Differentiability of the solution to the parametrix integral equation

Lectures on the parametrix method II Sensitivities and approximations

Alex Kulik

Wroc law University of Science and Technology

NOMP II, 23.03.2021

Differentiability of the solution to the parametrix integral equation

Recall that we have started from the Cauchy problem

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

ps,t(x, y) → δy(x), s % t, (1)

with the 2nd order PDO

Ltf (x) = a(t, x) · ∇f (x) + 1

2b(t, x) · ∇²f (x), which we have replaced by the integral equation

ps,t(x, y) = p⁰s,t(x, y) + Z t

s

Z

R^d

ps,r(x, z)Υr,t(z, y) dzdr, (2) Υs,t(x, y) = (∂s+ Ls,x)p⁰s,t(x, y).

The solution to (2) is given by

ps,t(x, y) = p⁰s,t(x, y) + Z t

s

Z

R^d

p⁰s,r(x, z)Ξr,t(z, y) dzdr, (3)

Ξ(x, y) =X

Υ^~k(x, y).

The further aim is to show that the solution to the integral equation (2) is the unique solution to (1)

Naive explanation: p⁰_s,t(x, y) is C¹in s and C² in x. Differentiating in (3) formally, we get

∇xps,t(x, y) = ∇xp⁰s,t(x, y) + Zt

s

Z

R^d

∇xp⁰s,r(x, z)Ξr,t(z, y) dzdr (4)

∇²_xxps,t(x, y) = ∇²_xxp⁰_s,t(x, y) + Z t

s

Z

R^d

∇²_xxp⁰_s,r(x, z)Ξr,t(z, y) dzdr (5)

∂sps,t(x, y) = ∂sp⁰s,t(x, y) − Ξs,t(x, y) + Z t

s

Z

R^d

∂sp⁰s,r(x, z)Ξr,t(z, y) dzdr (6) Combining ∇x, ∇²_xxwith coefficients we get then

(∂sps,t(x, y) + Ls,xps,t(x, y))

= Υs,t(x, y) − Ξs,t(x, y) + Zt

s

Z

R^d

Υs,r(x, z)Ξr,t(z, y) dzdr

= Υ −

∞

XΥ^~k+

∞

XΥ ~ Υ^~k

!

(x, y) = 0.

Warm up: parametrix-based expansions for sensitivities w.r.t. external parameters

Let coefficients of the diffusion depend on external parameter θ = (α, β):

a = a(α; t, x), b = b(β; t, x).

Then

ps,t(θ; x, y) = ps,t(θ; x, y) +

∞

X

k=1

(p⁰~ Υ^~k)s,t(θ; x, y) and, provided that the corresponding series converge,

∇θps,t(θ; x, y) = ∇θps,t(θ; x, y) +

∞

X

k=1

∇θ(p⁰~ Υ^~k)s,t(θ; x, y)

with

∇θ



p⁰~ Υ^~k

= ∇θp⁰
~ Υ^~k+

k−1

X

j=0

p⁰~ Υ^~j~ (∇θΥ) ~ Υ^~(k−j−1).

UsingFact 3we get that

∂αp⁰s,t(θ; x, y) = (t − s)X

i

∂αai(α; t, y)Φⁱt−s(a(θ, t, y), b(θ, t, y); x, y)

∂βp⁰_s,t(θ; x, y) = t − s 2

X

i,j

∂βbij(β; t, y)Φ^i,j_t−s(a(θ, t, y), b(θ, t, y); x, y)

have orders (t − s)^1/2and (t − s)⁰, respectively. Similar but more cumbersome calculations give that ∂αΥs,t(θ; x, y), ∂βΥs,t(θ; x, y) have orders (t − s)^−1/2+δand (t − s)^−1+δ, and the parametrix series for ∇θps,t(θ; x, y) converges.

∂αps,t(θ; x, y) ≈_O(t−s)1/2+δ (t − s)X

i

∂αai(α; s, x)Φⁱ_t−s(a(s, x), b(s, x); x, y)

∂βps,t(θ; x, y) ≈_O(t−s)δ

t − s 2

X

i,j

∂βbij(β; s, x)Φ^i,j_t−s(a(θ, s, x), b(θ, s, x); x, y) The higher order approximations are also available.

