Refracted Lvy processes Ronnie Loeffen 1 Johann Radon Institute for - - PowerPoint PPT Presentation

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Refracted Lévy processes

Ronnie Loeffen1

Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences

December 4, 2008

1based on joint work with Andreas Kyprianou, University of Bath

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Outline

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Spectrally negative Lévy processes

• We let X on (Ω, {Ft : t ≥ 0}, P) be a Lévy process. We use

Px(·) = P(·|X0 = x).

• In this talk we consider only the case when X is spectrally

negative, i.e. X has only jumps downwards and X does not have monotone increasing or monotone decreasing paths.

• The Laplace exponent of X, ψ(λ) := log E(eλX1) is well

defined for λ ≥ 0 and is given by ψ(λ) = γλ + 1 2σ2λ2 −

• (0,∞)

(1 − e−λx − λx1{x≤1})ν(dx). Here (γ, σ, ν) is the Lévy triplet of X, where γ ∈ R, σ ≥ 0 and ν is a measure on (0, ∞) satisfying

• (0,∞)

(1 ∧ x2)ν(dx) < ∞.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Spectrally negative Lévy processes

• We let X on (Ω, {Ft : t ≥ 0}, P) be a Lévy process. We use

Px(·) = P(·|X0 = x).

• In this talk we consider only the case when X is spectrally

negative, i.e. X has only jumps downwards and X does not have monotone increasing or monotone decreasing paths.

• The Laplace exponent of X, ψ(λ) := log E(eλX1) is well

defined for λ ≥ 0 and is given by ψ(λ) = γλ + 1 2σ2λ2 −

• (0,∞)

(1 − e−λx − λx1{x≤1})ν(dx). Here (γ, σ, ν) is the Lévy triplet of X, where γ ∈ R, σ ≥ 0 and ν is a measure on (0, ∞) satisfying

• (0,∞)

(1 ∧ x2)ν(dx) < ∞.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Spectrally negative Lévy processes

• We let X on (Ω, {Ft : t ≥ 0}, P) be a Lévy process. We use

Px(·) = P(·|X0 = x).

• In this talk we consider only the case when X is spectrally

negative, i.e. X has only jumps downwards and X does not have monotone increasing or monotone decreasing paths.

• The Laplace exponent of X, ψ(λ) := log E(eλX1) is well

defined for λ ≥ 0 and is given by ψ(λ) = γλ + 1 2σ2λ2 −

• (0,∞)

(1 − e−λx − λx1{x≤1})ν(dx). Here (γ, σ, ν) is the Lévy triplet of X, where γ ∈ R, σ ≥ 0 and ν is a measure on (0, ∞) satisfying

• (0,∞)

(1 ∧ x2)ν(dx) < ∞.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Spectrally negative Lévy processes (continued)

• Lévy-Itô decomposition:

ψ(λ) = γλ + 1 2σ2λ2

• Brownian motion plus drift

−

• (1,∞)

(1 − e−λx)ν(dx)

• compound Poisson process

−

• (0,1]

(1 − e−λx − λx)ν(dx)

• limit of compensated cpp

.

• The process X has sample paths of unbounded variation if

and only if σ > 0 or

• (0,1] xν(dx) = ∞.
• If X is of bounded variation, then we can write

Xt = ct +

• 0<s≤t

∆Xs, where c = γ +

• (0,1] xν(dx) > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Spectrally negative Lévy processes (continued)

• Lévy-Itô decomposition:

ψ(λ) = γλ + 1 2σ2λ2

• Brownian motion plus drift

−

• (1,∞)

(1 − e−λx)ν(dx)

• compound Poisson process

−

• (0,1]

(1 − e−λx − λx)ν(dx)

• limit of compensated cpp

.

• The process X has sample paths of unbounded variation if

and only if σ > 0 or

• (0,1] xν(dx) = ∞.
• If X is of bounded variation, then we can write

Xt = ct +

• 0<s≤t

∆Xs, where c = γ +

• (0,1] xν(dx) > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Spectrally negative Lévy processes (continued)

• Lévy-Itô decomposition:

ψ(λ) = γλ + 1 2σ2λ2

• Brownian motion plus drift

−

• (1,∞)

(1 − e−λx)ν(dx)

• compound Poisson process

−

• (0,1]

(1 − e−λx − λx)ν(dx)

• limit of compensated cpp

.

