When given expected value is finite?
$begingroup$
I consider the stochastic process $(X_t)_{tgeq 0}$ that satisfies:
$$
dX_t = a(t, X_t) dt + b(t, X_t) dB_t.
$$
We assume that solution of the above SDE is unique non-exploding solution, e.g. $(X_t)_{tgeq 0}$ can be a Brownian Motion $left(a(t, X_t)=0 text{ and } b(t, X_t) = 1right)$ or geometric Brownian motion $left(a(t, X_t)=r X_t text{ and } b(t, X_t) = sigma X_tright)$.
Let us consider the expected value
$$
E[e^{-int_t^T X_sds}].
$$
I wonder when this expected value is finite.
I suppose that for some functions $a(t, X_t)$ and $b(t, X_t)$ I can encounter inifite expected value (I suppose the problem arises when $(X_t)_{tgeq 0}$ will be 'very' negative).
My question is: Are there any necessary conditions on $a(t, X_t)$ and $b(t, X_t)$ which guarantee that the above expectation is finite.
probability-theory stochastic-processes expected-value
asked Jan 7 at 9:39
MathMenMathMen
1267
$endgroup$
add a comment |
$begingroup$
I consider the stochastic process $(X_t)_{tgeq 0}$ that satisfies:
$$
dX_t = a(t, X_t) dt + b(t, X_t) dB_t.
$$
We assume that solution of the above SDE is unique non-exploding solution, e.g. $(X_t)_{tgeq 0}$ can be a Brownian Motion $left(a(t, X_t)=0 text{ and } b(t, X_t) = 1right)$ or geometric Brownian motion $left(a(t, X_t)=r X_t text{ and } b(t, X_t) = sigma X_tright)$.
Let us consider the expected value
$$
E[e^{-int_t^T X_sds}].
$$
I wonder when this expected value is finite.
I suppose that for some functions $a(t, X_t)$ and $b(t, X_t)$ I can encounter inifite expected value (I suppose the problem arises when $(X_t)_{tgeq 0}$ will be 'very' negative).
My question is: Are there any necessary conditions on $a(t, X_t)$ and $b(t, X_t)$ which guarantee that the above expectation is finite.
probability-theory stochastic-processes expected-value
asked Jan 7 at 9:39
MathMenMathMen
1267
$endgroup$
add a comment |
$begingroup$
I consider the stochastic process $(X_t)_{tgeq 0}$ that satisfies:
$$
dX_t = a(t, X_t) dt + b(t, X_t) dB_t.
$$
We assume that solution of the above SDE is unique non-exploding solution, e.g. $(X_t)_{tgeq 0}$ can be a Brownian Motion $left(a(t, X_t)=0 text{ and } b(t, X_t) = 1right)$ or geometric Brownian motion $left(a(t, X_t)=r X_t text{ and } b(t, X_t) = sigma X_tright)$.
Let us consider the expected value
$$
E[e^{-int_t^T X_sds}].
$$
I wonder when this expected value is finite.
I suppose that for some functions $a(t, X_t)$ and $b(t, X_t)$ I can encounter inifite expected value (I suppose the problem arises when $(X_t)_{tgeq 0}$ will be 'very' negative).
My question is: Are there any necessary conditions on $a(t, X_t)$ and $b(t, X_t)$ which guarantee that the above expectation is finite.
probability-theory stochastic-processes expected-value
asked Jan 7 at 9:39
MathMenMathMen
1267
$endgroup$
I consider the stochastic process $(X_t)_{tgeq 0}$ that satisfies:
$$
dX_t = a(t, X_t) dt + b(t, X_t) dB_t.
$$
We assume that solution of the above SDE is unique non-exploding solution, e.g. $(X_t)_{tgeq 0}$ can be a Brownian Motion $left(a(t, X_t)=0 text{ and } b(t, X_t) = 1right)$ or geometric Brownian motion $left(a(t, X_t)=r X_t text{ and } b(t, X_t) = sigma X_tright)$.
Let us consider the expected value
$$
E[e^{-int_t^T X_sds}].
$$
I wonder when this expected value is finite.
I suppose that for some functions $a(t, X_t)$ and $b(t, X_t)$ I can encounter inifite expected value (I suppose the problem arises when $(X_t)_{tgeq 0}$ will be 'very' negative).
My question is: Are there any necessary conditions on $a(t, X_t)$ and $b(t, X_t)$ which guarantee that the above expectation is finite.
probability-theory stochastic-processes expected-value
probability-theory stochastic-processes expected-value
asked Jan 7 at 9:39
MathMenMathMen
1267
asked Jan 7 at 9:39
MathMenMathMen
1267
asked Jan 7 at 9:39
MathMenMathMen
1267
asked Jan 7 at 9:39
MathMenMathMen
1267
asked Jan 7 at 9:39
MathMenMathMen
1267
1267
add a comment |
add a comment |
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