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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Lévy process, Lévy driven SDE,and quasi-likelihood estimation ∗
Hiroki Masuda
Kyushu UniversityJST CREST
YUIMA Summer School 2019
Brixen-Bressanone, Italy
June 25–28, 2019
∗This version: June 27, 2019Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 1 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Contents, June 27 a.m.
1 Lévy process: basics and simulationBasicsSimulation in YUIMA
2 Lévy driven SDE: basics and simulationBasicsSimulation in YUIMA
3 Quasi-likelihood estimation of Lévy driven SDEIntroduction and backgroundAsymptotics
4 Quasi-likelihood estimation of Lévy driven SDE (YUIMA demo)
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 2 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
5
Objective
Objective
Objective
Paul Lévy(1886-1971)
Kiyosi Itô(1915-2008)
Joseph L. Doob(1910-2004)
Norbert Wiener(1894-1964)
Martingale limit theorem Ito-stochastic calculus
Asymptotic / Non-asymptoticStatistics
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 3 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
1 Lévy process: basics and simulation
2 Lévy driven SDE: basics and simulation
3 Quasi-likelihood estimation of Lévy driven SDE
4 Quasi-likelihood estimation of Lévy driven SDE (YUIMA demo)
Standard references:
[Applebaum, 2009][Bertoin, 1996][Protter, 2005, (2nd.) Chapter I.4][Sato, 1999]
[Iacus and Yoshida, 2018] for many YUIMA examples
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 4 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Discrete-time random walk S1, S2, . . .
Sn :=
n∑j=1
ϵj , S0 := 0
ϵ1, ϵ2, . . . : i.i.d. random variables
Independent and stationary increments
Sk − Sl =k∑
j=l+1
ϵj
1 Sj1 − Sj0 , Sj2 − Sj1 , . . . , Sjn − Sjn−1 independent (n ∈ N)2 Sjk − Sjk−1 ∼ Sjk−jk−1 (k ∈ N)
Natural continuous-time counterpart?
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 5 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Lévy process: Continuous-time random walk
Xt =
n∑j=1
(Xtj −Xtj−1) =:n∑
j=1
∆jX (X0 = 0 a.s.)
Definition
1 Independent and stationary increments (0 = t0 < t1 < · · · < tn; n ∈ N)
Xt1 −Xt0 , Xt2 −Xt1 , . . . , Xtn −Xtn−1 are independent.Xtj −Xtj−1 ∼ Xtj−tj−1
2 Continuity in probability: Xsp−→ Xt as s→ t.
No pre-assigned jump time: P(|∆Xt| > 0) = 0 for each t > 0.W.l.g. we may suppose that t 7→ Xt(ω) is càdlàg.
∃Lévy process X s.t. X1 ∼ F ⇐⇒ F is infinitely divisibleF is infinitely divisible
def.⇐⇒ ∀n∃Fn, F = F ∗nn (:= Fn ∗ · · · ∗ Fn)Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 6 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Real data may be often leptokurtic: higher kurtosis than the normal (NYSEminutes data); maybe also skewed (energy consumption data).
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Density fits: hyperbolic vs normal
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Density fits: stable vs normal
Energy consumption data: Gaussian fit
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Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 7 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Two prominent cases
If X is a counting process:
Xt =∑j∈N
I(s ≤ τj) where 0 < τ1 < τ2 < . . . are event-occurrence times,
then X is necessarily a Poisson process with intensity λ:
∃λ > 0, Xt ∼ Pois(λt), t ∈ R+.
If X has continuous sample paths, then X is necessarily a Wiener process:
∃µ ∈ R ∃σ ≥ 0, Xt ∼ N(µt, σ2t), t ∈ R+,
i.e. we may writeXt = µt+ σwt
for a standard Wiener process w (wt ∼ N(0, t)).
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 8 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Lévy-Khintchine representation‡
Form of the Fourier transform:
φXt(u) := E(eiuXt) = exp{tψ(u)}, u ∈ R,
with the characteristic exponent function
ψ(u) := iuµ1 −1
2σ2u2 +
∫ (eiuz − 1− iuzI(|z| ≤ 1)
)ν(dz).