• The process X has sample paths of unbounded variation if

and only if σ > 0 or

• (0,1] xν(dx) = ∞.
• If X is of bounded variation, then we can write

Xt = ct +

• 0<s≤t

∆Xs, where c = γ +

• (0,1] xν(dx) > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Scale functions

• For each q ≥ 0 there exists a function W (q) : R → [0, ∞),

called the (q-)scale function of X, which satisfies W (q)(x) = 0 for x < 0 and is characterized on [0, ∞) as a strictly increasing and continuous function whose Laplace transform is given by ∞ e−λxW (q)(x)dx = 1 ψ(λ) − q for λ > Φ(q), where Φ(q) = sup{λ ≥ 0 : ψ(λ) = q} is the right-inverse of ψ.

Φ(0) Φ(q) λ ψ q

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Scale functions (continued)

• Scale functions appear in various fluctuation identities for

spectrally negative Lévy processes, e.g. the two-sided exit problem: Let τ +

a = inf{t > 0 : Xt > a}, τ − 0 = inf{t > 0 : Xt < 0} and

x ∈ [0, a]. Then Ex

• e−qτ +

a 1{τ + a <τ − 0 }

• = W (q)(x)

W (q)(a).

• Providing the scale function is smooth enough, we have

ΓW (q)(x) = qW (q)(x), where Γ is the infinitesimal generator of the process X.

• Drawback: There is a limited number of examples where

the scale function can be given in closed-form, see however Hubalek & Kyprianou (2007).

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Scale functions (continued)

• Scale functions appear in various fluctuation identities for

spectrally negative Lévy processes, e.g. the two-sided exit problem: Let τ +

a = inf{t > 0 : Xt > a}, τ − 0 = inf{t > 0 : Xt < 0} and

x ∈ [0, a]. Then Ex

• e−qτ +

a 1{τ + a <τ − 0 }

• = W (q)(x)

W (q)(a).

• Providing the scale function is smooth enough, we have

ΓW (q)(x) = qW (q)(x), where Γ is the infinitesimal generator of the process X.

• Drawback: There is a limited number of examples where

the scale function can be given in closed-form, see however Hubalek & Kyprianou (2007).

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Scale functions (continued)

• Scale functions appear in various fluctuation identities for

spectrally negative Lévy processes, e.g. the two-sided exit problem: Let τ +

a = inf{t > 0 : Xt > a}, τ − 0 = inf{t > 0 : Xt < 0} and

x ∈ [0, a]. Then Ex

• e−qτ +

a 1{τ + a <τ − 0 }

• = W (q)(x)

W (q)(a).

• Providing the scale function is smooth enough, we have

ΓW (q)(x) = qW (q)(x), where Γ is the infinitesimal generator of the process X.

• Drawback: There is a limited number of examples where

the scale function can be given in closed-form, see however Hubalek & Kyprianou (2007).

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What are scale functions used for these days?

• They serve as building blocks for obtaining more

sophisticated identities of (modified) spectrally negative Lévy processes.

• They have been used as key tools for, amongst others,

solving optimal stopping/control problems and deriving smooth pasting principles involving spectrally negative Lévy processes.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What are scale functions used for these days?

• They serve as building blocks for obtaining more

sophisticated identities of (modified) spectrally negative Lévy processes.

• They have been used as key tools for, amongst others,

solving optimal stopping/control problems and deriving smooth pasting principles involving spectrally negative Lévy processes.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Defining/constructing a refracted Lévy process

• Let X be a spectrally negative Lévy process of bounded
• variation. Recall that we can write

X t = ct +

• 0<s≤t

∆X s, c>0.

• If we let X model the capital reserves of an insurance

company, then c represents the premium rate and the jumps of X represent the claims.

• We now want to modify the process X. Let 0 < δ < c and

define Y by Y t = X t − δt. Note that Y is still a spectrally negative Lévy process.

• Let b be the threshold level.
• The refracted Lévy process U is then constructed as

follows.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Defining/constructing a refracted Lévy process

• Let X be a spectrally negative Lévy process of bounded
• variation. Recall that we can write

X t = ct +

• 0<s≤t

∆X s, c>0.

• If we let X model the capital reserves of an insurance

company, then c represents the premium rate and the jumps of X represent the claims.

• We now want to modify the process X. Let 0 < δ < c and

define Y by Y t = X t − δt. Note that Y is still a spectrally negative Lévy process.

