OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Notes on Schrodinger’s equation, Bessel bridgesand first-passage time problems

Gerardo Hernandez-del-Valle

March 3, 2010

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Overview of the first passage time problem

Integral equationsDiscrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

PDE approachGirsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Derivation of the density?Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

The problem

Given that X is a one-dimensional Markov process and f is a twicedifferentiable function. Let

T := inf {t ≥ 0|Xt = f (t)}

be the first time that X reaches the moving boundary f .

I What is the density ϕ of T ?

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

The problem

Given that X is a one-dimensional Markov process and f is a twicedifferentiable function. Let

T := inf {t ≥ 0|Xt = f (t)}

be the first time that X reaches the moving boundary f .

I What is the density ϕ of T ?

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Let (Xn)n≥0 be discrete-time, homogeneous Markov process. Thenthe Chapman-Kolmogorov equation describes:

0 2 4 6 8 10

02

46

810

x

y

z

k n

Px(Xn = z) =∑y

P(Xn−k = z)Px(Xk = y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Let (Xn)n≥0 be discrete-time, homogeneous Markov process. Thenthe Chapman-Kolmogorov equation describes:

0 2 4 6 8 10

02

46

810

x

y

z

k n

Px(Xn = z) =∑y

P(Xn−k = z)Px(Xk = y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Let (Xn)n≥0 be discrete-time, homogeneous Markov process. Thenthe Chapman-Kolmogorov equation describes:

0 2 4 6 8 10

02

46

810

x

y

z

k n

Px(Xn = z) =∑y

P(Xn−k = z)Px(Xk = y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Chapman-Kolmogorov and first passage time τ

If (Xn)n≥0 is a discrete-time, homogeneous Markov process (takingvalues in a countable setS); x and z be given and fixed in S ,g : N→ S , and

τ := inf{k ≥ 1|Xk = g(k)}

be the first-passage time of X over g . Then by theChapman-Kolmogorov equation it follows that:

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Chapman-Kolmogorov and first passage time τ

If (Xn)n≥0 is a discrete-time, homogeneous Markov process (takingvalues in a countable setS); x and z be given and fixed in S ,g : N→ S , and

τ := inf{k ≥ 1|Xk = g(k)}

be the first-passage time of X over g . Then by theChapman-Kolmogorov equation it follows that:

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Chapman-Kolmogorov and first passage time τ (cont.)

0 2 4 6 8 10

02

46

810

x

gz

k n

Px(Xn = z) =n∑

k=1

Pg(k)(Xn−k = z)Px(τ = k).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Chapman-Kolmogorov and first passage time τ (cont.)

0 2 4 6 8 10

02

46

810

x

gz

k n

Px(Xn = z) =n∑

k=1

Pg(k)(Xn−k = z)Px(τ = k).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Chapman-Kolmogorov and first passage time τ (cont.)

0 2 4 6 8 10

02

46

810

x

gz

k n

Px(Xn = z) =n∑

k=1

Pg(k)(Xn−k = z)Px(τ = k).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Theorem. Let (Xt)t≥0 be a strong, time-homogenous Markovprocess with continuous sample paths started at x , letg : (0,∞)→ R be a continuous function satisfying g(0+) ≥ x ,and

τ := inf{t > 0|Xt ≥ g(t)}

be the first-passage time of X over g , and let F = Fx denote thedistribution of τ . Then:

Px(Xt ∈ G ) =

∫ t

0Pg(s)(Xt−s ∈ G )F (ds)

for each measurable set G contained in [g(t),∞).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Theorem. Let (Xt)t≥0 be a strong, time-homogenous Markovprocess with continuous sample paths started at x , letg : (0,∞)→ R be a continuous function satisfying g(0+) ≥ x ,and

τ := inf{t > 0|Xt ≥ g(t)}

be the first-passage time of X over g , and let F = Fx denote thedistribution of τ .

Then:

Px(Xt ∈ G ) =

∫ t

0Pg(s)(Xt−s ∈ G )F (ds)

for each measurable set G contained in [g(t),∞).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Theorem. Let (Xt)t≥0 be a strong, time-homogenous Markovprocess with continuous sample paths started at x , letg : (0,∞)→ R be a continuous function satisfying g(0+) ≥ x ,and

τ := inf{t > 0|Xt ≥ g(t)}

be the first-passage time of X over g , and let F = Fx denote thedistribution of τ . Then:

Px(Xt ∈ G ) =

∫ t

0Pg(s)(Xt−s ∈ G )F (ds)

for each measurable set G contained in [g(t),∞).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Theorem. Let (Xt)t≥0 be a strong, time-homogenous Markovprocess with continuous sample paths started at x , letg : (0,∞)→ R be a continuous function satisfying g(0+) ≥ x ,and

τ := inf{t > 0|Xt ≥ g(t)}

be the first-passage time of X over g , and let F = Fx denote thedistribution of τ . Then:

Px(Xt ∈ G ) =

∫ t

0Pg(s)(Xt−s ∈ G )F (ds)

for each measurable set G contained in [g(t),∞).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Observations

I The previous equation, which links the distribution of theprocess X with the distribution of the random time τ is aChapman-Kolmogorv equation of Volterra type.

I This equation may be related to a partial differential equationof the forward or backward type.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Observations

I The previous equation, which links the distribution of theprocess X with the distribution of the random time τ is aChapman-Kolmogorv equation of Volterra type.

I This equation may be related to a partial differential equationof the forward or backward type.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Observations

I The previous equation, which links the distribution of theprocess X with the distribution of the random time τ is aChapman-Kolmogorv equation of Volterra type.