The element (µ1, σ2, ν) is called the generating triplet of Z:
1 µ1 is the drift (location-shift),2 σ2 ≥ 0 is the Gaussian variance,3 ν is the Lévy measure (roughly, expected jump frequency).∫
(1 ∧ |z|2)ν(dz)
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Remarks
∀q > 0: “E(|Xt|q) 1
|z|qν(dz) 1
zν(dz),
var(Xt) = i−2tψ′′(0) = tσ2 + t
∫z2ν(dz).
kth cumulant of Xt: if φXt is of Ck-class (k ≥ 3), then
i−k∂ku logφXt(0) = t
∫zkν(dz).
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 10 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
1
tlogφXt(u) = iuµ1 −
1
2σ2u2 +
∫ (eiuz − 1− iuzI(|z| ≤ 1)
)ν(dz)
In general, we should not do something like∫ (eiuz − 1− iuzI(|z| ≤ 1)
)ν(dz)
=
∫ (eiuz − 1
)ν(dz)− iu
∫|z|≤1
zν(dz).
The generating triplet uniquely determines the law of the process X, sothat it determines e.g.
L(
supt∈[0,1]
|Xt|), L
(inf{t ≥ 0 : |Xt| > 1}
), L
(∫ 10
Xsds
).
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 11 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Lévy-Itô decomposition of sample path
Sum of independent Gaussian and non-Gaussian factors:
Xt = µ1t+ σ wt + Jt
More formally [Applebaum, 2009]:
Xt = tµ1 + σwt
+
∫ t0
∫|z|>1
zµ(ds, dz) +
∫ t0
∫|z|≤1
z(µ− ν)(ds, dz). (1)
Poisson random measure µ((0, t], A) :=∑
0
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Toward generating discrete-time sample
Want to generate sample Xt1 , Xt2 , . . . , Xtn from
Xt = µt+ σ wt + Jt,
where 0 = t0 < t1 < · · · < tn = t are (fine) sampling time points.
Enough to be able to generate Xt for any t > 0:
Xt =
n∑j=1
(Xtj −Xtj−1) =n∑
j=1
∆jX, ∆jX ∼ Xtj−tj−1
Simulator list in YUIMA [Brouste et al., 2014]:
Help documents of rng function in YUIMA.[Iacus and Yoshida, 2018, Chapter 4]
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 13 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: Compound Poisson process
Xt =
Nt∑j=1
ϵj
Nt ∼ Pois(λt): Poisson process with intensity λ > 0.ϵ1, ϵ2, · · · ∼ i.i.d. with P(ϵ1 = 0) = 0, independent of N .
Any Lévy process can be a weak limit of a compound Poisson process.
[Sato, 1999, Corollary 8.8]
▷ yss2019 hm demo.html
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 14 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: Inverse-Gaussian subordinator
The density of Xt ∼ IG(δt, γ), δ, γ > 0, is
x 7→ δteδtγ
√2π
x−3/2 exp
{− 1
2
((δt)2
x+ γ2x
)}, x > 0.
▷ yss2019 hm demo.html
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 15 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: Normal inverse Gaussian Lévy process
Normal variance-mean mixture, i.e. subordination §:
Xt = µt+ βτt + wτt
τt ∼ IG(δt, γ),Standard Wiener process w independent of τ .
The density of Xt ∼ NIG(α, β, δt, µt) is
x 7→ αδt exp{δt√α2 − β2 + β(x− µt)}K1(αψ(x; δt, µt))
πψ(x; δt, µt)
α2 := γ2 + β2
ψ(x; δt, µt) :=√
(δt)2 + (x− µt)2
▷ yss2019 hm demo.html
§General subordination formulae for probability and Lévy densities are available (τ → X);see [Iacus and Yoshida, 2018, Sect 4.8.3]; [Sato, 1999, chap 6] for general account.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 16 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Put simply
φXt(u) := E(eiuXt) = exp{tψ(u)}, u ∈ R,
ψ(u) := iuµ1 −1
2σ2u2 +
∫ (eiuz − 1− iuzI(|z| ≤ 1)
)ν(dz).
Lévy process is completely characterized by the generating triplet, which
sometimes crucial in calculations,while sometimes does not matter at all.
