Part I: Introduction
Part II: Results
Markov chain approximations to scale functions
of Lévy processes
Matija Vidmar1 (joint work with A. Mijatović and S. Jacka)
Ljubljana, UL FMF
January, 2014
1 The
support of the Slovene Human Resources Development and Scholarship Fund
under contract number 11010-543/2011 is recognized.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
TOC
1 Part I: Introduction
Spectrally negative Lévy processes and their scale functions
Scale functions are important . . .
Evaluating scale functions
2 Part II: Results
Abstract
The algorithm . . .
. . . and the idea behind it
Rates of convergence
Numerical illustrations
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
Spectrally negative Lévy processes and their scale functions
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
Spectrally negative Lévy processes and their scale functions
Setting: filtered probability space (Ω, F, F = (Ft )t≥0 , P) under the
standard assumptions supporting a Lévy process X .
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
Spectrally negative Lévy processes and their scale functions
Setting: filtered probability space (Ω, F, F = (Ft )t≥0 , P) under the
standard assumptions supporting a Lévy process X .
Definition (Lévy process)
A continuous-time F-adapted stochastic process X = (Xt )t≥0 , with state
space R, is a Lévy process on the stochastic basis (F, P), if it starts at 0,
a.s., Xt−s ∼ Xt − Xs ⊥ Fs (0 ≤ s ≤ t; stationary independent
increments) and it is càdlàg off a null set.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
Spectrally negative Lévy processes and their scale functions
Setting: filtered probability space (Ω, F, F = (Ft )t≥0 , P) under the
standard assumptions supporting a Lévy process X .
Definition (Lévy process)
A continuous-time F-adapted stochastic process X = (Xt )t≥0 , with state
space R, is a Lévy process on the stochastic basis (F, P), if it starts at 0,
a.s., Xt−s ∼ Xt − Xs ⊥ Fs (0 ≤ s ≤ t; stationary independent
increments) and it is càdlàg off a null set.
A Lévy process is called spectrally negative, if:
1
it has no positive jumps (a.s.) and
2
does not have a.s. monotone paths.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
Spectrally negative Lévy processes and their scale functions
Setting: filtered probability space (Ω, F, F = (Ft )t≥0 , P) under the
standard assumptions supporting a Lévy process X .
Definition (Lévy process)
A continuous-time F-adapted stochastic process X = (Xt )t≥0 , with state
space R, is a Lévy process on the stochastic basis (F, P), if it starts at 0,
a.s., Xt−s ∼ Xt − Xs ⊥ Fs (0 ≤ s ≤ t; stationary independent
increments) and it is càdlàg off a null set.
A Lévy process is called spectrally negative, if:
1
it has no positive jumps (a.s.) and
2
does not have a.s. monotone paths.
For this class of Lévy processes, fluctuation theory in terms of the two
families of scale functions, (W (q) )q∈[0,∞) and (Z (q) )q∈[0,∞) , has been
developed.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
(cont’d)
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
(cont’d)
Notation: Fix henceforth X , a spectrally negative Lévy process with Laplace exponent
ψ, ψ(β) := log E[e βX1 ] (β ∈ {γ ∈ C : <γ ≥ 0} =: C→ ).
Note: ψ is given by [(σ 2 , λ, µ)c̃ – characteristic triplet; c̃ := idR 1[−V ,0) , V ∈ {0, 1}]:
Z
1
e βy − βc̃(y ) − 1 λ(dy ), β ∈ C→ .
ψ(β) = σ 2 β 2 + µβ +
2
(−∞,0)
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
(cont’d)
Notation: Fix henceforth X , a spectrally negative Lévy process with Laplace exponent
ψ, ψ(β) := log E[e βX1 ] (β ∈ {γ ∈ C : <γ ≥ 0} =: C→ ).
Note: ψ is given by [(σ 2 , λ, µ)c̃ – characteristic triplet; c̃ := idR 1[−V ,0) , V ∈ {0, 1}]:
Z
1
e βy − βc̃(y ) − 1 λ(dy ), β ∈ C→ .