Derivatives in x

Lemma

The identity (4) for ∇xps,t(x, y) holds true and

|∇xps,t(x, y)| ≤ C(t − s)^−1/2ϕc(t−s)(x, y), ϕt(x, y) = ϕt(y − x).

The key estimate:

∂x_ip⁰_s,t(x, y) = Φⁱ_t−s(a(s, x), b(s, x); x, y) has order (t − s)^−1/2, hence the integral

Z t s

Z

R^d

|∇xp⁰s,r(x, z)||Ξr,t(z, y)| dzdr is finite.

Lemma

The identity (5) for ∇²_xxps,t(x, y) holds true and

|∇²xxps,t(x, y)| ≤ C(t − s)⁻¹ϕc(t−s)(x, y) Here the situation is more subtle because

∂x²_ix_jp⁰s,t(x, y) = Φ^i,j_t−s(a(t, y), b(t, y); x, y) has order (t − s)⁻¹, and the integral

Z t s

Z

R^d

|∂²x_ix_jp⁰s,r(x, z)||Ξr,t(z, y)| dzdr

has no means to be finite. Before proving the lemma, let us first establish certain H¨older continuity properties of the kernels involved into the parametrix construction.

Proposition For any δ < γ,

|p⁰t(x1, y) − p⁰t(x2, y)| ≤ C|x1− x2|^2δ t^δ



ϕct(x1, y) + ϕct(x2, y)

 ,

|Υ(x1, y) − Υt(x2, y)| ≤ C|x1− x2|^2δt^−1+γ−δ



ϕct(x1, y) + ϕct(x2, y)



|Ξ(x1, y) − Ξt(x2, y)| ≤ C|x1− x2|^2δt^−1+γ−δ

ϕct(x1, y) + ϕct(x2, y)

|pt(x1, y) − pt(x2, y)| ≤ C|x1− x2|^2δ t^δ



ϕct(x1, y) + ϕct(x2, y) . Similar statement holds true for the increments w.r.t. y.

The proof follows the same ‘parametrix’ strategy we have already discussed.

Proof of the Lemma

We have

∂x²_ix_jp⁰s,t(x, y) = Φ^i,j_t−s(a(t, y), b(t, y); x, y) and the following two facts:

Φ^i,j_t−s(a(t, y), b(t, y); x, y) − Φ^i,j_t−s(a(t, x), b(t, x); x, y) has order (t − s)^−1+γ;

Z

R^d

Φ^i,j_t−s(a(t, x), b(t, x); x, z) dz = 0.

These facts combined give     Z

R^d

Φ^i,j_t−s(a(t, z), b(t, z); x, z) dz    

≤ C(t − s)^−1+γ.

Then     Z

R^d

∇²xxp⁰_s,r(x, z)Ξr,t(z, y) dz    

≤     Z

R^d

∇²xxp⁰_s,r(x, z)Ξr,t(x, y) dz     +

    Z

R^d

∇²xxp⁰s,r(x, z)(Ξr,t(x, y) − Ξr,t(z, y)) dz    

≤ C(r − s)^−1+δ(t − r)^−1+γ−δϕc(t−r)(x, y)

≤ ˜C(r − s)^−1+δ(t − r)^−1+γ−δϕc(t−s)/2(x, y) for r < ^t+s₂ , and

    Z

R^d

∇²xxp⁰s,r(x, z)Ξr,t(z, y) dz    

≤ C(r − s)⁻¹(t − r)^−1+γϕc(t−s)(x, y) for r > ^t+s₂ . These estimates provide

Z t s

    Z

R^d

∇²xxp⁰_s,r(x, z)Ξr,t(z, y) dz    

dr ≤ C(t − s)^−1+γϕ_c(t−s)(x, y).

PMP and the properties of the evolutionary family of operators

We have proved the following Proposition

For any f ∈ C∞(R^d) and t ∈ R, function u(s, x) = Ps,tf (x) satisfies

 (∂s+ Ls,x)u(s, x) = 0, s ∈ (−∞, t), u(s, x) → f (x) s % t.