Given any infinitely divisible distribution F , there essentially uniquelycorresponds a Lévy process X such that X1 ∼ F .
rng has several slots for generating specific Lévy process (see help file)
Approximate “inputting-ν(dz)” way, yet to be implemented.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 17 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
1 Lévy process: basics and simulation
2 Lévy driven SDE: basics and simulation
3 Quasi-likelihood estimation of Lévy driven SDE
4 Quasi-likelihood estimation of Lévy driven SDE (YUIMA demo)
A reader-friendly and comprehensive monograph is [Applebaum, 2009].
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 18 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Diffusion process is an SDE driven by a Wiener process:
dXt = a(Xt)dt+ b(Xt)dwt,
a strong solution X realized as a functional form
X = F (X0, w).
e.g. Geometric Brownian motion:
dXt = Xt(µdt+ σdwt),
Xt = X0 exp
{σwt +
(µ− σ
2
2
)t
}.
The driving Wiener process w could be replaced by a Lévy process.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 19 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Lévy driven Stochastic Differential Equation (SDE)
dXt = a(Xt)dt+ b(Xt)dwt + c(Xt−)dJt(Xt = x0 +
∫ t0
a(Xs)ds+
∫ t0
b(Xs)dws +
∫ t0
c(Xs−)dJs
)Initial variable X0 ∈ Rd, possibly random.Driving noises:
d′-dimensional standard Wiener process w = (wj)d′
j=1;
d′′-dimensional pure-jump Lévy process J = (Jj)d′′
j=1 of the form
Jt :=
∫ t0
∫|z|≤1
z(µ− ν)(ds, dz) +∫ t0
∫|z|>1
zµ(ds, dz).
Coefficient functions:
Drift coefficient a(x) = {ak(x)}k≤d : Rd → Rd
Diffusion coefficient b(x) = {bkl(x)}k≤d; l≤d′ : Rd → Rd ⊗ Rd′
Jump coefficient c(x) = {ckl(x)}k≤d; l≤d′′ : Rd → Rd × Rd′′
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 20 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
dXt = a(Xt)dt+ b(Xt)dwt + c(Xt−)dJt(Xt = x0 +
∫ t0
a(Xs)ds+
∫ t0
b(Xs)dws +
∫ t0
c(Xs−)dJs
)
The stochastic integrations:∫ t0
Ys−dJs := limn→∞
n∑j=1
Y(j−1)t/n(Jjt/n − J(j−1)t/n
)=
∫ t0
∫|z|>1
Yszµ(ds, dz) +
∫ t0
∫|z|≤1
Ysz(µ− ν)(ds, dz)
with the notation in (1), [Applebaum, 2009, Section 6.2];
L2-stochastic integration theory for small-jump part,Pathwise interlacing of large-jump component for large-jump part.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 21 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
dXt = a(Xt)dt+ b(Xt)dwt + c(Xt−)dJt(Xt = x0 +
∫ t0
a(Xs)ds+
∫ t0
b(Xs)dws +
∫ t0
c(Xs−)dJs
)
Globally Lipschitz (a, b, c): ∃K > 0, ∀x1, x2 ∈ Rd,
|a(x1)− a(x2)|+ |b(x1)− b(x2)|+ |c(x1)− c(x2)| ≤ K|x1 − x2|,
leads to existence of unique strong solution (a (w, J)-Lévy functional)
X =: F (x0, w, J).
[Applebaum, 2009, Theorems 6.2.9 and 6.4.6].
The simplest but widely applicable device to approximate a solution processto is the Euler(-Maruyama) scheme [Platen and Bruti-Liberati, 2010].