I This equation may be related to a partial differential equationof the forward or backward type.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Theorem Let B be one-dimensional standard Brownian motion, letf be a continuously differentiable function with f (0) > 0, and

T := inf {t ≥ |Bt = f (t)}

be the first passage time of B over the moving boundary f . Thenthe density ϕf of T satisfies the following Volterra integralequation of the second kind:

ϕf (t) =f (t)

tφ

(f (t)√

t

)−∫ t

0

f (t)− f (s)

(t − s)φ

(f (t)− f (s)√

t − s

)ϕf (s)ds

where φ is the density of the standard Normal r.v.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Theorem Let B be one-dimensional standard Brownian motion, letf be a continuously differentiable function with f (0) > 0, and

T := inf {t ≥ |Bt = f (t)}

be the first passage time of B over the moving boundary f .

Thenthe density ϕf of T satisfies the following Volterra integralequation of the second kind:

ϕf (t) =f (t)

tφ

(f (t)√

t

)−∫ t

0

f (t)− f (s)

(t − s)φ

(f (t)− f (s)√

t − s

)ϕf (s)ds

where φ is the density of the standard Normal r.v.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Theorem Let B be one-dimensional standard Brownian motion, letf be a continuously differentiable function with f (0) > 0, and

T := inf {t ≥ |Bt = f (t)}

be the first passage time of B over the moving boundary f . Thenthe density ϕf of T satisfies the following Volterra integralequation of the second kind:

ϕf (t) =f (t)

tφ

(f (t)√

t

)−∫ t

0

f (t)− f (s)

(t − s)φ

(f (t)− f (s)√

t − s

)ϕf (s)ds

where φ is the density of the standard Normal r.v.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Theorem Let B be one-dimensional standard Brownian motion, letf be a continuously differentiable function with f (0) > 0, and

T := inf {t ≥ |Bt = f (t)}

be the first passage time of B over the moving boundary f . Thenthe density ϕf of T satisfies the following Volterra integralequation of the second kind:

ϕf (t) =f (t)

tφ

(f (t)√

t

)−∫ t

0

f (t)− f (s)

(t − s)φ

(f (t)− f (s)√

t − s

)ϕf (s)ds

where φ is the density of the standard Normal r.v.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Comments

I Ricciardi et. al.’s equation is not the only integral equationwhich may be derived from Schrodinger’s generalrepresentation.

I Volterra type integral equations for the density of τ can beexplicitly solved only in the case in which the boundary is“linear”.

I We will show the density φ has a partial differentialrepresentation.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Comments

I Ricciardi et. al.’s equation is not the only integral equationwhich may be derived from Schrodinger’s generalrepresentation.

I Volterra type integral equations for the density of τ can beexplicitly solved only in the case in which the boundary is“linear”.

I We will show the density φ has a partial differentialrepresentation.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Comments

I Ricciardi et. al.’s equation is not the only integral equationwhich may be derived from Schrodinger’s generalrepresentation.

I Volterra type integral equations for the density of τ can beexplicitly solved only in the case in which the boundary is“linear”.

I We will show the density φ has a partial differentialrepresentation.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Discrete caseContinuous case, Schrodinger (1915)Brownian motion case, Ricciardi, et. al. (1984)

Comments

I Ricciardi et. al.’s equation is not the only integral equationwhich may be derived from Schrodinger’s generalrepresentation.

I Volterra type integral equations for the density of τ can beexplicitly solved only in the case in which the boundary is“linear”.

I We will show the density φ has a partial differentialrepresentation.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Girsanov’s theorem tells us how stochastic processes behave underchanges in measure.In particular, suppose that f is a twice differentiable function suchthat: f (0) > 0 and f ′′(t) ≥ 0, then it follows from Girsanov’stheorm that:

P PB· B.M. B.M. +

∫ ·0 f ′(u)du

B· B.M.−∫ ·0 f ′(u)du B.M.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Girsanov’s theorem tells us how stochastic processes behave underchanges in measure.

In particular, suppose that f is a twice differentiable function suchthat: f (0) > 0 and f ′′(t) ≥ 0, then it follows from Girsanov’stheorm that:

P PB· B.M. B.M. +

∫ ·0 f ′(u)du

B· B.M.−∫ ·0 f ′(u)du B.M.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Girsanov’s theorem tells us how stochastic processes behave underchanges in measure.In particular, suppose that f is a twice differentiable function suchthat: f (0) > 0 and f ′′(t) ≥ 0, then it follows from Girsanov’stheorm that:

P PB· B.M. B.M. +

∫ ·0 f ′(u)du

B· B.M.−∫ ·0 f ′(u)du B.M.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Girsanov’s theorem tells us how stochastic processes behave underchanges in measure.In particular, suppose that f is a twice differentiable function suchthat: f (0) > 0 and f ′′(t) ≥ 0, then it follows from Girsanov’stheorm that:

P PB· B.M. B.M. +

∫ ·0 f ′(u)du

B· B.M.−∫ ·0 f ′(u)du B.M.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Where the measure P and P are related through theRadon-Nikodym derivative:(

dPd P

)t

:= exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}.

which in fact is a ”martingale” and induces the followingrelationship:

P(Bt ∈ A) = E[(

dPd P

)t

I(Bt∈A)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Where the measure P and P are related through theRadon-Nikodym derivative:(

dPd P

)t

:= exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}.

which in fact is a ”martingale”

and induces the followingrelationship:

P(Bt ∈ A) = E[(

dPd P

)t

I(Bt∈A)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Where the measure P and P are related through theRadon-Nikodym derivative:(

dPd P

)t

:= exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}.

which in fact is a ”martingale” and induces the followingrelationship:

P(Bt ∈ A) = E[(

dPd P

)t

I(Bt∈A)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Where the measure P and P are related through theRadon-Nikodym derivative:(

dPd P

)t

:= exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}.

which in fact is a ”martingale” and induces the followingrelationship:

P(Bt ∈ A) = E[(

dPd P

)t

I(Bt∈A)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In particular: (back to our problem)

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]where

P PT inf {t ≥ 0|Bt = f (t)} inf

{t ≥ 0|Bt = f (0)

}.