• Let b be the threshold level.
• The refracted Lévy process U is then constructed as

follows.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Defining/constructing a refracted Lévy process

• Let X be a spectrally negative Lévy process of bounded
• variation. Recall that we can write

X t = ct +

• 0<s≤t

∆X s, c>0.

• If we let X model the capital reserves of an insurance

company, then c represents the premium rate and the jumps of X represent the claims.

• We now want to modify the process X. Let 0 < δ < c and

define Y by Y t = X t − δt. Note that Y is still a spectrally negative Lévy process.

• Let b be the threshold level.
• The refracted Lévy process U is then constructed as

follows.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Defining/constructing a refracted Lévy process

• Let X be a spectrally negative Lévy process of bounded
• variation. Recall that we can write

X t = ct +

• 0<s≤t

∆X s, c>0.

• If we let X model the capital reserves of an insurance

company, then c represents the premium rate and the jumps of X represent the claims.

• We now want to modify the process X. Let 0 < δ < c and

define Y by Y t = X t − δt. Note that Y is still a spectrally negative Lévy process.

• Let b be the threshold level.
• The refracted Lévy process U is then constructed as

follows.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Defining/constructing a refracted Lévy process

• Let X be a spectrally negative Lévy process of bounded
• variation. Recall that we can write

X t = ct +

• 0<s≤t

∆X s, c>0.

• If we let X model the capital reserves of an insurance

company, then c represents the premium rate and the jumps of X represent the claims.

• We now want to modify the process X. Let 0 < δ < c and

define Y by Y t = X t − δt. Note that Y is still a spectrally negative Lévy process.

• Let b be the threshold level.
• The refracted Lévy process U is then constructed as

follows.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

b T1 S1 T2 S2 T3

• The refracted process can be seen as an insurance risk

process with a two-step premium rate or as a risk process where the company pays out dividends according to a ‘threshold strategy’.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Related literature and objectives

• Insurance
• Lin, Pavlova (2006)
• Zhang, Zhou, Guo (2006)
• Gerber, Shiu (2006)
• etc.
• Queuing
• Bekker, Boxma, Resing (2007)
• Up until now only the case when the Lévy measure is a

finite measure has been studied.

• We want to extend the concept of a refracted Lévy

processes to the case where X is a general spectrally negative Lévy process.

• Further we want to find identities expressed in terms of

scale functions related to exit problems of refracted Lévy processes.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Related literature and objectives

• Insurance
• Lin, Pavlova (2006)
• Zhang, Zhou, Guo (2006)
• Gerber, Shiu (2006)
• etc.
• Queuing
• Bekker, Boxma, Resing (2007)
• Up until now only the case when the Lévy measure is a

finite measure has been studied.

• We want to extend the concept of a refracted Lévy

processes to the case where X is a general spectrally negative Lévy process.

• Further we want to find identities expressed in terms of

scale functions related to exit problems of refracted Lévy processes.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Related literature and objectives

• Insurance
• Lin, Pavlova (2006)
• Zhang, Zhou, Guo (2006)
• Gerber, Shiu (2006)
• etc.
• Queuing
• Bekker, Boxma, Resing (2007)
• Up until now only the case when the Lévy measure is a

finite measure has been studied.

• We want to extend the concept of a refracted Lévy

processes to the case where X is a general spectrally negative Lévy process.

• Further we want to find identities expressed in terms of

scale functions related to exit problems of refracted Lévy processes.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Related literature and objectives

• Insurance
• Lin, Pavlova (2006)
• Zhang, Zhou, Guo (2006)
• Gerber, Shiu (2006)
• etc.
• Queuing
• Bekker, Boxma, Resing (2007)
• Up until now only the case when the Lévy measure is a

finite measure has been studied.

• We want to extend the concept of a refracted Lévy

processes to the case where X is a general spectrally negative Lévy process.

• Further we want to find identities expressed in terms of

scale functions related to exit problems of refracted Lévy processes.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

The case when X has paths of unbounded variation

• The same construction as in the case when X is of

bounded variation, does not work anymore.

T1=S1=T2=S2=… b

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

How then to define a refracted Lévy process?

• We define a refracted Lévy process U as the solution to

the SDE: dUt = −δ1{Ut>b}dt + dXt

• r equivalently

Ut = −δ t 1{Us>b}ds + Xt.

• Question: what about existence and uniqueness?
• Uniqueness: easy argument.
• Existence: problem when X is of unbounded variation and

σ = 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

How then to define a refracted Lévy process?