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 22 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Euler-discretization scheme
As in the case of diffusions, for tj − tj−1 small enough,
Xtj+1 = Xtj +
∫ tj+1tj
a(Xs)ds+
∫ tj+1tj
b(Xs)dws +
∫ tj+1tj
c(Xs−)dJs
≈ Xtj +∫ tj+1tj
a(Xtj−1)ds+
∫ tj+1tj
b(Xtj−1)dws +
∫ tj+1tj
c(Xtj−1)dJs
≈ Xtj−1 + a(Xtj−1)(tj − tj−1) + b(Xtj−1)(wtj − wtj−1)+ c(Xtj−1)(Jtj − Jtj−1) (2)
Need to generate Jtj − Jtj−1(∼ ∆jJ)-random number at each step.For this, YUIMA internally use the rng slots.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 23 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Generation of discretized process
The Euler-discretized process X∆ =: (X∆t ), for ∆ > 0 small enough:
1 X∆t := X0 for t ∈ [0,∆).2 For t ∈ [j∆, (j + 1)∆), j ∈ N,
X∆t := X∆(j−1)∆ + aj−1∆+ bj−1∆jw + cj−1∆jJ.
fj−1 := f(X(j−1)∆)∆jx = ∆
nj x := xtj − xtj−1 : the jth increment of a process x
Then, we approximate as Xt ≈ X∆t over a period [0, T ];Having generated a finest-approximating process X∆,we can extract any thinned process, say Xk∆ for some k ≥ 2,which plays a role of discretely observed sample from X.
Strong and weak approximation errors are defined as in diffusion cases.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 24 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
May seem like:
. . . .
Introduction and Background
. . . . . . . .
Quasi-likelihood methods
. . . .
Simulations
. . . .
Further Topics
. .
Summary and Conclusion
Lévy Driven Models: Application Fields
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Inadvisability of Gaussian noise is common in some application fields:! Signal processing (detection, estimation)! Continuous-time system identification in engineering! Trend detection and analysis in the environmental sciences! Control and optimization through time-scale separation! Physical science such as turbulence
Hiroki Masuda, ISI 2011, Dublin 4/28
Diffusion
Lévy SDE
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 25 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: Diffusion with compound-Poisson jumps
For J begin a compound Poisson process with Γ(3, 3)-distributed jumps,
dXt = {sin(Xt)−Xt} dt+ 2dwt − dJt.
Downward jumps only.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 26 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
The recipe YUIMA uses for (Xt)t∈[0,T ]:
dXt = a(Xt)dt+ b(Xt)dwt + c(Xt−)dJt
0. Generate X0 and NT ← Pois(λT ) independently, and set j = 1.0.1. If NT = 0, then (Xt)t∈[0,T ] is a diffusion dXt = a(Xt)dt+ b(Zt)dwt.0.2. Otherwise, generate U1, . . . , UNT ∼ i.i.d.U(0, T );
Sort them as U(1) < U(2) < · · · < U(NT );For k ≤ NT , pick a jk ∈ {1, 2, . . . , [T/∆]} s.t. U(k) ∈ ((jk − 1)∆, jk∆]. ¶
1. Generate ηj ∼ Nd′(0, Id′) and then1.1. If j = jk for some k, then generate ζk ∼ F (dz) (jump law) and deliver
Xj∆ ← X(j−1)∆ + aj−1∆+ bj−1√∆ ηj + cj−1ζk.
1.2. Otherwise, Xj∆ = X(j−1)∆ + aj−1∆+ bj−1√∆ ηj .
2. Update j = j + 1 and return to step 1: repeat step 1 until j = [T/∆].
▷ yss2019 hm demo.html
¶Ignores the possibility of multiple ij : that’ll be negligoble for ∆ small enough.Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 27 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: Geometric Lévy process
SDE driven by a general Lévy process Xt = µt+ σ wt + Jt:
dYt = Yt−dXt, Y0 = 1,
Yt = exp
(Xt −
σ2
2t
) ∏0
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Itô’s formula (Univariate case)
dXt = a(Xt)dt+ b(Xt)dwt + c(Xt−)dJt
f(Xt) = f(X0) +
∫ t0
f ′(Xs−)dXs +1
2
∫ t0
f ′′(Xs−)d⟨Xc⟩s
+∑
0
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: Heavy-tailed SDE
Non-Gaussian infinite-variance Lévy process Jt ∼ Sα(β, σ, µ):
φJt(u) =
−(t1/ασ)α|u|α
(1− iβsign(u) tan απ
2
)+ iµtu, α ̸= 1
−tσ|u|(1 + i
2β
πsign(u) log |u|
)+ iµtu, α = 1
(α, β, σ, µ) ∈ (0, 2)× [−1, 1]× (0,∞)× R:α > 1 ⇒ Finite mean and infinite varianceα = 1 ⇒ Cauchy (possibly skewed)α < 1 ⇒ Infinite mean
The index α ∈ (0, 2) controls tail heaviness and small-jump activity.
