Lectures on the parametrix method I Basic constructions

Lectures on the parametrix method IBasic constructions

Alex Kulik

Wroc law University of Science and Technology

NOMP II, 22.03.2021

Alex Kulik Lectures on the parametrix method I 1/32 1 / 32

Plan of the course

I Basic constructionsMarkov and diffusion processes; Kolmogorov’s differential equationsThe parametrix method for 2nd order parabolic PDEs, I: construction of the solutionsSome probabilistic applications

II Sensitivities and approximationsThe parametrix method for 2nd order parabolic PDEs, II: differentiability of thesolutionsThe PMP and the semigroup propertiesBack to Processes: SDEs and Martingale ProblemsThe PMP revised: approximate fundamental solutions

III Non-local PDEs: new effects and methodsNon-local models with dominationSuper-critical driftsEssentially singular models


Markov processes: Glossary

Markov process: a stochastic process on a filtered probability space (Ω,F , Ft,P)taking values in a measurable state space (E, E) such that

P(Xt ∈ A|Fs) = Ps,t(Xs, A), t ≥ s, A ∈ E

Transition probability Ps,t(x, dy) and transition probability density

ps,t(x, y) =Ps,t(x, dy)

m(dy).

In what follows: E = Rd and m(dy) = dy, the Lebesgue measure.

Family of operators

Ps,tf(x) =

∫Rdf(y)Ps,t(x, dy) = Es,xf(Xt)

in a proper functional space, e.g. f ∈ Bb(Rd) (bounded measurable functions) orf ∈ C∞(Rd) (continuous functions vanishing at ∞).


The Chapman-Kolmogorov equation: for s ≤ r ≤ t,

Ps,t(Xs, A) =

∫RdPs,r(x, dy)Pr,s(y,A).

Evolutionary property: for s ≤ r ≤ t,

Ps,t = Ps,rPr,t.

In the time homogeneous setting Ps,t = Pt−s, the evolutionary family Ps,t reducesto a semigroup of operators Ptt≥0 with the semigroup property: for s, t ≥ 0,

Pt+s = PtPs.


Diffusion processes: Kolmogorov’s approach

A.Kolmogorov (1931) “Uber die analytischen Methoden in derWahrscheinlichkeitsrechnung”, Mathematische Annalen 104

Empirical model: for arbitrary ε > 0,

0. (Continuity condition)

Ps,t(x, y : |y − x| ≥ ε) = o(t− s), t→ s.

I. (Drift condition)∫|y−x|<ε

(y − x)Ps,t(x, dy) = a(s, x)(t− s) + o(t− s), t→ s.

II. (Diffusion condition)∫|y−x|<ε

(y − x)⊗2Ps,t(x, dy) = b(s, x)(t− s) + o(t− s), t→ s.


Differential equations: if a diffusion process has a smooth enough transition probabilitydensity ps,t(x, y), then this density satisfies

(the Backward Kolmogorov Equation)

∂sps,t(x, y) = −∑i

ai(s, x)∂xips,t(x, y)− 1

2

∑i,j

bij(s, x)∂2xixjps,t(x, y);

(the Forward Kolmogorov Equation)

∂tps,t(x, y) = −∑i

∂yi(ai(t, y)ps,t(x, y)

)+

1

2

∑i,j

∂2yiyj

(bij(t, y)ps,t(x, y)

).


For the operator family Ps,t:

−∂sPs,tf(x) = LsPs,tf(x), ∂tPs,tf(x) = Ps,tLtf(x)

with

Ltf(x) =∑i

ai(t, x)∂xif(x) +1

2

∑i,j

bij(t, x)∂2xixjf(x)

= a(t, x) · ∇f(x) +1

2b(t, x) · ∇2f(x),

the generating family of the evolutionary family Ps,t.

In the time homogeneous setting: Lt = L, the generator of the semigroup, and

∂tPtf = LPtf = PtLf.


How to construct a diffusion process?

I. The Ito-Levy stochastic calculus approach: Xt is a strong solution to the SDE

dXt = a(t,Xt) dt+ σ(t,Xt) dWt, σ(t, x)σ(t, x)∗ = b(t, x). (1)

Kiyosi Ito (1942) “Differential equations determining a Markoff process”, Zenkoku SizyoSugaku Danwakai-siWolfgang Doeblin (1940) “Sur l’equation de Kolmogoroff”, C. R. Acad. Sci. Paris, t.331, Serie I, p. 1059–1102, 2000

II. Analytic approach: ps,t(x, y) is the solution to the Backward Kolmogorov EquationXt. Corresponding Markov process is a weak solution to the SDE (1).