Note that under P the density of T is known, since the boundary isconstant!

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In particular: (back to our problem)

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]where

P PT inf {t ≥ 0|Bt = f (t)} inf

{t ≥ 0|Bt = f (0)

}.

Note that under P the density of T is known, since the boundary isconstant!

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In particular: (back to our problem)

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]where

P PT inf {t ≥ 0|Bt = f (t)} inf

{t ≥ 0|Bt = f (0)

}.

Note that under P the density of T is known, since the boundary isconstant!

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In particular: (back to our problem)

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]where

P PT inf {t ≥ 0|Bt = f (t)} inf

{t ≥ 0|Bt = f (0)

}.

Note that under P the density of T is known, since the boundary isconstant!

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In particular: (back to our problem)

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]where

P PT inf {t ≥ 0|Bt = f (t)} inf

{t ≥ 0|Bt = f (0)

}.

Note that under P the density of T is known, since the boundary isconstant!

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Example: Linear boundary

Suppose the boundary is linear, i.e. f (t) = a + bt, for a, b > 0 andt ≥ 0 (f (0) = a). Then:

P(T < t) = E[

exp

{−bBt −

1

2bt

}I(T<t)

]= E

[exp

{−bBT −

1

2bT

}I(T<t)

]= E

[exp

{−ba− 1

2bT

}I(T<t)

]=

∫ t

0exp

{−ba− 1

2bs

}ϕa(s)ds

where ϕa is the density of the first passage time of a BM over thefixed boundary a = f (0).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Example: Linear boundary

Suppose the boundary is linear, i.e. f (t) = a + bt, for a, b > 0 andt ≥ 0 (f (0) = a). Then:

P(T < t) = E[

exp

{−bBt −

1

2bt

}I(T<t)

]= E

[exp

{−bBT −

1

2bT

}I(T<t)

]= E

[exp

{−ba− 1

2bT

}I(T<t)

]=

∫ t

0exp

{−ba− 1

2bs

}ϕa(s)ds

where ϕa is the density of the first passage time of a BM over thefixed boundary a = f (0).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Example: Linear boundary

Suppose the boundary is linear, i.e. f (t) = a + bt, for a, b > 0 andt ≥ 0 (f (0) = a). Then:

P(T < t) = E[

exp

{−bBt −

1

2bt

}I(T<t)

]

= E[

exp

{−bBT −

1

2bT

}I(T<t)

]= E

[exp

{−ba− 1

2bT

}I(T<t)

]=

∫ t

0exp

{−ba− 1

2bs

}ϕa(s)ds

where ϕa is the density of the first passage time of a BM over thefixed boundary a = f (0).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Example: Linear boundary

Suppose the boundary is linear, i.e. f (t) = a + bt, for a, b > 0 andt ≥ 0 (f (0) = a). Then:

P(T < t) = E[

exp

{−bBt −

1

2bt

}I(T<t)

]= E

[exp

{−bBT −

1

2bT

}I(T<t)

]

= E[

exp

{−ba− 1

2bT

}I(T<t)

]=

∫ t

0exp

{−ba− 1

2bs

}ϕa(s)ds

where ϕa is the density of the first passage time of a BM over thefixed boundary a = f (0).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Example: Linear boundary

Suppose the boundary is linear, i.e. f (t) = a + bt, for a, b > 0 andt ≥ 0 (f (0) = a). Then:

P(T < t) = E[

exp

{−bBt −

1

2bt

}I(T<t)

]= E

[exp

{−bBT −

1

2bT

}I(T<t)

]= E

[exp

{−ba− 1

2bT

}I(T<t)

]

=

∫ t

0exp

{−ba− 1

2bs

}ϕa(s)ds

where ϕa is the density of the first passage time of a BM over thefixed boundary a = f (0).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Example: Linear boundary

Suppose the boundary is linear, i.e. f (t) = a + bt, for a, b > 0 andt ≥ 0 (f (0) = a). Then:

P(T < t) = E[

exp

{−bBt −

1

2bt

}I(T<t)

]= E

[exp

{−bBT −

1

2bT

}I(T<t)

]= E

[exp

{−ba− 1

2bT

}I(T<t)

]=

∫ t

0exp

{−ba− 1

2bs

}ϕa(s)ds

where ϕa is the density of the first passage time of a BM over thefixed boundary a = f (0).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Example: Linear boundary

Suppose the boundary is linear, i.e. f (t) = a + bt, for a, b > 0 andt ≥ 0 (f (0) = a). Then:

P(T < t) = E[

exp

{−bBt −

1

2bt

}I(T<t)

]= E

[exp

{−bBT −

1

2bT

}I(T<t)

]= E

[exp

{−ba− 1

2bT

}I(T<t)

]=

∫ t

0exp

{−ba− 1

2bs

}ϕa(s)ds

where ϕa is the density of the first passage time of a BM over thefixed boundary a = f (0).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In general, if a := f (0), and f ′′ ≥ 0. Then:

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]= E

[e−f ′(t)Bt+

R t0 f ′′(u)Budu− 1

2

R t0 (f ′(u))2duI(T<t)

]= E

[e−f ′(T )BT +

R T0 f ′′(u)Budu− 1

2

R T0 (f ′(u))2duI(T<t)