SDE driven by a Lévy process J1 ∼ Stable(1.3, 0, 1, 0):
dXt = −Xt√1 +X2t
dt+ dJt
▷ yss2019 hm demo.html
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 30 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: Multidimensional nonlinear SDE
The same recipe as in the one-dimensional case (2):
Xtj+1d×1
= Xtj +
∫ tj+1tj
a(Xs)d×1
ds+
∫ tj+1tj
b(Xs)d×r
dwsr×1
+
∫ tj+1tj
c(Xs−)d×m
dJsm×1
≈ Xtj−1 + a(Xtj−1)(tj − tj−1) + b(Xtj−1)(wtj − wtj−1)+ c(Xtj−1)(Jtj − Jtj−1)
Just matrix multiplications, keeping the form:
(Predictable coefficient)×(Noise increment)
YUIMA has slots for exact generation of multidimensional Jtj − Jtj−1 .
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 31 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Two-dim. SDE X = (X1, X2) driven by a 2-dim. NIG Lévy process:
d
(X1tX2t
)=
(−2X1t
0.3X1t − 1/√
1 + (X2t )2
)dt+
(1/√
1 + (X1t )2 −0.5
0 1
)dJt
▷ yss2019 hm demo.html
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 32 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Put simply
dXt = a(Xt)dt+ b(Xt)dwt + c(Xt−)dJt
Jt =
∫ t0
∫|z|≤1
z(µ− ν)(ds, dz) +∫ t0
∫|z|>1
zµ(ds, dz)
Good (a, b, c) leads to the existence of unique strong solution.
simulate in YUIMA can generate X, as soon as YUIMA can generate Jh.
YUIMA has several options for L(Jh)-random numbers.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 33 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
1 Lévy process: basics and simulation
2 Lévy driven SDE: basics and simulation
3 Quasi-likelihood estimation of Lévy driven SDE
4 Quasi-likelihood estimation of Lévy driven SDE (YUIMA demo)
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 34 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Discrete-time location-scale time series model:
Xn = b(Xn−1, β) + a(Xn−1, α)ϵn
ϵ1, ϵ2, · · · ∼ i.i.d. (0, 1)θ = (α, β): Statistical parameter, to be estimated from (X1, . . . , Xn).
A natural continuous-time counterpart is a
dXt = a(Xt−, α)dZt + b(Xt, β)dt, θ = (α, β)
Z is a standard Lévy process: E(Zt) = 0 and var(Zt) = t.Estimate θ from (Xt0 , Xt1 , . . . , Xtn).
aNote: the coefficient notation got changed! (a↔ b)
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
High-frequency sampling can provide us with unified inference strategies,which generally cannot be shared with the discrete-time framework.
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Setup: joint asymptotics
Univariate parametric Stochastic differential equation (SDE):
dXt = a(Xt−, α)dZt + b(Xt, β)dt, θ = (α, β)
Available data (Xtj )nj=0; tj = jhn = jh, Tn := nh→ ∞, nh2 → 0.
Driving Lévy process s.t. E(Zt) = 0, var(Zt) = t:
Zt = σWt +
∫ t0
∫z (µ− ν)(ds, dz),
A nuisance element.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 37 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Regularity conditions
dXt = a(Xt−, α)dZt + b(Xt, β)dt
Correctly specified ∥ smooth coefficients, known up to θ := (α, β).
Stability ((Exponential) Ergodicity and bounded moments) ∗∗:
1
T
∫ T0
f(Xs)dsp−→∫f(x)π(dx), T → ∞, (3)
∀q > 0, supt∈R+
E (|Xt|q)
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Remark on the stability assumption
dXt = a(Xt)dt+ b(Xt)dwt + c(Xt−)dJt.