W. Feller (1936) Zur Theorie der stochastischen Prozesse. (Existenz- undEindeutigkeitssatze). Mathematische Annalen 113Translated and reprinted:Schilling, R.L., Vondracek, Z., Wojczynski, W.: William Feller. Selected Papers I, II.Springer, Cham 2015.


From diffusions to Levy-type processes

In the empirical model, the ‘Continuity condition’ 0. should be changed to

0’. (Jump intensity condition): for arbitrary ε > 0 and Γ ⊂ y : |y − x| ≥ ε,

Ps,t(x,Γ) = ν(s, x,Γ)(t− s) + o(t− s), t→ s.


Generating family consists of non-local integro-differential operators

Ltf(x) = a(t, x) · ∇f(x) +1

2b(t, x) · ∇2f(x)

+

∫Rd

(f(y)− f(x)− (y − x) · ∇f(x)1|y−x|≤1) ν(t, x, dy)(2)

with a Levy kernel ν(t, x, dy):∫Rd

(|y − x|2 ∧ 1

)ν(t, x, dy) <∞.

The well known Courrege theorem states that for any Feller process with the generatorwell defined on C2

∞(Rd), this generator has the form

Lf(x) = a(x) · ∇f(x) +1

2b(x) · ∇2f(x)

+

∫Rd

(f(y)− f(x)− (y − x) · ∇f(x)1|y−x|≤1) ν(x, dy).

Close to the Levy-Khinchin characterization of the Levy processes; state-dependenttriplet a(x), b(x), ν(x, dy).


The parametrix method for diffusions

We have (∂s + Ls,x)ps,t(x, y) = 0, s ∈ (−∞, t),ps,t(x, y)→ δy(x), s t,

(3)

i.e. the transition probability density ps,t(x, y) is a fundamental solution to the Cauchyproblem for the parabolic 2nd order PDO ∂s + Ls.

E.E. Levi (1907) Sulle equazioni lineari totalmente ellittiche alle derivate parziali.Rendiconti del Circolo Matematico di Palermo 24.M. Gevrey (1913,1914) Sur les equations aux derivees partielles du type parabolique.Journal des Mathematiques Pures et Appliquees 9,10

A. Friedman (1964) Partial differential equations of parabolic type. Prentice-Hall,New-York.S.D. Eidel’man (1969) Parabolic Systems. North-Holland & Wolters-Noordhoff,Amsterdam.


Integral equation for the fundamental solution, I: derivation using volumepotentials

For given f(x), g(s, x) define the volume potential

V (s, x) =

∫Rdps,t(x, y) f(y) dy +

∫ t

s

∫Rdps,r(x, y) g(r, y) dy. (4)

If ps,t(x, y) is a fundamental solution and f(x), g(s, x) possess certain mild regularity,V (s, x) satisfies

(∂s + Ls)V (s, x) = −g(s, x), s < t,V (s, x)→ f(x), s t.

Pick up p0s,t(x, y), an explicit “zero order approximation” to unknown ps,t(x, y).

p0s,t(x, y) is C1 in s, C2 in x,p0s,t(x, y)→ δy(x), s t,


DenoteΥs,t(x, y) = (∂s + Ls,x)p0

s,t(x, y),

the “differential error of approximation” of ps,t(x, y) by p0s,t(x, y). Then

Rs,t(x, y) = ps,t(x, y)− p0s,t(x, y)

satisfies (∂s + Ls,x)Rs,t(x, y) = −Υs,t(x, y), s < t,rs,t(x, y)→ 0, s t.

Applying the volume potential formula (4), we write

Rs,t(x, y) =

∫ t

s

∫Rdps,r(x, z)Υr,t(z, y) dydr,

or equivalently

ps,t(x, y) = p0s,t(x, y) +

∫ t

s

∫Rdps,r(x, z)Υr,t(z, y) dzdr (5)

Integral equation (5) is the cornerstone of the parametrix construction.