• We define a refracted Lévy process U as the solution to

the SDE: dUt = −δ1{Ut>b}dt + dXt

• r equivalently

Ut = −δ t 1{Us>b}ds + Xt.

• Question: what about existence and uniqueness?
• Uniqueness: easy argument.
• Existence: problem when X is of unbounded variation and

σ = 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

How then to define a refracted Lévy process?

• We define a refracted Lévy process U as the solution to

the SDE: dUt = −δ1{Ut>b}dt + dXt

• r equivalently

Ut = −δ t 1{Us>b}ds + Xt.

• Question: what about existence and uniqueness?
• Uniqueness: easy argument.
• Existence: problem when X is of unbounded variation and

σ = 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

How then to define a refracted Lévy process?

• We define a refracted Lévy process U as the solution to

the SDE: dUt = −δ1{Ut>b}dt + dXt

• r equivalently

Ut = −δ t 1{Us>b}ds + Xt.

• Question: what about existence and uniqueness?
• Uniqueness: easy argument.
• Existence: problem when X is of unbounded variation and

σ = 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Existence when X is of unbounded variation

• Let (X (n)) be a sequence of bounded variation spectrally

negative Lévy processes such that for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|X (n)

s

− Xs| = 0 a.s.

• Let U(n) be the refracted Lévy process corresponding to

X (n).

• One can show that for all n, m

sup

s∈[0,t]

|U(n)

s

− U(m)

s

| ≤ 2 sup

s∈[0,t]

|X (n)

s

− X (m)

s

| a.s.

• Therefore (U(n)) is a Cauchy sequence a.s. in the space of

cadlag functions on [0, ∞) (equipped with the local uniform topology).

• Hence there exists U(∞) such that

for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|U(n)

s

− U(∞)

s

| = 0 a.s.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Existence when X is of unbounded variation

• Let (X (n)) be a sequence of bounded variation spectrally

negative Lévy processes such that for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|X (n)

s

− Xs| = 0 a.s.

• Let U(n) be the refracted Lévy process corresponding to

X (n).

• One can show that for all n, m

sup

s∈[0,t]

|U(n)

s

− U(m)

s

| ≤ 2 sup

s∈[0,t]

|X (n)

s

− X (m)

s

| a.s.

• Therefore (U(n)) is a Cauchy sequence a.s. in the space of

cadlag functions on [0, ∞) (equipped with the local uniform topology).

• Hence there exists U(∞) such that

for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|U(n)

s

− U(∞)

s

| = 0 a.s.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Existence when X is of unbounded variation

• Let (X (n)) be a sequence of bounded variation spectrally

negative Lévy processes such that for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|X (n)

s

− Xs| = 0 a.s.

• Let U(n) be the refracted Lévy process corresponding to

X (n).

• One can show that for all n, m

sup

s∈[0,t]

|U(n)

s

− U(m)

s

| ≤ 2 sup

s∈[0,t]

|X (n)

s

− X (m)

s

| a.s.

• Therefore (U(n)) is a Cauchy sequence a.s. in the space of

cadlag functions on [0, ∞) (equipped with the local uniform topology).

• Hence there exists U(∞) such that

for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|U(n)

s

− U(∞)

s

| = 0 a.s.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Existence when X is of unbounded variation

• Let (X (n)) be a sequence of bounded variation spectrally

negative Lévy processes such that for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|X (n)

s

− Xs| = 0 a.s.

• Let U(n) be the refracted Lévy process corresponding to

X (n).

• One can show that for all n, m

sup

s∈[0,t]

|U(n)

s

− U(m)

s

| ≤ 2 sup

s∈[0,t]

|X (n)

s

− X (m)

s

| a.s.

• Therefore (U(n)) is a Cauchy sequence a.s. in the space of

cadlag functions on [0, ∞) (equipped with the local uniform topology).

• Hence there exists U(∞) such that

for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|U(n)

s

− U(∞)

s

| = 0 a.s.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Existence when X is of unbounded variation

• Let (X (n)) be a sequence of bounded variation spectrally

negative Lévy processes such that for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|X (n)

s

− Xs| = 0 a.s.

• Let U(n) be the refracted Lévy process corresponding to

X (n).

• One can show that for all n, m

sup

s∈[0,t]

|U(n)

s

− U(m)

s

| ≤ 2 sup

s∈[0,t]

|X (n)

s

− X (m)

s

| a.s.