In other words, u(s, x) is harmonic for (∂s+ Ls) on (−∞, t) × R^dand u(t, x) = f (x).

Each operator Lssatisfies the Positive Maximum Principle (PMP) on C∞² : f (x∗) ≥ f (x), x ∈ R^d, f (x∗) ≥ 0 =⇒ Lsf (x∗) ≥ 0.

Proposition

Let a function u(s, x) be harmonic for (∂s+ Ls) on (−∞, t) × R^dwith Ls satisfying the PMP.

Then u(t, x) ≥ 0 implies u(s, x) ≥ 0, s < t.

Corollary

I. For any f ≥ 0 and s ≤ t, Ps,tf (x) ≥ 0.

II. For any s ≤ r ≤ t, Ps,t= Ps,rPr,t. III. For any s ≤ t and f ∈ C∞²,

Ps,tf (x) = f (x) + Z t

s

Ps,rLrf (x) dr.

To prove, apply Proposition to I. u(s, x) = Ps,tf (x);

II. u(s, x) = ±Ps,tf (x) ∓ Ps,rPr,tf (x), s ≤ r;

III. u(s, x) = ±Ps,tf (x) ∓ f (x) ∓Rt

Ps,rLrf (x) dr.

Summary

Assume that

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 H¨older continuous: for some γ ∈ (0,¹₂],

|b(t, x) − b(t⁰, x)| ≤ C|t − t⁰|^γ, |b(t, x) − b(t, x⁰)| ≤ C|x − x⁰|^2γ. Then the kernel ps,t(x, y) is the unique Fundamental Solution to the Cauchy problem for

∂s+ Ls in the class of kernels fs,t(x, y) such that for any ε > 0, s < t

|fs,t(x, y)| ≤ Ce^ε|y−x|2.

The corresponding operator family {Ps,t} is positivity preserving, evolutionary, and satisfies Forward and Backward Kolmogorov’s equations

Back to Processes: SDEs and Martingale Problems

The kernel ps,t(x, y) is the transition density for a Markov process X = {Xt} with the generating family

Ltf (x) = a(t, x) · ∇f (x) + 1

2b(t, x) · ∇²f (x).

The process X is the unique weak solution to the SDE

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

Existence of some weak solution to SDE follows by compactness argument By the Itˆo formula, for any weak solution X and u ∈ C^1,2,

u(r, Xr) − Z r

s

(∂v+ Lv)u(v, Xv) dv is a martingale.

Taking u(r, x) = Pr,tf (x), we get that

u(r, Xr), r ∈ [s, t]

is a martingale, which yields

In the time homogeneous setting, the above argument is closely related to the notion of the martingale problem.

D. W. Stroock, S. R. S. Varadhan (1979), Multidimensional Diffusion Processes, Springer, Berlin

Ethier, S.N., Kurtz, T.G. (1986) Markov Processes: Characterization and Convergence.

Wiley, New York

Recall that a process X is said to be a solution to the martingale problem (L, C∞² (R^d)), if for every f ∈ C∞² (R^d) the process

f (Xt) − Z t

0

Lf (Xs) ds, t ≥ 0

is a martingale w.r.t. the natural filtration of X. The martingale problem is said to be well-posed, if

1 for any probability measure π on R^dthere exists a solution X to the martingale problem with X0∼ π, and

2 any two solutions with same π have the same distribution.

The above argument shows that the martingale problem for Lf (x) = a(x) · ∇f (x) +1

2b(x) · ∇²f (x), f ∈ C∞² (R^d)

Some other possibilities

We have

|(∇x)^kps,t(x, y)| ≤ C(t − s)^−k/2ϕc(t−s)(x, y), k = 1, 2, which yields

|∂sps,t(x, y)| ≤ C(t − s)⁻¹ϕc(t−s)(x, y).

I.In the time-homogeneous setting, this can be extended to

|(∂t)^mpt(x, y)| ≤ C(t − s)^−mϕ_c(t−s)(x, y), m ≥ 1.

