




stochastic-processes stochastic-analysis




share|cite|improve this question


















  • 1




    There is no common definition for "bounded process", I think; the definition varies from book to book. In this particular case you are mentioning I agree that 1) is more reasonable. Note that 2) does not ensure integrability of $X_t$ which is clearly a necessary ingredient for $X_t$ to be a submartingale.
    – saz
    Aug 19 at 9:07











  • Thanks saz! I will emphasize this in the notes that I am writing up so that it is explicit for me in the future.
    – Ceeerson
    Aug 19 at 9:12












up vote
1
down vote

favorite









up vote
1
down vote

favorite











I am currently reading through Protter's 'stochastic integration and differential equations' and Bass' 'stochastic processes' books.



Both seem to make use of "bounded" stochastic processes. I am unsure of whether this means



1) there exists $K > 0$ such that for each $(t, omega) in [0,infty) times Omega$, $|X(t,omega)| < K$



or



2) for each $omega in Omega$, there exists $K > 0$ such that for each $t in [0,infty)$, $|X(t, omega)| <K$.



I could not find the definition in either of these books, so I could be missing it, or it could be assumed to be known? If you know a page number then that would be great.



I suspect that definition 1) is the case, but I just wanted to make sure because I want to show that "a bounded increasing cadlag process is a submartingale" (which I think should be adapted as well?) (Bass pg 130)
Bass then goes on to use the Doob Meyer Decomposition on this process, but I think that it should be of class D. I think it would be easier to show it is of class D if the definition of "bounded" was that of 1).




stochastic-processes stochastic-analysis




share|cite|improve this question














I am currently reading through Protter's 'stochastic integration and differential equations' and Bass' 'stochastic processes' books.



Both seem to make use of "bounded" stochastic processes. I am unsure of whether this means



1) there exists $K > 0$ such that for each $(t, omega) in [0,infty) times Omega$, $|X(t,omega)| < K$



or



2) for each $omega in Omega$, there exists $K > 0$ such that for each $t in [0,infty)$, $|X(t, omega)| <K$.



I could not find the definition in either of these books, so I could be missing it, or it could be assumed to be known? If you know a page number then that would be great.



I suspect that definition 1) is the case, but I just wanted to make sure because I want to show that "a bounded increasing cadlag process is a submartingale" (which I think should be adapted as well?) (Bass pg 130)
Bass then goes on to use the Doob Meyer Decomposition on this process, but I think that it should be of class D. I think it would be easier to show it is of class D if the definition of "bounded" was that of 1).





stochastic-processes stochastic-analysis





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 19 at 9:34

























asked Aug 19 at 8:40









Ceeerson

434




434







  • 1




    There is no common definition for "bounded process", I think; the definition varies from book to book. In this particular case you are mentioning I agree that 1) is more reasonable. Note that 2) does not ensure integrability of $X_t$ which is clearly a necessary ingredient for $X_t$ to be a submartingale.
    – saz
    Aug 19 at 9:07











  • Thanks saz! I will emphasize this in the notes that I am writing up so that it is explicit for me in the future.
    – Ceeerson
    Aug 19 at 9:12












  • 1




    There is no common definition for "bounded process", I think; the definition varies from book to book. In this particular case you are mentioning I agree that 1) is more reasonable. Note that 2) does not ensure integrability of $X_t$ which is clearly a necessary ingredient for $X_t$ to be a submartingale.
    – saz
    Aug 19 at 9:07











  • Thanks saz! I will emphasize this in the notes that I am writing up so that it is explicit for me in the future.
    – Ceeerson
    Aug 19 at 9:12







1




1




There is no common definition for "bounded process", I think; the definition varies from book to book. In this particular case you are mentioning I agree that 1) is more reasonable. Note that 2) does not ensure integrability of $X_t$ which is clearly a necessary ingredient for $X_t$ to be a submartingale.
– saz
Aug 19 at 9:07





There is no common definition for "bounded process", I think; the definition varies from book to book. In this particular case you are mentioning I agree that 1) is more reasonable. Note that 2) does not ensure integrability of $X_t$ which is clearly a necessary ingredient for $X_t$ to be a submartingale.
– saz
Aug 19 at 9:07













Thanks saz! I will emphasize this in the notes that I am writing up so that it is explicit for me in the future.
– Ceeerson
Aug 19 at 9:12




Thanks saz! I will emphasize this in the notes that I am writing up so that it is explicit for me in the future.
– Ceeerson
Aug 19 at 9:12















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