• Therefore (U(n)) is a Cauchy sequence a.s. in the space of

cadlag functions on [0, ∞) (equipped with the local uniform topology).

• Hence there exists U(∞) such that

for all t ≥ 0 lim

n→∞ sup s∈[0,t]

|U(n)

s

− U(∞)

s

| = 0 a.s.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Does U(∞) solve the SDE?

U(n)

t

= X (n)

t

−δ t 1{U(n)

s

>b}ds

n → ∞  

• 



• 

 ? U(∞)

t

= Xt −δ t 1{U(∞)

s

>b}ds

• 1{U(n)

s

>b} → 1{U(∞)

s

>b}, provided U(∞) s

= b.

• Hence U(∞) solves the SDE and thus is the (unique)

refracted process corresponding to X if we can show Px(U(∞)

t

= b) = 0 for almost every t ≥ 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Does U(∞) solve the SDE?

U(n)

t

= X (n)

t

−δ t 1{U(n)

s

>b}ds

n → ∞  

• 



• 

 ? U(∞)

t

= Xt −δ t 1{U(∞)

s

>b}ds

• 1{U(n)

s

>b} → 1{U(∞)

s

>b}, provided U(∞) s

= b.

• Hence U(∞) solves the SDE and thus is the (unique)

refracted process corresponding to X if we can show Px(U(∞)

t

= b) = 0 for almost every t ≥ 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we say when σ > 0?

• Since U(∞) is a semi-martingale, we can use the following
• ccupation time formula

∞

−∞

Lt(x)f(x)dx = t f(U(∞)

s− )d[U(∞)]c s

a.s., where f is bounded and measurable and {Lt(x) : t ≥ 0} is the semi-martingale local time of U(∞) at level x.

• Choosing f(x) = 1{x=b} and using that [U(∞)]c

s = σ2s, we

get 0 = ∞

−∞

Lt(x)1{x=b}dx = σ2 t 1{U(∞)

s

=b}ds

a.s.

• Hence U(∞)

t

= b a.s. for almost every t ≥ 0 and so a refracted process exists when σ > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we say when σ > 0?

• Since U(∞) is a semi-martingale, we can use the following
• ccupation time formula

∞

−∞

Lt(x)f(x)dx = t f(U(∞)

s− )d[U(∞)]c s

a.s., where f is bounded and measurable and {Lt(x) : t ≥ 0} is the semi-martingale local time of U(∞) at level x.

• Choosing f(x) = 1{x=b} and using that [U(∞)]c

s = σ2s, we

get 0 = ∞

−∞

Lt(x)1{x=b}dx = σ2 t 1{U(∞)

s

=b}ds

a.s.

• Hence U(∞)

t

= b a.s. for almost every t ≥ 0 and so a refracted process exists when σ > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we say when σ > 0?

• Since U(∞) is a semi-martingale, we can use the following
• ccupation time formula

∞

−∞

Lt(x)f(x)dx = t f(U(∞)

s− )d[U(∞)]c s

a.s., where f is bounded and measurable and {Lt(x) : t ≥ 0} is the semi-martingale local time of U(∞) at level x.

• Choosing f(x) = 1{x=b} and using that [U(∞)]c

s = σ2s, we

get 0 = ∞

−∞

Lt(x)1{x=b}dx = σ2 t 1{U(∞)

s

=b}ds

a.s.

• Hence U(∞)

t

= b a.s. for almost every t ≥ 0 and so a refracted process exists when σ > 0.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0?

• For each ǫ > 0 there exists N > 0 such that for all n > N

{U(∞)

t

= b} ⊆ {U(n)

t

∈ (b − ǫ, b + ǫ)} a.s.

• Hence for each ǫ > 0

Px(U(∞)

t

= b) ≤ lim inf

n→∞ Px(U(n) t

∈ (b − ǫ, b + ǫ)).

• Note that Px(U(∞)

t

= b) = 0 for t ≥ 0 a.e. is equivalent to ∞

0 e−qtPx(U(∞) t

= b)dt = 0, where q > 0.

• By Fatou’s lemma

∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞

∞ e−qtPx(U(n)

t

∈ (b − ǫ, b + ǫ))dt.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0?

• For each ǫ > 0 there exists N > 0 such that for all n > N

{U(∞)

t

= b} ⊆ {U(n)

t

∈ (b − ǫ, b + ǫ)} a.s.