Knopova, V., Kulik, A. (2018) Parametrix construction of the transition probability density of the solution to an SDE driven by α-stable noise. Annales de l’Institut Henri Poincar´e: Probabilites et Statistiques 54

pt(x, y) = p⁰t(x, y) + Z t/2

0

Z

R^d

p⁰t−s(x, z)Ξs(z, y) dzds

+ Z t/2

0

Z

R^d

p⁰_s(x, z)Ξt−s(z, y) dz ds.

∂tΥ^~(k+1)_t (x, y) = Z t/2

0

Z

R^d

(∂tΥ^~k)t−s(x, z)Υs(z, y) dzds

+ Zt/2Z

Φ^~k_s (x, z)(∂tΥ)t−s(z, y) dzds

II.Provided that a, b are C^m in x,

|(∇x+ ∇y)^mps,t(x, y)| ≤ Cϕc(t−s)(x, y).

S.D. Eidel’man (1969) Parabolic Systems. North-Holland & Wolters-Noordhoff, Amsterdam.

Approximate fundamental solutions: the way to avoid (5)

Define for η > 0

ps,t,η(x, y) = p⁽⁰⁾_s,t+η(x, y) + Z t

s

(p⁽⁰⁾_s,r+η∗ Ξr+η,t+η)(x, y) dr.

Denote

Ps,t,ηf (x) = Z

R^d

ps,t,η(x, y)f (y) dy,

∆s,t,ηf (x) = (∂s+ Ls,x)Ps,t,ηf (x).

Lemma

Each Ps,t,ηf (x) is C¹ in t and C² in x, thus ∆s,t,ηf (x) is well defined. For each f ∈ C∞(R^d),

Ps,t,ηf → Ps,tf, η → 0 uniformly on 0 ≤ t − s ≤ T and

∆s,t,ηf → 0, η → 0

(∂sps,t,η(x, y) + Ls,xps,t,η(x, y))

= Υs,t+η(x, y) − (p⁰s,s+η∗ Ξs+η,t+η)(x, y) +

Z t s

(Υs,r+η∗ Ξr+η,t+η)(x, y) dr.

→ Υ −

∞

X

k=1

Υ^~k+

∞

X

k=1

Υ ~ Υ^~k

!

s,t

(x, y) = 0, η → 0.

We have proved the following Proposition

For any f ∈ C∞(R^d) and t ∈ R, sequence u^η(s, x) = Ps,t,ηf (x) satisfies

 (∂s+ Ls,x)uη(s, x) → 0, s ∈ (−∞, t − τ ), uη(s, x) → u(s, x) := Ps,tf (x) .

In such a case, we call function u(s, x) approximate harmonic

Rebuilding the theory using approximate harmonic functions

A. Kulik (2019) Approximation in law of locally α-stable L´evy-type processes by non-linear regressions. Electronic Journal of Probability 24

V. Knopova, A. Kulik, R. Schilling (2020) Construction and heat kernel estimates of general stable-like Markov processes, arXiv:2005.08491

Proposition

Let a function u(s, x) beapproximateharmonic for (∂s+ Ls) on (−∞, t) × R^dwith Ls

satisfying the PMP.

Then u(t, x) ≥ 0 implies u(s, x) ≥ 0, s < t.

Corollary

I. For any f ≥ 0 and s ≤ t, Ps,tf (x) ≥ 0.

II. For any s ≤ r ≤ t, Ps,t= Ps,rPr,t. III. For any s ≤ t and f ∈ C∞²,

Ps,tf (x) = f (x) + Z t

s

Ps,rLrf (x) dr.

Proposition

Let a function u(s, x) beapproximateharmonic for (∂s+ Ls) and Y be a weak solution to the SDE.

Then u(r, Yr), r ∈ [s, t] is a martingale.

Mr,η:= uη(r, Yr) − Z r

s

(∂v+ Lv)uη(v, Yv) dv → u(r, Yr), η → 0.

Taking u(r, x) = Ps,tf (x) we get that X, Y have the same laws.

Summary

I. ps,t(x, y) defines a Markov process X.

II. X is the unique weak solution to SDE.

III. ps,t(x, y) solves the Forward Equation.

IV. ps,t(x, y) solves the Backward Equation in the approximate sense.

These properties are based on the parametrix integral equation only, and allow easy extension to the L´evy-type PDOs

Thank you!