Integral equation for the fundamental solution, II: derivation using theForward Equation

W. Feller (1936): If a FS for the Backward Equation at the same time solves the ForwardEquation, then it is the unique solution to the Backward one.∫ t−ε

s+ε

∫Rdps,r(x, z)Υr,t(z, y) dzdr =

∫ t−ε

s+ε

∫Rdps,r(x, z)(∂r + Lr,x)p0

r,t(z, y) dzdr

= (I.B.P.) =

∫Rdps,t−ε(x, z)p

0t−ε,t(z, y) dz −

∫Rdps,s+ε(x, z)p

0s+ε,t(z, y) dz

+

∫ t−ε

s+ε

∫Rd

(−∂rps,r(x, z)p0r,t(z, y) + ps,r(x, z)Lr,xp

0r,t(z, y)) dz︸︷︷︸

−∂rPs,rϕ(x)+Ps,rLrϕ(x)=0,ϕ(z)=p0r,t(z,y)

dr

= (F.E.) =

∫Rdps,t−ε(x, z)p

0t−ε,t(z, y) dz −

∫Rdps,s+ε(x, z)p

0s+ε,t(z, y) dz

→ ps,t(x, y)− p0s,t(x, y), ε→ 0.

Initial value conditions:

ps,t(x, y)→ δx(y), t s, (F.E.)

p0s,t(x, y)→ δy(x), s t, (B.E.)


How to solve the integral equation, I: Outline

Write ∫Rdf(x, z)g(z, y) dz = (f ∗ g)(x, y),∫ t

s

∫Rdfs,r(x, z)gr,t(z, y) dzdr =

∫ t

s

(fs,r ∗ gr,t)(x, y) dr = (f ~ g)s,t(x, y),

then the equation can be shortly written as

ps,t(x, y) = p0s,t(x, y) + (p~Υ)s,t(x, y) = p0

s,t(x, y) +

∫ t

s

(ps,r ∗Υr,t)(x, y) dr. (6)

The last identity shows that this is a Volterra equation, and we can try to solve it usingthe successful iteration procedure, which would lead to the series representation

ps,t(x, y) = p0s,t(x, y) + (p0 ~Υ)s,t(x, y) + (p0 ~Υ~2)s,t(x, y) + . . . . (7)

The latter can be also written as

ps,t(x, y) = p0s,t(x, y) + (p0 ~ Ξ)s,t(x, y), (8)

Ξt(x, y) =∑k≥1

Υ~ks,t (x, y). (9)


We define the kernel ps,t(x, y) as the solution to the integral equation (5) given by theseries (7), hoping to show later that this is indeed the FS to (the Cauchy problem for)

the Backward Equation and the heat kernel for the diffusion process with the localcharacteristics a(t, x), b(t, x).

‘Good’ choice of p0t (x, y) estimates for Υs,t(x, y) convergence of the series (7), (9).


How to solve the integral equation, II: Choice of p0t (x, y)

The classic choice of p0s,t(x, y).

Recall that

Ltf(x) = a(t, x) · ∇f(x) +1

2b(t, x) · ∇2f(x),

and introduce the family of operators

L(ξ,τ)f(x) =1

2b(τ, ξ) · ∇2f(x).

Denote

p(ξ,τ)s,t (x, y) =

1

(2π(t− s))d/2(det b(τ, ξ))1/2exp

(− 1

2(t− s) (y − x)>b(τ, ξ)−1(y − x)

),

the Gaussian probability density N (x, (t− s)b(τ, ξ)).

p0s,t(x, y) := p

(y,t)s,t (x, y).


How to solve the integral equation, III: Gaussian kernel estimates

Denote

ϕt(x) =1

(2πt)d/2exp

(−|x|

2

2t

).

We say that a kernel fs,t(x, y) has order (t− s)p if there exist C, c > 0 such that

|fs,t(x, y)| ≤ C(t− s)pϕc(t−s)(y − x).

Lemma

Assume the following:

A.1 Coefficients a(t, x), b(t, x) are bounded.

A.2 Coefficient b(t, x) is uniformly elliptic: for some β > 0,

v>b(t, x)v ≥ β|v|2.

A.3 Coefficient b(t, x) is Holder continuous: for some γ ∈ (0, 12],

|b(t, x)− b(t′, x)| ≤ C|t− t′|γ , |b(t, x)− b(t, x′)| ≤ C|x− x′|2γ .

Then Υs,t(x, y) has order (t− s)−1+γ .