• Hence for each ǫ > 0

Px(U(∞)

t

= b) ≤ lim inf

n→∞ Px(U(n) t

∈ (b − ǫ, b + ǫ)).

• Note that Px(U(∞)

t

= b) = 0 for t ≥ 0 a.e. is equivalent to ∞

0 e−qtPx(U(∞) t

= b)dt = 0, where q > 0.

• By Fatou’s lemma

∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞

∞ e−qtPx(U(n)

t

∈ (b − ǫ, b + ǫ))dt.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0?

• For each ǫ > 0 there exists N > 0 such that for all n > N

{U(∞)

t

= b} ⊆ {U(n)

t

∈ (b − ǫ, b + ǫ)} a.s.

• Hence for each ǫ > 0

Px(U(∞)

t

= b) ≤ lim inf

n→∞ Px(U(n) t

∈ (b − ǫ, b + ǫ)).

• Note that Px(U(∞)

t

= b) = 0 for t ≥ 0 a.e. is equivalent to ∞

0 e−qtPx(U(∞) t

= b)dt = 0, where q > 0.

• By Fatou’s lemma

∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞

∞ e−qtPx(U(n)

t

∈ (b − ǫ, b + ǫ))dt.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0?

• For each ǫ > 0 there exists N > 0 such that for all n > N

{U(∞)

t

= b} ⊆ {U(n)

t

∈ (b − ǫ, b + ǫ)} a.s.

• Hence for each ǫ > 0

Px(U(∞)

t

= b) ≤ lim inf

n→∞ Px(U(n) t

∈ (b − ǫ, b + ǫ)).

• Note that Px(U(∞)

t

= b) = 0 for t ≥ 0 a.e. is equivalent to ∞

0 e−qtPx(U(∞) t

= b)dt = 0, where q > 0.

• By Fatou’s lemma

∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞

∞ e−qtPx(U(n)

t

∈ (b − ǫ, b + ǫ))dt.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0? (continued)

• Let

Rn(x, B) = ∞ e−qtPx(U(n)

t

∈ B)dt be the resolvent of U(n) with density rn(x, y). Then ∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞ Rn(x, (b − ǫ, b + ǫ))

= lim inf

n→∞

b+ǫ

b−ǫ

rn(x, y)dy = b+ǫ

b−ǫ

lim inf

n→∞ rn(x, y)dy

→ 0 as ǫ ↓ 0.

• The above calculation can be made rigorous by expressing

rn(x, y) in terms of scale functions.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0? (continued)

• Let

Rn(x, B) = ∞ e−qtPx(U(n)

t

∈ B)dt be the resolvent of U(n) with density rn(x, y). Then ∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞ Rn(x, (b − ǫ, b + ǫ))

= lim inf

n→∞

b+ǫ

b−ǫ

rn(x, y)dy = b+ǫ

b−ǫ

lim inf

n→∞ rn(x, y)dy

→ 0 as ǫ ↓ 0.

• The above calculation can be made rigorous by expressing

rn(x, y) in terms of scale functions.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0? (continued)

• Let

Rn(x, B) = ∞ e−qtPx(U(n)

t

∈ B)dt be the resolvent of U(n) with density rn(x, y). Then ∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞ Rn(x, (b − ǫ, b + ǫ))

= lim inf

n→∞

b+ǫ

b−ǫ

rn(x, y)dy = b+ǫ

b−ǫ

lim inf

n→∞ rn(x, y)dy

→ 0 as ǫ ↓ 0.

• The above calculation can be made rigorous by expressing

rn(x, y) in terms of scale functions.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0? (continued)

• Let

Rn(x, B) = ∞ e−qtPx(U(n)

t

∈ B)dt be the resolvent of U(n) with density rn(x, y). Then ∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞ Rn(x, (b − ǫ, b + ǫ))

= lim inf

n→∞

b+ǫ

b−ǫ

rn(x, y)dy = b+ǫ

b−ǫ

lim inf

n→∞ rn(x, y)dy

→ 0 as ǫ ↓ 0.

• The above calculation can be made rigorous by expressing

rn(x, y) in terms of scale functions.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0? (continued)

• Let

Rn(x, B) = ∞ e−qtPx(U(n)

t

∈ B)dt be the resolvent of U(n) with density rn(x, y). Then ∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞ Rn(x, (b − ǫ, b + ǫ))

= lim inf

n→∞

b+ǫ

b−ǫ

rn(x, y)dy = b+ǫ

b−ǫ

lim inf

n→∞ rn(x, y)dy

→ 0 as ǫ ↓ 0.