X is “exponentially” ergodic (hence (3)) and (4) if:1 (a, b, c) is of class C1(R) and globally Lipschitz, and (b, c) is bounded.2 Either one of the following conditions holds true:
(i) b(x′) ̸= 0 for some x′, c(x′′) ̸= 0 for every x′′, and there exists a constantϵ > 0 such that ν(−ϵ, 0) ∧ ν(0, ϵ) > 0 for every ϵ ∈ (0, ϵ);
(ii) b ≡ 0, c(x′′) ̸= 0 for every x′′, and we have the decompositionν = ν⋆ + ν♮
for two Lévy measures ν⋆ and ν♮, where the restriction of ν⋆ to some openset of the form (−ϵ, 0) ∪ (0, ϵ) admits a continuously differentiable positivedensity g⋆.
3 E(J1) = 0 and∫|z|>1 |z|
qν(dz) 0, and
lim sup|x|→∞
a(x)
x< 0.
See [Masuda, 2013, Sect 5] for details; another conditions are possible.
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
GQLF and GQMLE
dXt = a(Xt−, α)dZt + b(Xt, β)dt, θ = (α, β)
Gaussian approximation in small time (fj−1(θ) := f(Xtj−1 , θ)):
XtjPθ≈ Xtj−1 + aj−1(α)∆jZ + hbj−1(β)L(Pθ)≈ Xtj−1 + aj−1(α)N(0, h) + hbj−1(β)
Gaussian quasi-likelihood function (GQLF) and Gaussian QMLE (GQMLE)
Hn(θ) =n∑
j=1
log ϕ(Xtj ; Xtj−1 + hbj−1(β), ha
2j−1(α)
), (5)
θ̂n = (α̂n, β̂n) ∈ argmaxHn.
User’s input:Function forms of scale a(x, α) and drift b(x, β).Small sampling stepsize value h = hn.
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Asymptotic normality: joint asymptotics
dXt = a(Xt−, α)dZt + b(Xt, β)dt, θ = (α, β)
Diffusion case Z = w [Kessler, 1997]:(√n(α̂n − α0),
√Tn(β̂n − β0)
)L−→ Npα+pβ
(0, diag{I−1α (α0), I−1β (θ0)}
)In the presence of jumps [Masuda, 2013](√Tn(α̂n − α0),
√Tn(β̂n − β0)
)L−→ Npα+pβ
(0,
(ν4I−1α (θ0) sym.ν3Jαβ(θ0) I−1β (β0)
))
Straightforward to estimate Iα(θ0), Iβ(β0), and Jαβ(θ0) empirically.Non-Gaussian structure of Z appears in the asymptotic covariance:
νk :=
∫zkν(dz), k = 3, 4.
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Difference in magnitude in small time:
Xtj+1 = Xtj +
∫ tj+1tj
b(Xs, β)ds︸ ︷︷ ︸≈Op(h)
+
∫ tj+1tj
a(Xs−, α)dZs︸ ︷︷ ︸≈Op(
√h)
Suggests:
First estimate α with ignoring b(x, β);Then estimate β with plugging in α̂n,
even for general standard Lévy process Z;
See [Kamatani and Uchida, 2015] and the ref’s therein for the diffusion case.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 42 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Setup: stepwise asymptotics
Objective
Estimate true parameter θ0 = (α0, γ0) of
dXt = a(Xt, α, γ)dt+ c(Xt−, γ)dJt.
from discrete-time sample (Xtj )nj=1 for tj = jhn with h = hn s.t.
