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Expected exit times from intervals of diffusions using scale functions, speed measures and Green's function
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To compute expected exit times from intervals of the form $(a,b)$ for one dimensional diffusions $X = (X_t)_t geq 0$ started at $x in (a,b)$, one can use that
$$
E tau_a,b
=
int_a^b G_a,b(x,y) m(dy)
\
=
- int_a^x left( p(x) - p(y) right) m(dy)
+ fracp(x) - p(a)p(b) - p(a)
int_a^b (p(b) - p(y)) m(dy),
$$
where $p(x)$ is the scale function and $m(x)$ is the speed measure of the diffusion $X$, and $tau_a,b = inf t geq 0 :: X_t notin (a,b) $, with Green's function being defined as
$$
G_a,b(x,y)
=
frac
( p(x wedge y) - p(a) ) ( p(b) - p(x vee y )
p(b) - p(a) .
$$
This is from [1] p. 343.
For example, if $X$ is a standard Brownian motion, then the above yields the well-known identity $E tau_a,b = (b-x)(x-a)$. Here the scale function is $p(x) = x$ and the speed measure is $m(dx) = 2dx$.
I would like to use Green's function found in [2]. For the Brownian motion, using [2], p. 119, Green's function is
$$
G_alpha(x,y) = w_alpha^-1 e^- sqrt2 alpha x e^sqrt2 alpha y,
qquad x geq y,
$$
where $w_alpha = 2 sqrt2 alpha$ is the Wronskian, and I can compute
$$
int_a^b G_alpha(x,y) m(dy)
= frac
left( -e^sqrt2 alpha a + e^sqrt2 alpha b right) e^- sqrt2 alpha x
2 alpha,
$$
but this is not something I recognize.
What is the relationship between Green's function in [1] (which yields the desired result) and Green's function in [2]? How may I use Green's function in [2] to compute $E tau_a,b$ and what is the role of $alpha$?
[1] Karatzas & Shreve: Brownian Motion and Stochastic Calculus (1998).
[2] Borodin & Salminen: Handbook of Brownian Motion Facts and Formulae (2002).
Edit:
According to John Dawkins, we should use the correct Green function below:
$$
G_alpha(x,y)
= w_alpha^-1 sinh ((b-x)sqrt2 alpha) sinh ((y-a) sqrt2 alpha),
qquad b > x geq y > a,
$$
where $w_alpha = sqrt2 alpha sinh((b-a) sqrt2 alpha)$ is the Wronskian. Then we let
$$
G_0(x,y) := lim_alpha to 0 G_alpha(x,y)
=
fracab - ax - by + xya-b, qquad b > x geq y > a
$$
and compute
$$
int_a^b 2 G_0(x,y) dy
= frac(b-x)(a-x)^2-(a-b),
$$
which is not quite $(b-x)(x-a)$. Where did I go wrong?
probability probability-theory stochastic-processes expectation brownian-motion
asked Jul 16 at 12:16
K. Brix
4661312
add a comment |Â
up vote
2
down vote
favorite
To compute expected exit times from intervals of the form $(a,b)$ for one dimensional diffusions $X = (X_t)_t geq 0$ started at $x in (a,b)$, one can use that
$$
E tau_a,b
=
int_a^b G_a,b(x,y) m(dy)
\
=
- int_a^x left( p(x) - p(y) right) m(dy)
+ fracp(x) - p(a)p(b) - p(a)
int_a^b (p(b) - p(y)) m(dy),
$$
where $p(x)$ is the scale function and $m(x)$ is the speed measure of the diffusion $X$, and $tau_a,b = inf t geq 0 :: X_t notin (a,b) $, with Green's function being defined as
$$
G_a,b(x,y)
=
frac
( p(x wedge y) - p(a) ) ( p(b) - p(x vee y )
p(b) - p(a) .
$$
This is from [1] p. 343.
For example, if $X$ is a standard Brownian motion, then the above yields the well-known identity $E tau_a,b = (b-x)(x-a)$. Here the scale function is $p(x) = x$ and the speed measure is $m(dx) = 2dx$.
I would like to use Green's function found in [2]. For the Brownian motion, using [2], p. 119, Green's function is
$$
G_alpha(x,y) = w_alpha^-1 e^- sqrt2 alpha x e^sqrt2 alpha y,
qquad x geq y,
$$
where $w_alpha = 2 sqrt2 alpha$ is the Wronskian, and I can compute
$$
int_a^b G_alpha(x,y) m(dy)
= frac
left( -e^sqrt2 alpha a + e^sqrt2 alpha b right) e^- sqrt2 alpha x
2 alpha,
$$
but this is not something I recognize.