]=

∫ t

0E[e−f ′(s)a+

R s0 f ′′(u)Budu− 1

2

R s0 (f ′(u))2du

∣∣∣T = s]ϕa(s)ds

=

∫ t

0e−f ′(s)a− 1

2

R s0 (f ′(u))2du E

[e

R s0 f ′′(u)Budu

∣∣∣T = s]ϕa(s)ds

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In general, if a := f (0), and f ′′ ≥ 0. Then:

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]

= E[e−f ′(t)Bt+

R t0 f ′′(u)Budu− 1

2

R t0 (f ′(u))2duI(T<t)

]= E

[e−f ′(T )BT +

R T0 f ′′(u)Budu− 1

2

R T0 (f ′(u))2duI(T<t)

]=

∫ t

0E[e−f ′(s)a+

R s0 f ′′(u)Budu− 1

2

R s0 (f ′(u))2du

∣∣∣T = s]ϕa(s)ds

=

∫ t

0e−f ′(s)a− 1

2

R s0 (f ′(u))2du E

[e

R s0 f ′′(u)Budu

∣∣∣T = s]ϕa(s)ds

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In general, if a := f (0), and f ′′ ≥ 0. Then:

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]= E

[e−f ′(t)Bt+

R t0 f ′′(u)Budu− 1

2

R t0 (f ′(u))2duI(T<t)

]

= E[e−f ′(T )BT +

R T0 f ′′(u)Budu− 1

2

R T0 (f ′(u))2duI(T<t)

]=

∫ t

0E[e−f ′(s)a+

R s0 f ′′(u)Budu− 1

2

R s0 (f ′(u))2du

∣∣∣T = s]ϕa(s)ds

=

∫ t

0e−f ′(s)a− 1

2

R s0 (f ′(u))2du E

[e

R s0 f ′′(u)Budu

∣∣∣T = s]ϕa(s)ds

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In general, if a := f (0), and f ′′ ≥ 0. Then:

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]= E

[e−f ′(t)Bt+

R t0 f ′′(u)Budu− 1

2

R t0 (f ′(u))2duI(T<t)

]= E

[e−f ′(T )BT +

R T0 f ′′(u)Budu− 1

2

R T0 (f ′(u))2duI(T<t)

]

=

∫ t

0E[e−f ′(s)a+

R s0 f ′′(u)Budu− 1

2

R s0 (f ′(u))2du

∣∣∣T = s]ϕa(s)ds

=

∫ t

0e−f ′(s)a− 1

2

R s0 (f ′(u))2du E

[e

R s0 f ′′(u)Budu

∣∣∣T = s]ϕa(s)ds

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In general, if a := f (0), and f ′′ ≥ 0. Then:

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]= E

[e−f ′(t)Bt+

R t0 f ′′(u)Budu− 1

2

R t0 (f ′(u))2duI(T<t)

]= E

[e−f ′(T )BT +

R T0 f ′′(u)Budu− 1

2

R T0 (f ′(u))2duI(T<t)

]=

∫ t

0E[e−f ′(s)a+

R s0 f ′′(u)Budu− 1

2

R s0 (f ′(u))2du

∣∣∣T = s]ϕa(s)ds

=

∫ t

0e−f ′(s)a− 1

2

R s0 (f ′(u))2du E

[e

R s0 f ′′(u)Budu

∣∣∣T = s]ϕa(s)ds

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

In general, if a := f (0), and f ′′ ≥ 0. Then:

P(T < t) = E[

exp

{−∫ t

0f ′(u)dBu −

1

2

∫ t

0(f ′(u))2du

}I(T<t)

]= E

[e−f ′(t)Bt+

R t0 f ′′(u)Budu− 1

2

R t0 (f ′(u))2duI(T<t)

]= E

[e−f ′(T )BT +

R T0 f ′′(u)Budu− 1

2

R T0 (f ′(u))2duI(T<t)

]=

∫ t

0E[e−f ′(s)a+

R s0 f ′′(u)Budu− 1

2

R s0 (f ′(u))2du

∣∣∣T = s]ϕa(s)ds

=

∫ t

0e−f ′(s)a− 1

2

R s0 (f ′(u))2du E

[e

R s0 f ′′(u)Budu

∣∣∣T = s]ϕa(s)ds

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

What is P(Bu ∈ A|T = s))?

0.0 0.2 0.4 0.6 0.8

−0.

050.

000.

05

s

a

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

What is P(Bu ∈ A|T = s))?

0.0 0.2 0.4 0.6 0.8

−0.

050.

000.

05

s

a

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

What is P(Bu ∈ A|T = s))?

0.0 0.2 0.4 0.6 0.8

−0.

050.

000.

05

s

−a

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

What is P(Bu ∈ A|T = s))?

0.0 0.2 0.4 0.6 0.8

−0.

050.

000.

05

s

a

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Such a process X = a− BsI{T=s} has the following dynamics:

dXt = dWt

+1

Xt

dt − Xt

s − tdt, 0 ≤ t ≤ s, X0 = a

Thus

E[e

R s0 f ′′(u)Budu

∣∣∣T = s]

= E[

exp

{∫ s

0f ′′(u)(a− Xu)du

}]= ef ′(s)a−f ′(0)a E

[exp

{−∫ t

0f ′′(u)Xudu

}];

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Such a process X = a− BsI{T=s} has the following dynamics:

dXt = dWt +1

Xt

dt

− Xt

s − tdt, 0 ≤ t ≤ s, X0 = a

Thus

E[e

R s0 f ′′(u)Budu

∣∣∣T = s]

= E[

exp

{∫ s

0f ′′(u)(a− Xu)du

}]= ef ′(s)a−f ′(0)a E

[exp

{−∫ t

0f ′′(u)Xudu

}];

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Such a process X = a− BsI{T=s} has the following dynamics:

dXt = dWt +1

Xt

dt − Xt

s − tdt,

0 ≤ t ≤ s, X0 = a

Thus

E[e

R s0 f ′′(u)Budu

∣∣∣T = s]