ψ(β) = σ 2 β 2 + µβ +
2
(−∞,0)
We then find in A. E. Kyprianou’s Introductory Lectures on Fluctuations of Lévy
Processes, the following characterization of scale functions.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Spectrally negative Lévy processes and their scale functions
(cont’d)
Notation: Fix henceforth X , a spectrally negative Lévy process with Laplace exponent
ψ, ψ(β) := log E[e βX1 ] (β ∈ {γ ∈ C : <γ ≥ 0} =: C→ ).
Note: ψ is given by [(σ 2 , λ, µ)c̃ – characteristic triplet; c̃ := idR 1[−V ,0) , V ∈ {0, 1}]:
Z
1
e βy − βc̃(y ) − 1 λ(dy ), β ∈ C→ .
ψ(β) = σ 2 β 2 + µβ +
2
(−∞,0)
We then find in A. E. Kyprianou’s Introductory Lectures on Fluctuations of Lévy
Processes, the following characterization of scale functions.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Scale functions are important . . .
Scale functions are important . . .
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Scale functions are important . . .
Scale functions are important . . .
There exist numerous identities concerning boundary crossing problems
and related path decompositions in which scale functions feature. For
example;
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Scale functions are important . . .
Scale functions are important . . .
There exist numerous identities concerning boundary crossing problems
and related path decompositions in which scale functions feature. For
example;
(i) Two-sided exit problem. For a ≥ 0 let Ta (respectively Ta− ) be
the first entrance time of X to [a, ∞) (respectively (−∞, −a)).
Then:
W (x)
P(Tx− > Ta ) =
,
W (a + x)
whenever {a, x} ⊂ (0, ∞) =: R+ .
(ii) Ruin probabilities. In the case that X drifts to +∞ we have for
x ∈ R+ , the generalised Cramér-Lundberg identity:
P(Tx− = ∞) = W (x)ψ 0 (0+).
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Scale functions are important . . .
(cont’d)
(iii) Continuous-state branching processes. Under mild conditions, the law of the
supremum of a continuous state branching process Y is given by the identity (for
x ∈ R+ , y ∈ R):
W (x − y )
,
Py (sup Ys ≤ x) =
W (x)
s≥0
where W is the scale function of the associated Lévy process.
(iv) Population biology. The typical branch length H between two consecutive
individuals alive at time t ∈ R+ , conditionally on there being at least two extant
individuals at said time, satisfies the identity:
P(H < s) =
1 − W (s)−1
,
1 − W (t)−1
whenever s ∈ (0, t], and with W the scale function associated to the jumping
chronological contour process.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Scale functions are important . . .
(cont’d)
(iii) Continuous-state branching processes. Under mild conditions, the law of the
supremum of a continuous state branching process Y is given by the identity (for
x ∈ R+ , y ∈ R):
W (x − y )
,
Py (sup Ys ≤ x) =
W (x)
s≥0
where W is the scale function of the associated Lévy process.
(iv) Population biology. The typical branch length H between two consecutive
individuals alive at time t ∈ R+ , conditionally on there being at least two extant
individuals at said time, satisfies the identity:
P(H < s) =
1 − W (s)−1
,
1 − W (t)−1
whenever s ∈ (0, t], and with W the scale function associated to the jumping
chronological contour process.
Miscellaneous other areas featuring scale functions W (q) (together with
their derivatives and the integrals Z (q) ), include queuing theory, optimal
stopping and control problems, fragmentation processes etc.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Evaluating scale functions
Evaluating scale functions
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Evaluating scale functions
Evaluating scale functions
• Central to applications to be able to evaluate scale functions
numerically for any spectrally negative Lévy process X .
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Evaluating scale functions
Evaluating scale functions
• Central to applications to be able to evaluate scale functions
numerically for any spectrally negative Lévy process X .
• Analytically, scale functions are characterized in terms of their
Laplace transforms.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Evaluating scale functions
Evaluating scale functions
• Central to applications to be able to evaluate scale functions
numerically for any spectrally negative Lévy process X .
• Analytically, scale functions are characterized in terms of their
Laplace transforms.