∃ϵ0 ∈ (0, 1), nh1+ϵ0 →∞nh2 → 0
Assumed
Smooth parametric coefficients known up to θ = (α, γ) ∈ Rp.Standardized pure-jump Lévy process J s.t. ∀q > 0, E(|Jt|q)
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
dXt = a(Xt, α, γ)dt+ c(Xt−, γ)dZt
XtjPθ≈ Xtj−1 + haj−1(α, γ) + cj−1(γ)∆jZ
Stepwise-estimation recipe [Uehara and Masuda, 2017], with Hn(α, γ) of (5)
1 L(Xtj |Xtj−1 = x)Pθ≈ N
(x, hc2(x, γ)
): γ̂n ∈ argminγ H1n(γ),
H1n(γ) :=n∑
j=1
log ϕ(Xtj ; Xtj−1 , hc
2j−1(γ)
)2 L(Xtj |Xtj−1 = x)
Pθ≈ N(x+ ha(x, α, γ), hc2(x, γ̂n)
): α̂n ∈ argminα Hn(α, γ̂n),
Hn(α, γ) :=n∑
j=1
log ϕ(Xtj ; Xtj−1 + haj−1(α, γ), hc
2j−1(γ)
)
Result: [Masuda and Uehara, 2017] & [Uehara and Masuda, 2017]
Joint asymptotic normality of (γ̂n, α̂n) at speed√Tn (Tn := nh)
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Some technical details
Asymptotics of γ̂n is the same as in the joint estimation, and as for α̂n:
Explicit stochastic expansions√Tn(α̂n − α0) = Î−1α,n
(Ân√Tn(γ̂n − γ0) + vn
)+ op(1)
vn :=1√Tn
n∑j=1
∂αaj−1(α0, γ0)
c2j−1(γ0)(∆jX − haj−1(α0, γ0)),
Îα,n :=1
n
n∑j=1
(∂αâj−1)⊗2
ĉ2j−1,
Ân :=1
n
n∑j=1
∂αâj−1 ⊗ ∂γ âj−1ĉ2j−1
p−→∫∂αa(x, α0, γ0)⊗ ∂γa(x, α0, γ0)
c2(x, γ0)π0(dx),
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Asymptotic normality√TnΣ̂
−1/2n
(Î−1α,n Î−1α,nÂnO Î−1γ,n
)(α̂n − α0γ̂n − γ0
)L→ Np(0, I)
Clarifies effect of simultaneous presence of γ in the coefficients.
Î−1γ,n :=1
n
n∑j=1
(∂γ ĉj−1)⊗2
ĉ2j−1, and Σ̂n is also given explicitly.
Readily provides us with an approximate (1− s)-confidence set:{(α, γ) :
∣∣∣∣√TnΣ̂−1/2n (Î−1α,n Î−1α,nÂnO Î−1γ,n)(
α̂n − αγ̂n − γ
)∣∣∣∣2 ≤ χ2(p; s)}
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 46 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Put simply
For the univariate parametric Stochastic differential equation (SDE):
dXt = a(Xt−, α)dZt + b(Xt, β)dt, θ = (α, β)
ordXt = c(Xt−, γ)dJt + a(Xt, α, γ)dt, θ = (α, γ).
for stepwise estimation, we can make use of the explicit GQLF
Hn(θ) =n∑
j=1
log ϕ(Xtj ; Xtj−1 + hbj−1(β), ha
2j−1(α)
).
Available data (Xtj )nj=0; tj = jhn = jh, Tn := nh→∞, nh2 → 0.
Driving Lévy process s.t. E(Zt) = 0, var(Zt) = t.
Stability (Ergodicity) is essential here.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 47 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
1 Lévy process: basics and simulation
2 Lévy driven SDE: basics and simulation
3 Quasi-likelihood estimation of Lévy driven SDE
4 Quasi-likelihood estimation of Lévy driven SDE (YUIMA demo)
The YUIMA function qmleLevy was composed by Dr. Yuma Uehara.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 48 / 56
https://sites.google.com/site/yumauehara1928/yuima-package
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
YUIMA demo: qmleLevy
dXt = a(Xt, α, γ)dt+ c(Xt−, γ)dZt
Usage
qmleLevy(yuima, start, lower, upper, joint = FALSE, third = FALSE)
Arguments
yuima a yuima objectlower a named list for specifying lower bounds of parameters.upper a named list for specifying upper bounds of parameters.start initial values to be passed to the optimizer.
joint perform joint estimation or two stage estimation?
by default joint=FALSE. If there exists an overlappingparameter, joint=TRUE currently does not work.
third perform third estimation?
by default third=FALSE. If there exists an overlappingparameter, third=TRUE currently does not work.
Value
first estimated values of first estimation (scale parameters)second estimated values of second estimation (drift parameters)third estimated values of third estimation (scale parameters)
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: bilateral gamma
dXt = −θ0Xtdt+θ1√
1 +X2tdZt,
Zt ∼ Bilateral gamma(t,√2, t,
√2), (θ0,0, θ1,0) = (1, 2)
▷ yss2019 hm demo.html
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 50 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
Example: Normal inverse Gaussian
dXt = −θ0Xtdt+θ1√
1 +X2tdZt,
Zt ∼ NIG (δ, 0, δt, 0, 1) with δ = 10, (θ0,0, θ1,0) = (1, 2).