= E[

exp

{∫ s

0f ′′(u)(a− Xu)du

}]= ef ′(s)a−f ′(0)a E

[exp

{−∫ t

0f ′′(u)Xudu

}];

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Such a process X = a− BsI{T=s} has the following dynamics:

dXt = dWt +1

Xt

dt − Xt

s − tdt, 0 ≤ t ≤ s, X0 = a

Thus

E[e

R s0 f ′′(u)Budu

∣∣∣T = s]

= E[

exp

{∫ s

0f ′′(u)(a− Xu)du

}]= ef ′(s)a−f ′(0)a E

[exp

{−∫ t

0f ′′(u)Xudu

}];

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Such a process X = a− BsI{T=s} has the following dynamics:

dXt = dWt +1

Xt

dt − Xt

s − tdt, 0 ≤ t ≤ s, X0 = a

Thus

E[e

R s0 f ′′(u)Budu

∣∣∣T = s]

= E[

exp

{∫ s

0f ′′(u)(a− Xu)du

}]= ef ′(s)a−f ′(0)a E

[exp

{−∫ t

0f ′′(u)Xudu

}];

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Such a process X = a− BsI{T=s} has the following dynamics:

dXt = dWt +1

Xt

dt − Xt

s − tdt, 0 ≤ t ≤ s, X0 = a

Thus

E[e

R s0 f ′′(u)Budu

∣∣∣T = s]

= E[

exp

{∫ s

0f ′′(u)(a− Xu)du

}]

= ef ′(s)a−f ′(0)a E[

exp

{−∫ t

0f ′′(u)Xudu

}];

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Such a process X = a− BsI{T=s} has the following dynamics:

dXt = dWt +1

Xt

dt − Xt

s − tdt, 0 ≤ t ≤ s, X0 = a

Thus

E[e

R s0 f ′′(u)Budu

∣∣∣T = s]

= E[

exp

{∫ s

0f ′′(u)(a− Xu)du

}]= ef ′(s)a−f ′(0)a E

[exp

{−∫ t

0f ′′(u)Xudu

}];

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Let g(a) = 1 for all a ∈ R and k(u, a) := f ′′(u)a,k(t, a) : [0, s]× R+ → [0,∞).

Then:

E[

exp

{−∫ t

0f ′′(u)Xudu

}]= E0,a

[g(Xs) exp

{−∫ s

0k(u, Xu)du

}]Theorem. (Feynman-Kac) Under some “conditions” supposethat we have the following stochastic representation:

v(t, a) = Et,a[g(XT ) exp

{−∫ s

tk(u,Xu)du

}+

∫ s

th(u,Xu) exp

{−∫ u

tk(θ,Xθ)dθ

}du]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Let g(a) = 1 for all a ∈ R and k(u, a) := f ′′(u)a,k(t, a) : [0, s]× R+ → [0,∞). Then:

E[

exp

{−∫ t

0f ′′(u)Xudu

}]=

E0,a

[g(Xs) exp

{−∫ s

0k(u, Xu)du

}]Theorem. (Feynman-Kac) Under some “conditions” supposethat we have the following stochastic representation:

v(t, a) = Et,a[g(XT ) exp

{−∫ s

tk(u,Xu)du

}+

∫ s

th(u,Xu) exp

{−∫ u

tk(θ,Xθ)dθ

}du]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Let g(a) = 1 for all a ∈ R and k(u, a) := f ′′(u)a,k(t, a) : [0, s]× R+ → [0,∞). Then:

E[

exp

{−∫ t

0f ′′(u)Xudu

}]= E0,a

[g(Xs) exp

{−∫ s

0k(u, Xu)du

}]

Theorem. (Feynman-Kac) Under some “conditions” supposethat we have the following stochastic representation:

v(t, a) = Et,a[g(XT ) exp

{−∫ s

tk(u,Xu)du

}+

∫ s

th(u,Xu) exp

{−∫ u

tk(θ,Xθ)dθ

}du]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Let g(a) = 1 for all a ∈ R and k(u, a) := f ′′(u)a,k(t, a) : [0, s]× R+ → [0,∞). Then:

E[

exp

{−∫ t

0f ′′(u)Xudu

}]= E0,a

[g(Xs) exp

{−∫ s

0k(u, Xu)du

}]Theorem. (Feynman-Kac) Under some “conditions” supposethat we have the following stochastic representation:

v(t, a) = Et,a[g(XT ) exp

{−∫ s

tk(u,Xu)du

}+

∫ s

th(u,Xu) exp

{−∫ u

tk(θ,Xθ)dθ

}du]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Then v(t, a) satisfies the following Cauchy problem

−∂v

∂t+ kv = At + h; in [0, s)× R+,

v(s, a) = g(a); a ∈ R+,

Definition. A fundamental solution of the second-order partialdifferential equation

− ∂u

∂t+ ku = Atu (1)

is a nonnegative function G (t, a; τ, b) defined on 0 ≤ t < τ < s,a, b ∈ R+, with the property that for every g ∈ C0(R), τ ∈ (0, s],the function

u(t, a) :=

∫ ∞0

G (t, a; τ, b)g(b)db; 0 ≤ t < τ, a ∈ R+

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

Then v(t, a) satisfies the following Cauchy problem

−∂v

∂t+ kv = At + h; in [0, s)× R+,

v(s, a) = g(a); a ∈ R+,

Definition. A fundamental solution of the second-order partialdifferential equation

