L evy process, L evy driven SDE, and quasi-likelihood estimation · 2019. 6. 27. · L evy process L evy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo Contents, June

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


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


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



1 Lévy process: basics and simulation

2 Lévy driven SDE: basics and simulation

3 Quasi-likelihood estimation of Lévy driven SDE


Standard references:

[Applebaum, 2009][Bertoin, 1996][Protter, 2005, (2nd.) Chapter I.4][Sato, 1999]

[Iacus and Yoshida, 2018] for many YUIMA examples



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?



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


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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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)).



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)


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).



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

).



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


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]



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



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.




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


§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.



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.







A reader-friendly and comprehensive monograph is [Applebaum, 2009].



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.



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′′




∫ 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.




∫ 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].



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.



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.



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



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.



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/∆].


¶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


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


Itô’s formula (Univariate case)


f(Xt) = f(X0) +

∫ t0

f ′(Xs−)dXs +1

2

∫ t0

f ′′(Xs−)d⟨Xc⟩s

+∑

0


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




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 .



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




Put simply


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.



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)



High-frequency sampling can provide us with unified inference strategies,which generally cannot be shared with the discrete-time framework.



Setup: joint asymptotics

Univariate parametric Stochastic differential equation (SDE):


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.



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)


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.



GQLF and GQMLE


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.



Asymptotic normality: joint asymptotics


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.



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.



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)


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)



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),



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)}



Put simply

For the univariate parametric Stochastic differential equation (SDE):


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.







The YUIMA function qmleLevy was composed by Dr. Yuma Uehara.


https://sites.google.com/site/yumauehara1928/yuima-package


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)



Example: bilateral gamma

dXt = −θ0Xtdt+θ1√

1 +X2tdZt,

Zt ∼ Bilateral gamma(t,√2, t,

√2), (θ0,0, θ1,0) = (1, 2)




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).




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).




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).




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.



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.



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.


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)

Documents

L evy process, L evy driven SDE, and quasi-likelihood estimation · 2019. 6. 27. · L evy process L evy driven SDE Quasi-likelihood estimation qmleLevy YUIMA demo Contents, June