▷ yss2019 hm demo.html
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
2-dim. Example: variance gamma
d
(X1,tX2,t
)=
(1− θ0X1,t −X2,t
−θ1X2,t
)dt+
(θ2
1+X21,t+ 1 0
1 1
)dZt,
Zt ∼ Variance gamma(
12t, 1,
(00
),
(00
),
(1 00 1
)), (θ0, θ1, θ2) = (1, 2, 3).
▷ yss2019 hm demo.html
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Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
2-dim. Example: Normal inverse Gaussian
d
(X1,tX2,t
)=
(1− θ0X1,t−θ1X2,t
)dt+
exp(− θ2
1+X21,t
)0
1 exp
(− θ3√
1+X22,t
) dZt,
Zt ∼ NIG2(
1√πt, 1,
(00
),
(00
),
(1 00 1
)), (θ0, θ1, θ2, θ3) = (1, 2, 3, 4).
▷ yss2019 hm demo.html
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 53 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
References I
Applebaum, D. (2009).
Lévy processes and stochastic calculus, volume 116 of Cambridge Studies in Advanced Mathematics.Cambridge University Press, Cambridge, second edition.
Bertoin, J. (1996).
Lévy processes, volume 121 of Cambridge Tracts in Mathematics.Cambridge University Press, Cambridge.
Brouste, A., Fukasawa, M., Hino, H., Iacus, S. M., Kamatani, K., Koike, Y., Masuda, H., Nomura,
R., Ogihara, T., Shimizu, Y., Uchida, M., and Yoshida, N. (2014).The yuima project: A computational framework for simulation and inference of stochastic differentialequations.Journal of Statistical Software, 57(4):1–51.
Iacus, S. M. and Yoshida, N. (2018).
Simulation and inference for stochastic processes with YUIMA.Use R! Springer, Cham.A comprehensive R framework for SDEs and other stochastic processes.
Kamatani, K. and Uchida, M. (2015).
Hybrid multi-step estimators for stochastic differential equations based on sampled data.Stat. Inference Stoch. Process., 18(2):177–204.
Kessler, M. (1997).
Estimation of an ergodic diffusion from discrete observations.Scand. J. Statist., 24(2):211–229.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 54 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
References II
Kulik, A. (2018).
Ergodic behavior of Markov processes, volume 67 of De Gruyter Studies in Mathematics.De Gruyter, Berlin.With applications to limit theorems.
Masuda, H. (2013).
Convergence of Gaussian quasi-likelihood random fields for ergodic Lévy driven SDE observed at highfrequency.Ann. Statist., 41(3):1593–1641.
Masuda, H. and Uehara, Y. (2017).
Two-step estimation of ergodic Lévy driven SDE.Stat. Inference Stoch. Process., 20(1):105–137.
Platen, E. and Bruti-Liberati, N. (2010).
Numerical solution of stochastic differential equations with jumps in finance, volume 64 of StochasticModelling and Applied Probability.Springer-Verlag, Berlin.
Protter, P. E. (2005).
Stochastic integration and differential equations, volume 21 of Stochastic Modelling and AppliedProbability.Springer-Verlag, Berlin.Second edition. Version 2.1, Corrected third printing.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 55 / 56
Lévy process Lévy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo
References III
Sato, K.-i. (1999).
Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in AdvancedMathematics.Cambridge University Press, Cambridge.Translated from the 1990 Japanese original, Revised by the author.
Uehara, Y. and Masuda, H. (2017).
Stepwise estimation of a Lévy driven stochastic differential equation.Proc. Inst. Statist. Math. (Japanese), 65(1):21–38.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June 27, 2019 56 / 56
Lévy process: basics and simulationBasicsSimulation in YUIMA
Lévy driven SDE: basics and simulationBasicsSimulation in YUIMA
Quasi-likelihood estimation of Lévy driven SDEIntroduction and backgroundAsymptotics
Quasi-likelihood estimation of Lévy driven SDE (YUIMA demo)
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