• Typically not possible to perform the inversion explicitly; user is
faced with a Laplace inversion algorithm involving complex
numerical integration of a function of the Laplace exponent of X .
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Evaluating scale functions
Evaluating scale functions
• Central to applications to be able to evaluate scale functions
numerically for any spectrally negative Lévy process X .
• Analytically, scale functions are characterized in terms of their
Laplace transforms.
• Typically not possible to perform the inversion explicitly; user is
faced with a Laplace inversion algorithm involving complex
numerical integration of a function of the Laplace exponent of X .
• The latter, however
(a) says little about the dependence of the scale function on the Lévy
triplet of X and
(b) fails to ensure that the computed values of the scale function are
probabilistically meaningful.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Abstract
Abstract
A copy-paste from the paper:
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
The algorithm . . .
The algorithm. . .
To compute W (x) for some x > 0, choose small h > 0 such that x/h is
an integer and define the approximation Wh (x) by the formula:
pWh (y + h) = Wh (y ) −
y /h
X
qk Wh (y − kh),
Wh (0) = (pγh)−1 ,
k=1
for y = 0, h, 2h, . . . , x − h, where the coefficients p, (qk )k≥1 and γ are
given explicitly in terms of (σ 2 , λ, µ)c̃ and h.
Remark: These coefficients also have a probabilistic interpretation in
terms of a process X h , which is used to weakly approximate X (details to
follow (!)), and Wh is nothing else than the scale function of X h .
Main result:
1
shows that, as h ↓ 0, the pointwise convergence of the approximating
scale functions to those of the original Lévy process holds and
2
establishes the sharp rate at which this convergence transpires on
Zh := hZ, the latter under a weak condition on the Lévy measure.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
The algorithm . . .
(cont’d)
Key attractions:
(a) consistency : for each fixed h > 0, algorithm calculates precisely (i.e.
without numerical error, only modulo rounding) the values of the
scale function Wh for a process X h , which weakly approximates X ;
(b) conceptual simplicity : a weak approximation of X (as a Markov
process) by a continuous time Markov chain X h , which is skip-free to
the right, provides a natural way of encoding the underlying
probabilistic structure of the problem in the design of the algorithm;
(c) robustness: method valid for all spectrally negative Lévy processes;
(d) straightforwardness of the algorithm: implementation requires only to
solve a lower triangular system of linear equations;
(e) convergence: rates of convergence have been established under mild
assumptions, together with their optimality.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
. . . and the idea behind it
. . . and the idea behind it
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
. . . and the idea behind it
. . . and the idea behind it
(q)
Note: similar recursions obtain for functions Wh
approximating W (q) and Z (q) , respectively.
Markov chain approximations to scale functions of Lévy processes
(q)
and Zh
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
. . . and the idea behind it
. . . and the idea behind it
(q)
(q)
Note: similar recursions obtain for functions Wh and Zh
approximating W (q) and Z (q) , respectively.
A two step programme, which yields these formulae, as follows:
(i) approximate the spectrally negative Lévy process X by a
continuous-time Markov chain X h with state space
Zh := {hk: k ∈ Z} (h ∈ (0, h? ) for some h? > 0):
(a) Brownian motion with drift → asymmetric random walk (using two
different schemes, according as to whether σ 2 > 0 or not);
(b) Lévy measure becomes concentrated on Zh ;
(c) + details for the part of the Lévy measure around the origin (when
the latter infinite).
[See Mijatović, Vidmar, Saul: Markov chain approximations for transition
densities of Lévy processes, EJP 19 (2014).]
(q)
(q)
(ii) find an algorithm for computing the scale functions Wh and Zh
of the chain X h . Requires a fluctuation theory for what are
continuous-time embedded right-continuous random walks. [See
Vidmar: Fluctuation theory for upwards skip-free Levy chains.]