What is the relationship between Green's function in [1] (which yields the desired result) and Green's function in [2]? How may I use Green's function in [2] to compute $E tau_a,b$ and what is the role of $alpha$?
[1] Karatzas & Shreve: Brownian Motion and Stochastic Calculus (1998).
[2] Borodin & Salminen: Handbook of Brownian Motion Facts and Formulae (2002).
Edit:
According to John Dawkins, we should use the correct Green function below:
$$
G_alpha(x,y)
= w_alpha^-1 sinh ((b-x)sqrt2 alpha) sinh ((y-a) sqrt2 alpha),
qquad b > x geq y > a,
$$
where $w_alpha = sqrt2 alpha sinh((b-a) sqrt2 alpha)$ is the Wronskian. Then we let
$$
G_0(x,y) := lim_alpha to 0 G_alpha(x,y)
=
fracab - ax - by + xya-b, qquad b > x geq y > a
$$
and compute
$$
int_a^b 2 G_0(x,y) dy
= frac(b-x)(a-x)^2-(a-b),
$$
which is not quite $(b-x)(x-a)$. Where did I go wrong?
probability probability-theory stochastic-processes expectation brownian-motion
asked Jul 16 at 12:16
K. Brix
4661312
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
To compute expected exit times from intervals of the form $(a,b)$ for one dimensional diffusions $X = (X_t)_t geq 0$ started at $x in (a,b)$, one can use that
$$
E tau_a,b
=
int_a^b G_a,b(x,y) m(dy)
\
=
- int_a^x left( p(x) - p(y) right) m(dy)
+ fracp(x) - p(a)p(b) - p(a)
int_a^b (p(b) - p(y)) m(dy),
$$
where $p(x)$ is the scale function and $m(x)$ is the speed measure of the diffusion $X$, and $tau_a,b = inf t geq 0 :: X_t notin (a,b) $, with Green's function being defined as
$$
G_a,b(x,y)
=
frac
( p(x wedge y) - p(a) ) ( p(b) - p(x vee y )
p(b) - p(a) .
$$
This is from [1] p. 343.
For example, if $X$ is a standard Brownian motion, then the above yields the well-known identity $E tau_a,b = (b-x)(x-a)$. Here the scale function is $p(x) = x$ and the speed measure is $m(dx) = 2dx$.
I would like to use Green's function found in [2]. For the Brownian motion, using [2], p. 119, Green's function is
$$
G_alpha(x,y) = w_alpha^-1 e^- sqrt2 alpha x e^sqrt2 alpha y,
qquad x geq y,
$$
where $w_alpha = 2 sqrt2 alpha$ is the Wronskian, and I can compute
$$
int_a^b G_alpha(x,y) m(dy)
= frac
left( -e^sqrt2 alpha a + e^sqrt2 alpha b right) e^- sqrt2 alpha x
2 alpha,
$$
but this is not something I recognize.
What is the relationship between Green's function in [1] (which yields the desired result) and Green's function in [2]? How may I use Green's function in [2] to compute $E tau_a,b$ and what is the role of $alpha$?
[1] Karatzas & Shreve: Brownian Motion and Stochastic Calculus (1998).
[2] Borodin & Salminen: Handbook of Brownian Motion Facts and Formulae (2002).
Edit:
According to John Dawkins, we should use the correct Green function below:
$$
G_alpha(x,y)
= w_alpha^-1 sinh ((b-x)sqrt2 alpha) sinh ((y-a) sqrt2 alpha),
qquad b > x geq y > a,
$$
where $w_alpha = sqrt2 alpha sinh((b-a) sqrt2 alpha)$ is the Wronskian. Then we let
$$
G_0(x,y) := lim_alpha to 0 G_alpha(x,y)
=
fracab - ax - by + xya-b, qquad b > x geq y > a
$$
and compute
$$
int_a^b 2 G_0(x,y) dy
= frac(b-x)(a-x)^2-(a-b),
$$
which is not quite $(b-x)(x-a)$. Where did I go wrong?
probability probability-theory stochastic-processes expectation brownian-motion
asked Jul 16 at 12:16
K. Brix
4661312
To compute expected exit times from intervals of the form $(a,b)$ for one dimensional diffusions $X = (X_t)_t geq 0$ started at $x in (a,b)$, one can use that
$$
E tau_a,b
=
int_a^b G_a,b(x,y) m(dy)
\
=
- int_a^x left( p(x) - p(y) right) m(dy)
+ fracp(x) - p(a)p(b) - p(a)
int_a^b (p(b) - p(y)) m(dy),
$$
where $p(x)$ is the scale function and $m(x)$ is the speed measure of the diffusion $X$, and $tau_a,b = inf t geq 0 :: X_t notin (a,b) $, with Green's function being defined as
$$
G_a,b(x,y)
=
frac
( p(x wedge y) - p(a) ) ( p(b) - p(x vee y )
p(b) - p(a) .
$$
This is from [1] p. 343.