− ∂u

∂t+ ku = Atu (1)

is a nonnegative function G (t, a; τ, b) defined on 0 ≤ t < τ < s,a, b ∈ R+, with the property that for every g ∈ C0(R), τ ∈ (0, s],the function

u(t, a) :=

∫ ∞0

G (t, a; τ, b)g(b)db; 0 ≤ t < τ, a ∈ R+

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Girsanov’s Theorem (1960)Girsanov’s and boundary problems3-dimensional Bessel bridge, boundary problemFeynman-Kac, PDE’s vs. SDE’s

is bounded, satisfies (1) and

limt↑τ

u(t, a) = g(a); a ∈ R+,

for fixed (τ, b) ∈ (0, s]× R+, the function

ϕ(t, a) := G (t, a; τ, b)

satisfies the backward Kolmogorov equation in the backwardvariables (t, a). And, for fixed (t, a) ∈ [0, s)× R+ the function

ψ(τ, b) := G (t, a; τ, b)

satisfies the forward Kolmogorov equation in the forward variables(τ, b).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

As has been pointed out, to derive the density of T we must firstcompute the expected value of the 3-dimensional Bessel bridgefunctional. We will proceed by first studying solutions of thefollowing PDE’s:

(b) − ∂ϕ

∂t(t, a) + f ′′(t)aϕ(t, a) =

1

2

∂2ϕ

∂a2(t, a)

+

(1

a− a

s − t

)∂ϕ

∂a(t, a);

(f )∂ψ

∂τ(τ, b) + f ′′k (τ)bψ(τ, b) =

1

2

∂2ψ

∂b2(τ, b)

− ∂

∂b

[(1

b− b

s − τ

)ψ(τ, b)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

As has been pointed out, to derive the density of T we must firstcompute the expected value of the 3-dimensional Bessel bridgefunctional. We will proceed by first studying solutions of thefollowing PDE’s:

(b) − ∂ϕ

∂t(t, a) + f ′′(t)aϕ(t, a) =

1

2

∂2ϕ

∂a2(t, a)

+

(1

a− a

s − t

)∂ϕ

∂a(t, a);

(f )∂ψ

∂τ(τ, b) + f ′′k (τ)bψ(τ, b) =

1

2

∂2ψ

∂b2(τ, b)

− ∂

∂b

[(1

b− b

s − τ

)ψ(τ, b)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

As has been pointed out, to derive the density of T we must firstcompute the expected value of the 3-dimensional Bessel bridgefunctional. We will proceed by first studying solutions of thefollowing PDE’s:

(b) − ∂ϕ

∂t(t, a) + f ′′(t)aϕ(t, a) =

1

2

∂2ϕ

∂a2(t, a)

+

(1

a− a

s − t

)∂ϕ

∂a(t, a);

(f )∂ψ

∂τ(τ, b) + f ′′k (τ)bψ(τ, b) =

1

2

∂2ψ

∂b2(τ, b)

− ∂

∂b

[(1

b− b

s − τ

)ψ(τ, b)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

(Deriving a solution) Step 1):

(b) − ∂ϕ1

∂t(t, a) + f ′′(t)aϕ1(t, a) =

1

2

∂2ϕ1

∂a2(t, a)

+

(1

a−

)∂ϕ1

∂a(t, a);

(f )∂ψ1

∂τ(τ, b) + f ′′k (τ)bψ1(τ, b) =

1

2

∂2ψ1

∂b2(τ, b)

− ∂

∂b

[(1

b−

)ψ1(τ, b)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

(Deriving a solution) Step 1):

(b) − ∂ϕ1

∂t(t, a) + f ′′(t)aϕ1(t, a) =

1

2

∂2ϕ1

∂a2(t, a)

+

(1

a−

)∂ϕ1

∂a(t, a);

(f )∂ψ1

∂τ(τ, b) + f ′′k (τ)bψ1(τ, b) =

1

2

∂2ψ1

∂b2(τ, b)

− ∂

∂b

[(1

b−

)ψ1(τ, b)

]

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 2) (Related to Schrodinger’s eq. for time dependent “linear”potential):

(b) − ∂ϕ

∂t(t, a) + f ′′(t)aϕ(t, a) =

1

2

∂2ϕ

∂a2(t, a)

(f )∂ψ

∂τ(τ, b) + f ′′(τ)bψ(τ, b) =

1

2

∂2ψ

∂b2(τ, b)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 2) (Related to Schrodinger’s eq. for time dependent “linear”potential):

(b) − ∂ϕ

∂t(t, a) + f ′′(t)aϕ(t, a) =

1

2

∂2ϕ

∂a2(t, a)

(f )∂ψ

∂τ(τ, b) + f ′′(τ)bψ(τ, b) =

1

2

∂2ψ

∂b2(τ, b)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Example. Quadratic boundary If f (t) = a + bt2, fora, b > 0,

then f ′′(t) = 2b. Hence

(b) − ∂ϕ

∂t(t, a) + f ′′(t)aϕ(t, a) =

1

2

∂2ϕ

∂a2(t, a)

−∂ϕ∂t

(t, a) + 2baϕ(t, a) =1

2

∂2ϕ

∂a2(t, a)

Applying the Laplace transform L with respect to t we have

−λL[ϕ(·, a)] + 2baL[ϕ(·, a)]− 1

2

∂2

∂a2L[ϕ(·, a)] = ϕ(0, a).

The independent solutions to this second order non-homogeneousO.D.E. are Airy’s functions [Martin-Lof (1998)].

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Example. Quadratic boundary If f (t) = a + bt2, fora, b > 0,then f ′′(t) = 2b. Hence

(b) − ∂ϕ

∂t(t, a) + f ′′(t)aϕ(t, a) =

1

2

∂2ϕ

∂a2(t, a)

−∂ϕ∂t

(t, a) + 2baϕ(t, a) =1

2

∂2ϕ

∂a2(t, a)

Applying the Laplace transform L with respect to t we have

−λL[ϕ(·, a)] + 2baL[ϕ(·, a)]− 1

2

∂2

∂a2L[ϕ(·, a)] = ϕ(0, a).