Proof of the Gaussian kernel estimates

Denote for a symmetric positive definite matrix b the Hermite function of the order 0 by

Φt(b;x, y) =1

(2πt)d/2(detb)1/2exp

(− 1

2t(y − x)>b−1(y − x)

),

and the family of higher order Hermite functions

Φi1,...,ikt (b;x, y) = ∂x1 . . . ∂xkΦt(b;x, y).

Thenp0s,t(x, y) = Φt−s(b(t, y);x, y)

Fact 1 If fs,t(x, y) has order (t− s)p, then |y − x|afs,t(x, y) has order (t− s)p+a2 .

Fact 2 Φi1,...,ikt−s (b;x, y) has order (t− s)−k/2.


We have(∂s + Lξ,τx )p

(ξ,τ)s,t (x, y) = 0,

henceΥs,t(x, y) = (∂s + Ls,x)p0

s,t(x, y)

= ∂s(p(y,t)s,t (x, y) + Ls,x)p

(y,t)s,t (x, y)

= −Ly,tx p(y,t)s,t (x, y) + Ls,xp

(y,t)s,t (x, y)

= a(s, x) · ∇xp(y,t)s,t (x, y)︸︷︷︸

(Φis,t(b(t,y);x,y)),order(t−s)−1/2

=1

2

(b(s, x)− b(t, y)

)· ∇2

xxp(y,t)s,t (x, y)︸︷︷︸

(Φi,js,t(b(t,y);x,y)),order(t−s)−1


How to solve the integral equation, IV: Estimates for the convolutions

We have|Υs,t(x, y)| ≤ C(t− s)−1+γϕc(t−s)(y − x)

andϕc(r−s) ∗ ϕc(t−r) = ϕc(t−s).

Then

|Υ~2s,t(x, y)| ≤

∫ t

s

|(Υs,r ∗Υr,t)(x, y)| dr

≤ C2

∫ t

s

(r − s)−1+γ(t− r)−1+γϕc(r−s) ∗ ϕc(t−r)(y − x) dr

= C2B(δ, δ)(t− s)−1+2δϕc(t−s)(y − x)

=(CΓ(δ))2

Γ(2δ)(t− s)−1+2δϕc(t−s)(y − x).

Similarly, by induction

|Υ~ks,t (x, y)| ≤ (CΓ(δ))k

Γ(kδ)(t− s)−1+kδϕc(t−s)(y − x)


How to solve the integral equation, V: Summary

Theorem

Let

A.1 Coefficients a(t, x), b(t, x) be bounded.

A.2 Coefficient b(t, x) be uniformly elliptic: for some β > 0,

v>b(t, x)v ≥ β|v|2.

A.3 Coefficient b(t, x) be Holder continuous: for some γ > 0,

|b(t, x)− b(t′, x)| ≤ C|t− t′|γ , |b(t, x)− b(t, x′)| ≤ C|x− x′|2γ .

Then the solution ps,t(x, y) for the integral equation (5) equals

ps,t(x, y) = p0s,t(x, y) +Rs,t(x, y),

where

Rs,t(x, y) = (p0 ~ Ξ)s,t(x, y) =∞∑k=1

(p0 ~Υ~k)s,t(x, y)

and|Rs,t(x, y)| ≤ C(t− s)δϕc(t−s)(y − x).


Heat kernel estimates and structure

From the parametrix construction, we get

Gaussian upper bound:

ps,t(x, y) ≤ Cϕc(t−s)(y − x);

a semi-explicit representation

ps,t(x, y) ≈ Φt−s(b(t, y);x, y)

with the error bounded by

C(t− s)δϕc(t−s)(y − x)

The probabilistic drawback is that the ‘principal part’ in this representation is not aprobability density: while

Φt−s(b(τ, ξ);x, y) ∼ N (0, (t− s)b(τ, ξ))

is a probability density for any values of parameters τ, ξ, the kernel Φt−s(b(t, y);x, y) isnot.


Other possible choices of p0s,t(x, y)

I. W. Feller (1936): Let d = 1 and V (x) solve V ′(x) = b(V (x))1/2, then

p0s,t(x, y) =

1√2π(t− s)V ′(V −1(y))

exp

(− 1

2(t− s) |V−1(y)− V −1(x)|2

).

In the SDEs language: use transformation X = V (Y ) to reduce the SDE

dXt = a(Xt) dt+ b(Xt)1/2 dWt

to the SDEdYt = a(Yt) dt+ dWt,

and use the distribution of V (Wt) as the principal one.

Requires: b ∈ C1, in higher dimensions: the Frobenius (integrability) condition on√b(x).