For example, if $X$ is a standard Brownian motion, then the above yields the well-known identity $E tau_a,b = (b-x)(x-a)$. Here the scale function is $p(x) = x$ and the speed measure is $m(dx) = 2dx$.
I would like to use Green's function found in [2]. For the Brownian motion, using [2], p. 119, Green's function is
$$
G_alpha(x,y) = w_alpha^-1 e^- sqrt2 alpha x e^sqrt2 alpha y,
qquad x geq y,
$$
where $w_alpha = 2 sqrt2 alpha$ is the Wronskian, and I can compute
$$
int_a^b G_alpha(x,y) m(dy)
= frac
left( -e^sqrt2 alpha a + e^sqrt2 alpha b right) e^- sqrt2 alpha x
2 alpha,
$$
but this is not something I recognize.
What is the relationship between Green's function in [1] (which yields the desired result) and Green's function in [2]? How may I use Green's function in [2] to compute $E tau_a,b$ and what is the role of $alpha$?
[1] Karatzas & Shreve: Brownian Motion and Stochastic Calculus (1998).
[2] Borodin & Salminen: Handbook of Brownian Motion Facts and Formulae (2002).
Edit:
According to John Dawkins, we should use the correct Green function below:
$$
G_alpha(x,y)
= w_alpha^-1 sinh ((b-x)sqrt2 alpha) sinh ((y-a) sqrt2 alpha),
qquad b > x geq y > a,
$$
where $w_alpha = sqrt2 alpha sinh((b-a) sqrt2 alpha)$ is the Wronskian. Then we let
$$
G_0(x,y) := lim_alpha to 0 G_alpha(x,y)
=
fracab - ax - by + xya-b, qquad b > x geq y > a
$$
and compute
$$
int_a^b 2 G_0(x,y) dy
= frac(b-x)(a-x)^2-(a-b),
$$
which is not quite $(b-x)(x-a)$. Where did I go wrong?
probability probability-theory stochastic-processes expectation brownian-motion
asked Jul 16 at 12:16
K. Brix
4661312
edited Jul 19 at 15:37
asked Jul 16 at 12:16
K. Brix
4661312
asked Jul 16 at 12:16
K. Brix
4661312
asked Jul 16 at 12:16
K. Brix
4661312
4661312
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
The formula you have taken from [2] is for the $alpha$-order Green function, the density of the $alpha$-resolvent with respect to $m$, for Brownian motion on all of $Bbb R$. What you want is the formula in [2], in the later section 6, for "Brownian motion on $(a,b)$ killed at $a$ or $b$", on page 137 in the Second Edition. Let $alpha$ go to $0$ in that formula and you'll get the same formula as in [1].
answered Jul 18 at 20:56
John Dawkins
12.5k1917
Hi @JohnDawkins, thank you for your help. I have made an edit following your input. Can you take a look at it? Where did I go wrong? Is it some computation error on my part?
– K. Brix
Jul 19 at 15:39
1
My computattion of $G_0$ gives $(b-x)(y-a)/(b-a)$, $b>xge y > a$ by using the fact that $sinh(u) = u + o(u^2)$ as $uto 0$.
– John Dawkins
Jul 19 at 16:46
Hi again John, thank you for your patience and clearing up my confusion, I now get the desired result. Cheers!
– K. Brix
Jul 20 at 8:07
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The formula you have taken from [2] is for the $alpha$-order Green function, the density of the $alpha$-resolvent with respect to $m$, for Brownian motion on all of $Bbb R$. What you want is the formula in [2], in the later section 6, for "Brownian motion on $(a,b)$ killed at $a$ or $b$", on page 137 in the Second Edition. Let $alpha$ go to $0$ in that formula and you'll get the same formula as in [1].
answered Jul 18 at 20:56
John Dawkins
12.5k1917
Hi @JohnDawkins, thank you for your help. I have made an edit following your input. Can you take a look at it? Where did I go wrong? Is it some computation error on my part?
– K. Brix
Jul 19 at 15:39
1
My computattion of $G_0$ gives $(b-x)(y-a)/(b-a)$, $b>xge y > a$ by using the fact that $sinh(u) = u + o(u^2)$ as $uto 0$.
– John Dawkins
Jul 19 at 16:46
Hi again John, thank you for your patience and clearing up my confusion, I now get the desired result. Cheers!