• The above calculation can be made rigorous by expressing

rn(x, y) in terms of scale functions.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What can we do when σ = 0? (continued)

• Let

Rn(x, B) = ∞ e−qtPx(U(n)

t

∈ B)dt be the resolvent of U(n) with density rn(x, y). Then ∞ e−qtPx(U(∞)

t

= b)dt ≤ lim inf

n→∞ Rn(x, (b − ǫ, b + ǫ))

= lim inf

n→∞

b+ǫ

b−ǫ

rn(x, y)dy = b+ǫ

b−ǫ

lim inf

n→∞ rn(x, y)dy

→ 0 as ǫ ↓ 0.

• The above calculation can be made rigorous by expressing

rn(x, y) in terms of scale functions.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

The resolvent measure in case of bounded variation

Assume X has paths of bounded variation and let U be the corresponding refracted process. Then the resolvent measure

• f U is given by Rq(x, B) =
• B∩[b,∞)

eΦ(q)(x−b) + δΦ(q)e−Φ(q)b x

b

eΦ(q)zW (q)(x − z)dz

• · Φ(q) − Φ(q)

δΦ(q) e−Φ(q)(y−b) − W (q)(x − y)

• dy

+

• B∩(−∞,b)

eΦ(q)(x−b) + δΦ(q)e−Φ(q)b x

b

eΦ(q)zW (q)(x − z)dz

• · Φ(q) − Φ(q)

Φ(q) eΦ(q)b ∞

b

e−Φ(q)zW (q)′(z − y)dz −

• W (q)(x − y) + δ

x

b

W (q)(x − z)W (q)′(z − y)dz dy.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

The resolvent measure in case of bounded variation

Assume X has paths of bounded variation and let U be the corresponding refracted process. Then the resolvent measure

• f U is given by Rq(x, B) =
• B∩[b,∞)

eΦ(q)(x−b) + δΦ(q)e−Φ(q)b x

b

eΦ(q)zW (q)(x − z)dz

• · Φ(q) − Φ(q)

δΦ(q) e−Φ(q)(y−b) − W (q)(x − y)

• dy

+

• B∩(−∞,b)

eΦ(q)(x−b) + δΦ(q)e−Φ(q)b x

b

eΦ(q)zW (q)(x − z)dz

• · Φ(q) − Φ(q)

Φ(q) eΦ(q)b ∞

b

e−Φ(q)zW (q)′(z − y)dz −

• W (q)(x − y) + δ

x

b

W (q)(x − z)W (q)′(z − y)dz dy.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Calculating exit problems for refracted Lévy processes

• In the case of bounded variation identities for exit problems
• f U can be obtained by rewriting them in terms of (known)

identities for the individual Lévy processes X and Y, hereby making use of the strong Markov property.

• Plug in the scale functions.
• Once an identity is obtained for the bounded variation

case, we can take limits to upgrade this identity to hold also for the case of unbounded variation.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Calculating exit problems for refracted Lévy processes

• In the case of bounded variation identities for exit problems
• f U can be obtained by rewriting them in terms of (known)

identities for the individual Lévy processes X and Y, hereby making use of the strong Markov property.

• Plug in the scale functions.
• Once an identity is obtained for the bounded variation

case, we can take limits to upgrade this identity to hold also for the case of unbounded variation.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Calculating exit problems for refracted Lévy processes

• In the case of bounded variation identities for exit problems
• f U can be obtained by rewriting them in terms of (known)

identities for the individual Lévy processes X and Y, hereby making use of the strong Markov property.

• Plug in the scale functions.
• Once an identity is obtained for the bounded variation

case, we can take limits to upgrade this identity to hold also for the case of unbounded variation.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Some examples

• Two sided exit problem. Let κ+

a = inf{t > 0 : Ut > a},

κ−

0 = inf{t > 0 : Ut < 0} and b ∈ (0, a). Then

Ex

• e−qκ+

a 1{κ+ a <κ− 0 }

• = W (q)(x) + δ

x

b W (q)(x − y)W (q)′(y)dy

W (q)(a) + δ a

b W (q)(a − y)W (q)′(y)dy

.