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Rates of convergence
Main results
Fx q ≥ 0; let K , G be bounded subset of (0, ∞); K bounded away from
zero when σ 2 = 0; and define:
∆K
W (h) :=
sup
x∈Zh ∩K
(q)
Wh (x − δ 0 h) − W (q) (x) and ∆G
Z (h) :=
sup
x∈Zh ∩G
(q)
Zh (x) − Z (q) (x) ,
0
where δ equals 0 if X has sample paths of finite variation and 1
otherwise. Further introduce:
Z
κ(δ) :=
|y |λ(dy ),
for any δ ≥ 0.
[−1,−δ)
If the jump part of X has paths of infinite variation, assume:
Assumption
There exists ∈ (1, 2) with:
(1) lim supδ↓0 δ λ(−1, −δ) < ∞ and
R
(2) lim inf δ↓0 [−δ,0) x 2 λ(dx)/δ 2− > 0.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Rates of convergence
(cont’d)
Theorem
Then the rates of convergence of the scale functions are summarized by
the following table:
λ(R) = 0
0 < λ(R) & κ(0) < ∞
κ(0) = ∞
2
G
∆K
W (h) = O(h ) and ∆Z (h) = O(h)
K
G
∆W (h) + ∆Z (h) = O(h)
G
2−
)
∆K
W (h) + ∆Z (h) = O(h
Moreover, the rates so established are sharp in the sense that for each of
the three entries in the table above, examples of spectrally negative Lévy
processes are constructed for which the rate of convergence is no better
than stipulated.
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Numerical illustrations
Some numerical experiments
Note: Algorithm computes values recursively, and so, together with
Wh (x), we obtain automatically and necessarily Wh |[0,x] !
1
W
0.9
W (⋅−1)
1
0.8
W1/2(⋅−1/2)
0.7
W1/4(⋅−1/4)
0.6
W1/8(⋅−1/8)
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure: Convergence of Wh (· − h) to W (as h ↓ 0) on the interval [0, 1] for
Brownian motion with drift (σ 2 = µ = 1, λ = 0, V = 0).
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Numerical illustrations
2
2
σ =0, µ=2
σ =0, µ=1
0.7
2
W
W
W
W
1
1
1.8
W1/2
0.65
W1/2
W1/4
W1/4
1.6
W1/8
W1/8
0.6
1.4
0.55
0.5
1.2
0
0.2
0.4
0.6
0.8
1
1
0
0.2
σ2=1, µ=2
0.6
0.8
1
0.8
1
σ2=1, µ=1
0.7
1.4
0.6
W
W1/2(⋅−1/2)
1.2
0.5
W1/4(⋅−1/4)
1
0.4
0.8
0.3
0.6
0.2
0.4
0.1
0
0.4
W
W1(⋅−1)
W1/2(⋅−1/2)
W1/4(⋅−1/4)
0.2
0
0.2
0.4
0.6
0.8
1
0
0
0.2
0.4
0.6
Figure: Convergence of Wh (respectively Wh (· − h)) to W (as h ↓ 0) on the
interval [0, 1] for the compound Poisson process with drift µ ∈ {2, 1} (V = 0),
exponential jumps (λ(dy ) = aρe ρy 1(−∞,0) (y )dy , a = ρ = 1) and with σ 2 = 0
(respectively σ 2 = 1).
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Numerical illustrations
0.5
W
0.45
W1(⋅−1)
0.4
W1/2(⋅−1/2)
W1/4(⋅−1/4)
0.35
W1/8(⋅−1/8)
0.3
W1/16(⋅−1/16)
W1/212(⋅−1/212)
0.25
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure: Convergence of Wh (· − h) to W (as h ↓ 0) on the interval [0, 1] for a
stable spectrally negative Lévy process (σ 2 = 0, µ = 1/(β − 1),
λ(dy ) = (−y1)1+β 1(−∞,0) (y )dy , β = 3/2, V = 1).
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Markov chain approximations to scale functions of Lévy processes
Part II: Results
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Thank you for your attention!
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)
Part I: Introduction
Part II: Results
Thank you for your attention!
[Further details; Mijatović, Vidmar, Saul: Markov chain approximations
to scale functions of Levy processes.]
Markov chain approximations to scale functions of Lévy processes
Matija Vidmar (joint work with A. Mijatović and S. Jacka)