Probability computation for highdimensional semilinear SDEs driven by isotropic -stable processes via mild Kolmogorov equations

2025-05-02 0 0 916.38KB 29 页 10玖币

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Probability computation for high–dimensional semilinear SDEs driven by

isotropic α−stable processes via mild Kolmogorov equations

Alessandro Bondi∗

Abstract

Semilinear, N−dimensional stochastic diﬀerential equations (SDEs) driven by additive Lévy noise are inves-

tigated. Speciﬁcally, given α∈1

2,1,the interest is on SDEs driven by 2α−stable, rotation–invariant pro-

cesses obtained by subordination of a Brownian motion. An original connection between the time–dependent

Markov transition semigroup associated with their solutions and Kolmogorov backward equations in mild

integral form is established via regularization–by–noise techniques. Such a link is the starting point for

an iterative method which allows to approximate probabilities related to the SDEs with a single batch of

Monte Carlo simulations as several parameters change, bringing a compelling computational advantage over

the standard Monte Carlo approach. This method also pertains to the numerical computation of solutions

to high–dimensional integro–diﬀerential Kolmogorov backward equations. The scheme, and in particular

the ﬁrst order approximation it provides, is then applied for two nonlinear vector ﬁelds and shown to oﬀer

satisfactory results in dimension N= 100.

Keywords: Kolmogorov equations, Semilinear SDEs, Iterative scheme, Isotropic α−stable Lévy processes.

MSC2020: 60G52, 60H50, 65C20, 45K05, 47D07.

1. Introduction

In this paper, we are concerned with the study of quantities related to the N−dimensional, semilinear

stochastic diﬀerential equation (SDE)

(dXt= (AXt+B0(t, Xt)) dt +√Q dWLt, t ∈[s, T ],

Xs=x∈RN,(1)

with a speciﬁc interest in the case Nhigh. Here, given α∈1

2,1,Lis an α−stable subordinator (i.e., an

increasing Lévy process) independent from (βn)n=1,...,N , which in turn are independent Brownian motions;

we write W=β1, . . . , βN>. All these processes are deﬁned in a common complete probability space

(Ω,F,P),which we endow with the minimal augmented ﬁltration generated by the subordinated Brownian

motion WL. Moreover, T > 0is a ﬁnite time horizon and s∈[0, T ]is the initial time. As for A, Q ∈RN×N,

they are diagonal matrices with Anegative–deﬁnite and Qpositive–deﬁnite. For our numerical experiments

we will consider Q=σ2Id,being Id ∈RN×Nthe identity matrix, so that σ > 0is a parameter describing

the strength of the noise. Finally, the nonlinear bounded vector ﬁeld B0: [0, T ]×RN→RNis subject to

suitable regularity conditions which will be speciﬁed in the sequel and guarantee, among other things, the

existence of a pathwise unique solution of (1): it will be denoted by Xs,x = (Xs,x

t)t∈[s,T ].

Connected to the SDE in (1), we have the following Kolmogorov backward equation:











∂su(s, x) = −Ax +B0(s, x),∇>u(s, x)

−RRNus, x +√Qz−u(s, x)−1D(z)∇u(s, x)√Qzν(dz), s ∈[0, t),

u(t, x) = φ(x), x ∈RN,

(2)

∗Classe di Scienze, Scuola Normale Superiore di Pisa, 56126 Pisa, Italy. Email: alessandro.bondi@sns.it

arXiv:2210.03004v1 [math.PR] 6 Oct 2022

where φ:RN→R,D=z∈RN,|z| ≤ 1is the closed unit ball and we ﬁx t∈[0, T ]. Here ν(dz)is the

Lévy measure of WL, and up to a positive multiplicative constant is of the form ν(dz) = |z|−(N+2α)dz (see,

e.g., [20, Theorem 30.1]). The link between the equations in (1) and (2) is provided by Theorem 7 (ii)below

(see also the book [15] for related results), where it is shown that the time–dependent Markov transition

semigroup E[φ(Xs,x

t)] associated with (1) satisﬁes (2) in the closed interval [0, t]for every φ∈C3

bRN.