• Value of the (discounted) dividends. Let b > 0. Then

Ex κ− e−qtδ1{Ut>b}dt

• = −δ

x−b W (q)(y)dy + W (q)(x) + δ x

b W (q)(x − y)W (q)′(y)dy

Φ(q) ∞

0 e−Φ(q)yW (q)′(y + b)dy

.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

Some examples

• Two sided exit problem. Let κ+

a = inf{t > 0 : Ut > a},

κ−

0 = inf{t > 0 : Ut < 0} and b ∈ (0, a). Then

Ex

• e−qκ+

a 1{κ+ a <κ− 0 }

• = W (q)(x) + δ

x

b W (q)(x − y)W (q)′(y)dy

W (q)(a) + δ a

b W (q)(a − y)W (q)′(y)dy

.

• Value of the (discounted) dividends. Let b > 0. Then

Ex κ− e−qtδ1{Ut>b}dt

• = −δ

x−b W (q)(y)dy + W (q)(x) + δ x

b W (q)(x − y)W (q)′(y)dy

Φ(q) ∞

0 e−Φ(q)yW (q)′(y + b)dy

.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

A related optimal control problem

• Suppose the reserves of an insurance company evolves as

a spectrally negative Lévy process.

• The company now has the option to pay out dividends from

its reserves to its shareholders.

• The company is only allowed to pay out dividends at a

(dynamic) rate which does not exceed a pre-specified ceiling rate δ.

• The company wants to maximize the expected total paid
• ut (discounted) dividends until ruin.
• What is the optimal dividend strategy?

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

A related optimal control problem

• Suppose the reserves of an insurance company evolves as

a spectrally negative Lévy process.

• The company now has the option to pay out dividends from

its reserves to its shareholders.

• The company is only allowed to pay out dividends at a

(dynamic) rate which does not exceed a pre-specified ceiling rate δ.

• The company wants to maximize the expected total paid
• ut (discounted) dividends until ruin.
• What is the optimal dividend strategy?

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

A related optimal control problem

• Suppose the reserves of an insurance company evolves as

a spectrally negative Lévy process.

• The company now has the option to pay out dividends from

its reserves to its shareholders.

• The company is only allowed to pay out dividends at a

(dynamic) rate which does not exceed a pre-specified ceiling rate δ.

• The company wants to maximize the expected total paid
• ut (discounted) dividends until ruin.
• What is the optimal dividend strategy?

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

A related optimal control problem

• Suppose the reserves of an insurance company evolves as

a spectrally negative Lévy process.

• The company now has the option to pay out dividends from

its reserves to its shareholders.

• The company is only allowed to pay out dividends at a

(dynamic) rate which does not exceed a pre-specified ceiling rate δ.

• The company wants to maximize the expected total paid
• ut (discounted) dividends until ruin.
• What is the optimal dividend strategy?

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

A related optimal control problem

• Suppose the reserves of an insurance company evolves as

a spectrally negative Lévy process.

• The company now has the option to pay out dividends from

its reserves to its shareholders.

• The company is only allowed to pay out dividends at a

(dynamic) rate which does not exceed a pre-specified ceiling rate δ.

• The company wants to maximize the expected total paid
• ut (discounted) dividends until ruin.
• What is the optimal dividend strategy?

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What if the threshold strategy is applied?

• The threshold strategy at level b is an admissible strategy.

The reserves with the dividend payments included then evolves as a refracted Lévy process.

• Does the threshold strategy form an optimal strategy (for a

suitably chosen value of b)?

• Not in general, but results on related optimal dividend

problems indicate that this strategy is optimal under certain regularity conditions on the Lévy measure (completely monotone/log-convex density). However, a proof of this conjecture is still missing.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What if the threshold strategy is applied?

• The threshold strategy at level b is an admissible strategy.

The reserves with the dividend payments included then evolves as a refracted Lévy process.

• Does the threshold strategy form an optimal strategy (for a

suitably chosen value of b)?

• Not in general, but results on related optimal dividend

problems indicate that this strategy is optimal under certain regularity conditions on the Lévy measure (completely monotone/log-convex density). However, a proof of this conjecture is still missing.

Lévy processes and scale functions What is a refracted Lévy process and does it exist? Future work

What if the threshold strategy is applied?

• The threshold strategy at level b is an admissible strategy.

The reserves with the dividend payments included then evolves as a refracted Lévy process.

• Does the threshold strategy form an optimal strategy (for a

suitably chosen value of b)?

• Not in general, but results on related optimal dividend

problems indicate that this strategy is optimal under certain regularity conditions on the Lévy measure (completely monotone/log-convex density). However, a proof of this conjecture is still missing.