Mixing
Integral as: martingale or local martingale
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I wonder when a stochastic integral is a martingale or a local martingale. Let's assume that we have a process:
$X_t = X_0 + int_{0}^{t} a_s ds + int_0^t b_s dW_s$
Is this kind of integral a martingale or local martingale. Is this connected with the integral $int_{0}^{t} a_s ds$? (if it is present, the process is a martigale and in the opposite situation it is a local martingale?). What would happen if we have some proces $Y_t$ and our $X_t$ looks like that:
$X_t = X_0 + int_{0}^{t} f(Y_s) ds + int_0^t f(Y_s) dW_s$
I would love to read something more about those properties. I will appreciate any link/paper related to this topic.
probability stochastic-processes stochastic-calculus martingales stochastic-integrals
asked Jan 31 at 22:00
FNTEFNTE
1276
$endgroup$
add a comment |
$begingroup$
I wonder when a stochastic integral is a martingale or a local martingale. Let's assume that we have a process:
$X_t = X_0 + int_{0}^{t} a_s ds + int_0^t b_s dW_s$
Is this kind of integral a martingale or local martingale. Is this connected with the integral $int_{0}^{t} a_s ds$? (if it is present, the process is a martigale and in the opposite situation it is a local martingale?). What would happen if we have some proces $Y_t$ and our $X_t$ looks like that:
$X_t = X_0 + int_{0}^{t} f(Y_s) ds + int_0^t f(Y_s) dW_s$
I would love to read something more about those properties. I will appreciate any link/paper related to this topic.
probability stochastic-processes stochastic-calculus martingales stochastic-integrals
asked Jan 31 at 22:00
FNTEFNTE
1276
$endgroup$
$begingroup$
What if $a_s=s$?
$endgroup$
– d.k.o.
Jan 31 at 22:42
$begingroup$
Your "first" process $X_t$ is a martingale iff $a:=0$.
$endgroup$
– saz
Feb 1 at 7:10
add a comment |
$begingroup$
I wonder when a stochastic integral is a martingale or a local martingale. Let's assume that we have a process:
$X_t = X_0 + int_{0}^{t} a_s ds + int_0^t b_s dW_s$
Is this kind of integral a martingale or local martingale. Is this connected with the integral $int_{0}^{t} a_s ds$? (if it is present, the process is a martigale and in the opposite situation it is a local martingale?). What would happen if we have some proces $Y_t$ and our $X_t$ looks like that:
$X_t = X_0 + int_{0}^{t} f(Y_s) ds + int_0^t f(Y_s) dW_s$
I would love to read something more about those properties. I will appreciate any link/paper related to this topic.
probability stochastic-processes stochastic-calculus martingales stochastic-integrals
asked Jan 31 at 22:00
FNTEFNTE
1276
$endgroup$
I wonder when a stochastic integral is a martingale or a local martingale. Let's assume that we have a process:
$X_t = X_0 + int_{0}^{t} a_s ds + int_0^t b_s dW_s$
Is this kind of integral a martingale or local martingale. Is this connected with the integral $int_{0}^{t} a_s ds$? (if it is present, the process is a martigale and in the opposite situation it is a local martingale?). What would happen if we have some proces $Y_t$ and our $X_t$ looks like that:
$X_t = X_0 + int_{0}^{t} f(Y_s) ds + int_0^t f(Y_s) dW_s$
I would love to read something more about those properties. I will appreciate any link/paper related to this topic.
probability stochastic-processes stochastic-calculus martingales stochastic-integrals
probability stochastic-processes stochastic-calculus martingales stochastic-integrals
asked Jan 31 at 22:00
FNTEFNTE
1276
asked Jan 31 at 22:00
FNTEFNTE
1276
asked Jan 31 at 22:00
FNTEFNTE
1276
asked Jan 31 at 22:00
FNTEFNTE
1276
asked Jan 31 at 22:00
FNTEFNTE
1276
1276
$begingroup$
What if $a_s=s$?
$endgroup$
– d.k.o.
Jan 31 at 22:42
$begingroup$
Your "first" process $X_t$ is a martingale iff $a:=0$.
$endgroup$
– saz
Feb 1 at 7:10
add a comment |
$begingroup$
What if $a_s=s$?
$endgroup$
– d.k.o.
Jan 31 at 22:42
$begingroup$
Your "first" process $X_t$ is a martingale iff $a:=0$.
$endgroup$
– saz
Feb 1 at 7:10
$begingroup$
What if $a_s=s$?
$endgroup$
– d.k.o.
Jan 31 at 22:42
$begingroup$
What if $a_s=s$?
$endgroup$
– d.k.o.
Jan 31 at 22:42
$begingroup$
Your "first" process $X_t$ is a martingale iff $a:=0$.
$endgroup$
– saz
Feb 1 at 7:10
$begingroup$
Your "first" process $X_t$ is a martingale iff $a:=0$.
$endgroup$
– saz
Feb 1 at 7:10
add a comment |
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$begingroup$
What if $a_s=s$?
$endgroup$
– d.k.o.
Jan 31 at 22:42
$begingroup$
Your "first" process $X_t$ is a martingale iff $a:=0$.
$endgroup$
– saz
Feb 1 at 7:10