Moreover, we are able to extend the validity of this connection in [0, t)to every function φ∈ BbRN

through an original procedure based on regularization–by–noise and a mild, integral formulation of (2) (see

Remark 1).

In the present work, we are precisely interested in these expected values, with particular attention to

the case φ(x)=1{|x|>R}(for some threshold R > 0), where one has E[φ(Xs,x

t)] = P(|Xs,x

t|> R). Hence

we want to describe a method which allows to compute probabilities related to the solution of the SDE

(1). Trying to get an estimate of them by numerically solving the integro–diﬀerential equation (2) is a

typical example of curse of dimensionality (CoD), and since we intend to deal with a high dimension (in the

simulations we take N= 100), this is an unfeasible way to proceed. The canonical approach to tackle our

problem is the Monte Carlo method: several paths of Xs,x are simulated by the Euler–Maruyama scheme

with a ﬁne time step, and then the ﬁnal points of these trajectories are averaged to get an approximation of

the desired expected values by virtue of the strong law of large numbers. However, if we were to follow this

scheme (which is known to be free of the CoD), then we would have to start over the procedure every time

we change the starting point xand the starting time s, the noise strength σand even the nonlinearity B0, a

practice that is very common in a wide range of applications including weather forecasts and calibration of

ﬁnancial models (see [1] and references therein). In order to overcome this setback, we aim to extend to our

framework the ideas developed in the papers [9, 10] for the Gaussian case, namely we search for an iterative

scheme which relies on a single bulk of Monte Carlo simulations independent from the aforementioned

parameters. Speciﬁcally, to approximate the value of the iterates vn

s(t, x), n ∈N∪ {0},we just need

to simulate once and for all, using the Euler–Maruyama scheme, a large number of sample paths of the

subordinator Land of the stochastic convolution e

t=Rt

0e(t−r)AdWLr, t ∈[0, T ], which is the unique (up

to indistinguishability) solution of the linear SDE

t=Ae

tdt +dWLt,e

0= 0.

The main novelty of the approach that we propose consists in the structure of the noise WL, which is a

2α−stable, rotation–invariant Lévy process (cfr. [20, Example 30.6]). In particular, the introduction of L

considerably complicates the framework compared to the Brownian one treated in [9, 10]. This fact leads us

to develop an original procedure –essentially based on conditioning with respect to the σ−algebra generated

by the subordinator– to get an expression for the iterates which is suitable for applications. Moreover, the

theoretical foundation of the iterative method analyzed in this work, Theorem 3, has a remarkable interest

on its own. Indeed, it establishes a connection between the time–dependent Markov transition semigroup

associated with (1) and a mild, integral formulation of (2) (see Equation (11)) that, at the best of our

knowledge, is new when it comes to isotropic Lévy processes.

The paper is structured as follows. Section 2 describes the setting and recalls the main concepts that

will be widely used in the rest of the paper. In addition, it introduces the integral formulation of the

Kolmogorov equation (2) and shows its well–posedness. Next, in Section 3 (see Theorem 3) we provide

the probabilistic interpretation of (2) in mild form, along with other interesting regularization–by–noise

results for SDEs driven by subordinated Wiener processes. In Section 4 we deﬁne the iterative scheme

and prove its convergence to the expected values that we are trying to approximate. Next, Section 5 is

concerned with the computation of the ﬁrst iterate v1

s(t, x); it is divided into two subsections referring to

the deterministic and random time–shifts, respectively. Its results are used in Section 6 as the base case

for the induction argument that allows to calculate vn

s(t, x)(see Theorem 17). The last part (Section 7) is

devoted to numerical experiments in dimension N= 100 for two choices of the nonlinear vector ﬁeld B0,

with particular attention on the improvements provided by the ﬁrst iteration over the linear approximation

corresponding to the Ornstein–Uhlenbeck (hereafter OU) processes. Finally, Appendix A contains the proof

of Lemma 4.