The independent solutions to this second order non-homogeneousO.D.E. are Airy’s functions [Martin-Lof (1998)].

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Example. Quadratic boundary If f (t) = a + bt2, fora, b > 0,then f ′′(t) = 2b. Hence

(b) − ∂ϕ

∂t(t, a) + f ′′(t)aϕ(t, a) =

1

2

∂2ϕ

∂a2(t, a)

−∂ϕ∂t

(t, a) + 2baϕ(t, a) =1

2

∂2ϕ

∂a2(t, a)

Applying the Laplace transform L with respect to t we have

−λL[ϕ(·, a)] + 2baL[ϕ(·, a)]− 1

2

∂2

∂a2L[ϕ(·, a)] = ϕ(0, a).

The independent solutions to this second order non-homogeneousO.D.E. are Airy’s functions [Martin-Lof (1998)].

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Remark. Let us introduce the Airy solution to the heat equation

u(t, x) := exp

{2t2

3− xt

}Ai(t2 − x)

or

v(t, x) := exp

{2t2

3− (x − zj)t

}Ai(t2 − (x − zj))

[You may consult for instance: Airy functions and applications tophysics by Olivier Vallee & Manuel Soares] where zj is a zero of Ai .Then:

v(t, t2) = exp

{2t2

3− (t2 − zj)t

}Ai(zj)

= 0.

The usefulness of this Remark with become evident after a fewslides.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 4) (Heat equation):

(b) − ∂ϕ

∂t(t, a) =

1

2

∂2ϕ

∂a2(t, a)

(f )∂ψ

∂τ(τ, b) =

1

2

∂2ψ

∂b2(τ, b)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 4) (Heat equation):

(b) − ∂ϕ

∂t(t, a) =

1

2

∂2ϕ

∂a2(t, a)

(f )∂ψ

∂τ(τ, b) =

1

2

∂2ψ

∂b2(τ, b)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 1.

Proposition. Equation (b) satisfies the following relationship

ϕ(t, a; s) = ϕ1(t, a)[A(t; s) exp

(B(t; s)a2

)].

where

B(t; s) =1

2(s − t)

A(t; s) = c · (s − t)3/2

af ′′(t)ϕ1(t, a)− 1

2ϕ1

aa(t, a)− 1

aϕ1

a(t, a)− ϕ1t (t, a) = 0

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 2.

Letting ϕ1 = 1/aϕ2, we have

ϕ1t =

1

aϕ2

t

ϕ1a =

1

aϕ2

a −1

a2ϕ2

ϕ1aa =

1

aϕ2

aa −2

a2ϕ2

a +2

a3ϕ2.

Hence

af ′′(t)ϕ2(t, a)− 1

2ϕ2

aa(t, a)− ϕ2t (t, a) = 0.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 2.

Letting ϕ1 = 1/aϕ2, we have

ϕ1t =

1

aϕ2

t

ϕ1a =

1

aϕ2

a −1

a2ϕ2

ϕ1aa =

1

aϕ2

aa −2

a2ϕ2

a +2

a3ϕ2.

Hence

af ′′(t)ϕ2(t, a)− 1

2ϕ2

aa(t, a)− ϕ2t (t, a) = 0.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 2.

Letting ϕ1 = 1/aϕ2, we have

ϕ1t =

1

aϕ2

t

ϕ1a =

1

aϕ2

a −1

a2ϕ2

ϕ1aa =

1

aϕ2

aa −2

a2ϕ2

a +2

a3ϕ2.

Hence

af ′′(t)ϕ2(t, a)− 1

2ϕ2

aa(t, a)− ϕ2t (t, a) = 0.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 3.

Theorem. The backward Kolmogorov equation

af ′′(t)ϕ2(t, a)− 1

2ϕ2

aa(t, a)− ϕ2t (t, a) = 0.

has a solution:

ϕ2(t, a) = exp

{−1

2

∫ t

0(f ′(u))2du − f ′(t)a

}ω(τ − t, a +

∫f ′(t)dt)

where ω is any solution of the heat equation.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Step 3.

Theorem. The backward Kolmogorov equation

af ′′(t)ϕ2(t, a)− 1

2ϕ2

aa(t, a)− ϕ2t (t, a) = 0.

has a solution:

ϕ2(t, a) = exp

{−1

2

∫ t

0(f ′(u))2du − f ′(t)a

}ω(τ − t, a +

∫f ′(t)dt)

where ω is any solution of the heat equation.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Sketch of Proof (Theorem)

I First set ϕ2(t, a) = λ(t, a)eβ(t)a, (where β is determinedlater) and substitute into equation.

We get

−λt(t, a) =1

2λaa(t, a) +

1

2β2(t)λ(t, a) + β(t)λa(t, a)

+a(βt(t)− f ′′(t))λ(t, a)

I (Change of variable.) Let y = a− v(t) and letλ(t, a) = u(t, y)

−∂u

∂t(t, y) + vt(t)

∂u

∂y(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+β(t)∂u

∂y(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Sketch of Proof (Theorem)

I First set ϕ2(t, a) = λ(t, a)eβ(t)a, (where β is determinedlater) and substitute into equation.