II.(Math. folklore) Denote

Φt(a,b;x, y) =1

(2πt)d/2(detb)1/2exp

(− 1

2t(y − x− ta)>b−1(y − x− ta)

)∼ N (ta, tb)

and takep0s,t(x, y) = Φt−s((t− s)a(t, y), (t− s)b(t, y);x, y).

Looks similar to the distribution density

pEMs,t (x, y) = Φt−s((t− s)a(s, x), (t− s)b(s, x);x, y)

of the Euler-Maruyama (conditionally Gaussian) approximation

Xt −Xs ≈ a(s,Xs)4t+ σ(s,Xs)4W,

but still has the same drawback of the coefficients being ‘fixed at the endpoint’ y.


III. (Math. folklore) Take

p0s,t(x, y) = Φt−s((t− s)a(s, x), (t− s)b(s, x);x, y)

and repeat the parametrix construction using the Forward Equation. Recall that F.E.involves differentiation of the coefficients:

∂tps,t(x, y) = −∑i

∂yi(ai(t, y)ps,t(x, y)

)+

1

2

∑i,j

∂2yiyj

(bij(t, y)ps,t(x, y)

).

Requires a ∈ C1, b ∈ C2 in the space variable.


IV. Consider mollified coefficients

as(t, x) =(a(t, ·) ∗ ϕ(t−s)δ

)(x), bs(t, x) =

(b(t, ·) ∗ ϕ(t−s)δ

)(x)

and takep0s,t(x, y) = Φt−s((t− s)as(t, x), (t− s)bs(t, x);x, y).

Computations are principally the same, but considerably more cumbersome.

Xt −Xs ≈ as(t,Xs)4t+ σs(t,Xs)4W,

a sort of ‘mollified’ Euler-Maruyama approximation


Parametrix-based probabilistic expansions of the heat kernel

Theorem

Let A.1 - A.3 hold, then

ps,t(x, y) = pEMs,t (x, y) +REMs,t (x, y)

with|REMs,t (x, y)| ≤ C(t− s)δϕc(t−s)(y − x).


S. Bodnarchuk, D. Ivanenko, A. Kohatsu-Higa, A. Kulik (2019), Improved localapproximation for multidimensional diffusions: the G-rates. Theory of Probability andMathematical Statistics. 2019, v. 101

Theorem

Let A.1,A.2 hold, and in addition a ∈ Cδ,2δ, b ∈ C1/2+δ,1+2δ then

ps,t(x, y) = Φt−s(a(s, x), b(s, x);x, y)

+ (t− s)2d∑

i,j,k=1

cijk(s, x)Φ(i,j,k)t−s (a(s, x), b(s, x);x, y) +R1

s,t(x, y)

with

cijk(s, x) =1

4

d∑l=1

bkl(s, x)∂xlbij(s, x)

and|R1s,t(x, y)| ≤ C(t− s)1/2+δϕc(t−s)(y − x).

D. Ivanenko, A. Kohatsu-Higa, A. Kulik (2021+): expansions of arbitrary order


Fact 3∂aiΦ

i1,...,ikt (a,b;x, y) = tΦ

i,i1,...,ikt (a,b;x, y),

∂bijΦi1,...,ikt (a,b;x, y) =

t

2Φi,j,i1,...,ikt (a,b;x, y)

Fact 4Φi1,...,ikt (a,b; ·) ∗ Φj1,...,jls (a,b; ·) = Φ

i1,...,ik,j1,...,jlt+s (a,b; ·)

By Fact 3 and the Mean Value Theorem,

p0s,t(x, y)− pEMs,t (x, y) = Φt−s(a(t, y), b(t, y);x, y)− Φt−s(a(s, x), b(s, x);x, y)

= (t− s)∑i

(a(t, y)− a(s, x))iΦit−s(a,b;x, y)

+t− s

2

∑i,j

(b(t, y)− b(s, x))ijΦi,jt−s(a,b;x, y)

has order (t− s)δ.


Y. Ait-Sahahia (2008) Closed-Form Likelihood Expansions for Multivariate Diffusions,Annals of Statistics, 36: Hermite polynomial-based expansion for heat kernels ofdiffusions which can be made ‘flat’ (in the same sense as in the Feller’36 construction).Used in a construction of ‘quasi-MLE’ for unknown parameter θ.


Thank you!


Lectures on the parametrix method I Basic constructions

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