– K. Brix
Jul 20 at 8:07
add a comment |Â
up vote
1
down vote
accepted
The formula you have taken from [2] is for the $alpha$-order Green function, the density of the $alpha$-resolvent with respect to $m$, for Brownian motion on all of $Bbb R$. What you want is the formula in [2], in the later section 6, for "Brownian motion on $(a,b)$ killed at $a$ or $b$", on page 137 in the Second Edition. Let $alpha$ go to $0$ in that formula and you'll get the same formula as in [1].
answered Jul 18 at 20:56
John Dawkins
12.5k1917
Hi @JohnDawkins, thank you for your help. I have made an edit following your input. Can you take a look at it? Where did I go wrong? Is it some computation error on my part?
– K. Brix
Jul 19 at 15:39
1
My computattion of $G_0$ gives $(b-x)(y-a)/(b-a)$, $b>xge y > a$ by using the fact that $sinh(u) = u + o(u^2)$ as $uto 0$.
– John Dawkins
Jul 19 at 16:46
Hi again John, thank you for your patience and clearing up my confusion, I now get the desired result. Cheers!
– K. Brix
Jul 20 at 8:07
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The formula you have taken from [2] is for the $alpha$-order Green function, the density of the $alpha$-resolvent with respect to $m$, for Brownian motion on all of $Bbb R$. What you want is the formula in [2], in the later section 6, for "Brownian motion on $(a,b)$ killed at $a$ or $b$", on page 137 in the Second Edition. Let $alpha$ go to $0$ in that formula and you'll get the same formula as in [1].
answered Jul 18 at 20:56
John Dawkins
12.5k1917
The formula you have taken from [2] is for the $alpha$-order Green function, the density of the $alpha$-resolvent with respect to $m$, for Brownian motion on all of $Bbb R$. What you want is the formula in [2], in the later section 6, for "Brownian motion on $(a,b)$ killed at $a$ or $b$", on page 137 in the Second Edition. Let $alpha$ go to $0$ in that formula and you'll get the same formula as in [1].
answered Jul 18 at 20:56
John Dawkins
12.5k1917
answered Jul 18 at 20:56
John Dawkins
12.5k1917
answered Jul 18 at 20:56
John Dawkins
12.5k1917
answered Jul 18 at 20:56
John Dawkins
12.5k1917
12.5k1917
Hi @JohnDawkins, thank you for your help. I have made an edit following your input. Can you take a look at it? Where did I go wrong? Is it some computation error on my part?
– K. Brix
Jul 19 at 15:39
1
My computattion of $G_0$ gives $(b-x)(y-a)/(b-a)$, $b>xge y > a$ by using the fact that $sinh(u) = u + o(u^2)$ as $uto 0$.
– John Dawkins
Jul 19 at 16:46
Hi again John, thank you for your patience and clearing up my confusion, I now get the desired result. Cheers!
– K. Brix
Jul 20 at 8:07
add a comment |Â
Hi @JohnDawkins, thank you for your help. I have made an edit following your input. Can you take a look at it? Where did I go wrong? Is it some computation error on my part?
– K. Brix
Jul 19 at 15:39
1
My computattion of $G_0$ gives $(b-x)(y-a)/(b-a)$, $b>xge y > a$ by using the fact that $sinh(u) = u + o(u^2)$ as $uto 0$.
– John Dawkins
Jul 19 at 16:46
Hi again John, thank you for your patience and clearing up my confusion, I now get the desired result. Cheers!
– K. Brix
Jul 20 at 8:07
Hi @JohnDawkins, thank you for your help. I have made an edit following your input. Can you take a look at it? Where did I go wrong? Is it some computation error on my part?
– K. Brix
Jul 19 at 15:39
Hi @JohnDawkins, thank you for your help. I have made an edit following your input. Can you take a look at it? Where did I go wrong? Is it some computation error on my part?
– K. Brix
Jul 19 at 15:39
1
1
My computattion of $G_0$ gives $(b-x)(y-a)/(b-a)$, $b>xge y > a$ by using the fact that $sinh(u) = u + o(u^2)$ as $uto 0$.
– John Dawkins
Jul 19 at 16:46
My computattion of $G_0$ gives $(b-x)(y-a)/(b-a)$, $b>xge y > a$ by using the fact that $sinh(u) = u + o(u^2)$ as $uto 0$.
– John Dawkins
Jul 19 at 16:46
Hi again John, thank you for your patience and clearing up my confusion, I now get the desired result. Cheers!
– K. Brix
Jul 20 at 8:07
Hi again John, thank you for your patience and clearing up my confusion, I now get the desired result. Cheers!
– K. Brix
Jul 20 at 8:07
add a comment |Â
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