We get

−λt(t, a) =1

2λaa(t, a) +

1

2β2(t)λ(t, a) + β(t)λa(t, a)

+a(βt(t)− f ′′(t))λ(t, a)

I (Change of variable.) Let y = a− v(t) and letλ(t, a) = u(t, y)

−∂u

∂t(t, y) + vt(t)

∂u

∂y(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+β(t)∂u

∂y(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Sketch of Proof (Theorem)

I First set ϕ2(t, a) = λ(t, a)eβ(t)a, (where β is determinedlater) and substitute into equation. We get

−λt(t, a) =1

2λaa(t, a) +

1

2β2(t)λ(t, a) + β(t)λa(t, a)

+a(βt(t)− f ′′(t))λ(t, a)

I (Change of variable.) Let y = a− v(t) and letλ(t, a) = u(t, y)

−∂u

∂t(t, y) + vt(t)

∂u

∂y(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+β(t)∂u

∂y(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Sketch of Proof (Theorem)

I First set ϕ2(t, a) = λ(t, a)eβ(t)a, (where β is determinedlater) and substitute into equation. We get

−λt(t, a) =1

2λaa(t, a) +

1

2β2(t)λ(t, a) + β(t)λa(t, a)

+a(βt(t)− f ′′(t))λ(t, a)

I (Change of variable.) Let y = a− v(t) and letλ(t, a) = u(t, y)

−∂u

∂t(t, y) + vt(t)

∂u

∂y(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+β(t)∂u

∂y(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Set vt(t) = β(t). Then,

−∂u

∂t(t, y) + vt(t)

∂u

∂y(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+β(t)∂u

∂y(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

becomes

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Set vt(t) = β(t). Then,

−∂u

∂t(t, y) + vt(t)

∂u

∂y(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+β(t)∂u

∂y(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

becomes

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Set vt(t) = β(t). Then,

−∂u

∂t(t, y) + vt(t)

∂u

∂y(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+β(t)∂u

∂y(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

becomes

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Set βt(t) = f ′′(t). Then,

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

becomes

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Set βt(t) = f ′′(t). Then,

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

becomes

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Set βt(t) = f ′′(t). Then,

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

+(y + v(t))(βt(t)− f ′′(t))u(t, y)

becomes

−∂u

∂t(t, y) =

1

2

∂2u

∂y2(t, y) +

1

2β2(t)u(t, y)

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Finally

I Let G (u) = 1/2β2(u) and

ϕ3(t, y) = u(t, y)eR t0 G(u)du

Then

−∂ϕ3

∂t(t, y) =

1

2

∂2ϕ3

∂y2(t, y) �

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Finally

I Let G (u) = 1/2β2(u) and

ϕ3(t, y) = u(t, y)eR t0 G(u)du

Then

−∂ϕ3

∂t(t, y) =

1

2

∂2ϕ3

∂y2(t, y) �

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Finally

I Let G (u) = 1/2β2(u) and

ϕ3(t, y) = u(t, y)eR t0 G(u)du

Then

−∂ϕ3

∂t(t, y) =

1

2

∂2ϕ3

∂y2(t, y) �

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Proposition. Equation (f) satisfies the following relationship

ψ(τ, b, s) = ψ1(τ, b; s)[A(τ ; s) exp

(B(τ ; s)b2

)],

where

B(τ ; s) = − 1

2(s − τ)

A(t; s) = c · (s − τ)−1/2(bf ′′(τ)− 1

b2− 1

s − τ

)ψ1(τ, b)− 1

2ψ1

bb(τ, b) +1

bψ1

b(τ, b, s)

+ψ1τ (τ, a, s) = 0.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

We proceed by first setting ψ1 = b · ψ2: ⇒

(bf ′′(τ)− 1

s − τ

)ψ2(τ, b)− 1

2ψ2

bb(τ, b) + ψ2τ (τ, b) = 0.

Next ψ2 = 1/(s − τ) · ψ3: ⇒

bf ′′(τ)ψ3(τ, b)− 1

2ψ3

bb(τ, b) + ψ3τ (τ, b) = 0.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

We proceed by first setting ψ1 = b · ψ2: ⇒(bf ′′(τ)− 1

s − τ

)ψ2(τ, b)− 1

2ψ2

bb(τ, b) + ψ2τ (τ, b) = 0.

Next ψ2 = 1/(s − τ) · ψ3: ⇒

bf ′′(τ)ψ3(τ, b)− 1

2ψ3

bb(τ, b) + ψ3τ (τ, b) = 0.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

We proceed by first setting ψ1 = b · ψ2: ⇒(bf ′′(τ)− 1

s − τ

)ψ2(τ, b)− 1

2ψ2

bb(τ, b) + ψ2τ (τ, b) = 0.

Next ψ2 = 1/(s − τ) · ψ3: ⇒

bf ′′(τ)ψ3(τ, b)− 1

2ψ3

bb(τ, b) + ψ3τ (τ, b) = 0.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

We proceed by first setting ψ1 = b · ψ2: ⇒(bf ′′(τ)− 1

s − τ

)ψ2(τ, b)− 1

2ψ2

bb(τ, b) + ψ2τ (τ, b) = 0.

Next ψ2 = 1/(s − τ) · ψ3: ⇒

bf ′′(τ)ψ3(τ, b)− 1

2ψ3

bb(τ, b) + ψ3τ (τ, b) = 0.

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

Theorem. Given the forward equation as in previous slide:

∂ψ3

∂τ(τ, b) =

1

2

∂2ψ3

∂b2(τ, b)− bf ′′(τ)ψ3(τ, b)

we have the following solution

ψ3(τ, b) = exp

{1

2

∫ τ

0(f ′(u))2du + f ′(τ)b

}ω(τ, b +

∫f ′(τ)dτ)

(recall that ω is any solution of the heat equation).

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges

OutlineOverview of the first passage time problem

Integral equationsPDE approach

Derivation of the density?

Guideline of the proof (backward equation)Sketch of the proof (backward equation)Solution of the forward equationGreen’s function

The derivation and verification of a Green function will bepostponed to upcoming notes

Gerardo Hernandez-del-Valle Notes on Schrodinger’s equation, Bessel bridges