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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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)
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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?
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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Real data may be often leptokurtic: higher kurtosis than the
normal (NYSEminutes data); maybe also skewed (energy consumption
data).
0
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6
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dens
ity
linetypehyperbolic
normal
fillhistogram
Density fits: hyperbolic vs normal
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ity
linetypenormal
stable
fillhistogram
Density fits: stable vs normal
Energy consumption data: Gaussian fit
xx
Den
sity
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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)).
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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)
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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).
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
).
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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]
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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.
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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.
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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.
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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′′
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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.
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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].
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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.
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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.
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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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.
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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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 .
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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)
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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High-frequency sampling can provide us with unified inference
strategies,which generally cannot be shared with the discrete-time
framework.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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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
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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)
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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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.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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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.
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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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)
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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)
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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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),
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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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)
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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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
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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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
Hiroki Masuda (Kyushu Univ.) YSS 2019 Brixen-Bressanone, June
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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
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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
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Lévy process Lévy driven SDE Quasi-likelihood estimation
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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
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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.
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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)