Notation: Let d, m, n ∈N. In this paper, elements of Rdare columns vectors. For any u, v ∈Rd,

we denote by |u|the Euclidean norm and by hu, vi=u>vthe standard scalar product. For a matrix

A∈Rd×m,|A|= supx∈Rm:|x|=1 |Ax|is the operator norm. Given a vector ﬁeld B:Rd→Rm×n, the

uniform norm is kBk∞= supx∈Rd|B(x)|. In particular, if n= 1 then the Jacobian matrix is denoted by

DB ∈Rm×d, and DhB=DBh, h ∈Rd; if also m= 1 (so that Bis a scalar function) then the gradient

∇Bis a row vector and D2B∈Rd×drepresents the Hessian matrix. For an integer k∈N∪ {0}, the

space Ck

bRd;Rm×nis constituted by the continuous vector ﬁelds Bwhich are bounded, continuously

diﬀerentiable up to order kwith bounded derivatives. Taken h= 1, . . . , k and B∈Ck

bRd;Rm×n, we

write 

∂hB

∞= supi,j,hk∂hBi,j k∞, where B= (Bi,j ), i = 1, . . . , m, j = 1, . . . , n and h∈(N∪ {0})dis a

multi–index with length khk1=h.

2. Preliminaries and Kolmogorov backward equation in mild form

Fix N∈Nand a complete probability space (Ω,F,P). Consider Nindependent Brownian motions

(βn)n=1,...,N : we write W=β1, . . . , βN>. Moreover, for α∈(0,1) we take a strictly α−stable subordinator

L= (Lt)t≥0independent from (βn)n, and denote by FLthe augmented σ−algebra it generates, i.e., FL=

σFL

0∪ N, where FL

0is the natural σ−algebra generated by Land Nis the family of F−negligible sets.

In other words, Lis an increasing Lévy process with (cfr. [20, Example 24.12])

EeiuL1= exp n−¯γα|u|α1−itan πα

2sign uo, u ∈R,for some ¯γ > 0.(3)

Let us introduce the diagonal matrices A=−diag [λ1, . . . , λN]and Q=diag σ2

1, . . . , σ2

N, with 0< λ1≤

··· ≤ λNand σ2

n>0, n = 1, . . . , N. We endow Ωwith the minimal augmented ﬁltration F= (Ft)t≥0

generated by WL, which means Ft=σFWL

0,t ∪ Nfor t≥0, with FWL

0,t t≥0being the natural ﬁltration

of WL.

Given T > 0and a continuous function f: [0, T ]→RN, if x∈RNand 0≤s < T then Zs,x =

(Zs,x

t)t∈[s,T ]is the OU process starting from xat time s, i.e., it is the unique solution of the next linear SDE

dZs,x

t= (AZs,x

t+f(t)) dt +pQ dWLt, Zs,x

s=x. (4)

We denote by R= (Rs,t),0≤s≤t≤T , the time–dependent, Markov transition semigroup associated with

this family of processes:

Rs,tφ=EφZs,·

t,0≤s≤t < T, φ ∈ BbRd,(5)

where BbRNdenotes the space of real–valued, Borel measurable and bounded functions deﬁned on RN.

The Chapman–Kolmogorov equations ensure that

Rs,t (Rt,uφ) = Rs,uφ, 0≤s<t<u≤T, φ ∈ BbRN.(6)

For every 0≤s < t ≤Twe deﬁne Fs,t =Rt

se(t−r)Af(r)dr ∈RNand IL

s,t =Rt

se2(t−r)AQ dLr: Ω →

RN×N.An adaptation of [5, Theorem 6] guarantees that, for every φ∈ BbRN, the function Rs,tφis

diﬀerentiable at any point x∈RNin every direction h∈RN, with

∇>Rs,tφ(x), h=Ehφ(Zs,x

t)DIL

s,t−1e(t−s)Ah, Zs,x

t−e(t−s)Ax−Fs,tEi.(7)

Moreover, Rs,tφ∈C1

bRNand the following gradient estimate holds true for some constant cα>0:



∇>Rs,tφ

∞≤cαkφk∞sup

n=1,...,N 1

σn

2α

r2αλn

1−e−2αλn(t−s)e−λn(t−s)!,0≤s < t ≤T. (8)

In the sequel, for every x∈RNand t∈(0, T ]we are going to need the continuity of R·,tφ(x)in the interval

[0, t)[resp., in the closed interval [0, t]] when φ∈ BbRN[resp., φ∈CbRN]. In order to prove this

property, we ﬁrst note that a variation of constants formula lets us consider (from (4))

Zs,x

t=e(t−s)Ax+Zt

e(t−r)Af(r)dr +Zt

e(t−r)ApQ dWLr,0≤s≤t≤T, x ∈RN.(9)

This expression shows that the process (Zs,x

t)s∈[0,t]is stochastically continuous (in the variable s). As a

consequence, if φ∈CbRN, then we can easily deduce the continuity of R·,tφ(x)in [0, t]applying the

continuous mapping and Vitali’s convergence theorems to (5). In the general case φ∈ BbRN,one can

use the same argument combined with the regularizing property of Rand (6) to obtain the continuity of

R·,tφ(x)in [0, t), as desired. Finally, observe that there exists a constant C=C(α, A, Q)>0such that

cαsup

n=1,...,N 1

σn

2α

r2αλn

1−e−2αλn(t−s)e−λn(t−s)!≤C1

(t−s)1/(2α),0≤s < t ≤T.

We refer to [5, Remark 5] for a similar computation. Let us assume α∈1

2,1: in this way, denoting by

γ= 1/(2α), we have γ∈(0,1) and the bound in (8) entails



∇>Rs,tφ

∞≤Ckφk∞

(t−s)γ,0≤s < t ≤T, φ ∈ BbRN.(10)

For a given measurable and bounded vector ﬁeld B: [0, T ]×RN→RN, we are concerned with the

analysis of the following Kolmogorov backward equation in mild, integral form:

uφ

s(t, x) = Rs,tφ(x) + Zt

Rs,r B(r, ·),∇>uφ

r(t, ·)(x)dr, s ∈[0, t], x ∈RN,(11)

where t∈(0, T ]and φ∈ BbRN. We denote by kBk0,T = sup0≤t≤TkB(t, ·)k∞. In order to study (11), for

every 0< t1< t2≤T, we consider the Banach space Λγ

1[t1, t2],k·kΛγ

1[t1,t2]deﬁned by

Λγ

1[t1, t2] = nV: [t1, t2]×RN→Rmeasurable :V(·, x)∈C([t1, t2]) , x ∈RN;

V(s, ·)∈C1

bRN, s ∈[t1, t2] ; sup

s∈[t1,t2]

sγkV(s, ·)k1<∞o,

kVkΛγ

1[t1,t2]= sup

s∈[t1,t2]

sγkV(s, ·)k1,where kV(s, ·)k1=kV(s, ·)k∞+

∂1V(s, ·)

∞.

When t1= 0, we are careful to remove the left–end point of the interval [t1, t2]in the previous deﬁnitions,

so that we will be working with the space Λγ

1(0, t2],k·kΛγ

1(0,t2]. The following lemma proves the well–

posedness of (11). We refer to [8, Theorem 9.24] for an analogous result concerning the Kolmogorov forward

equation in mild form associated with OU processes in inﬁnite dimension corresponding to Brownian motions.

Theorem 1. Let α∈1

2,1and B: [0, T ]×RN→RNbe a measurable and bounded vector ﬁeld. Then for

every φ∈ BbRNand 0< t ≤T, there exists a unique solution uφ

s(t, x), s ∈[0, t], x ∈RN,of (11) such

that uφ

t− (t, ·)∈Λγ

1(0, t], where γ= 1/(2α).

Proof. Let us ﬁx φ∈ BbRN, t ∈(0, T ],¯s∈(0, t]and introduce the map Γ1: Λγ

1(0,¯s]→Λγ

1(0,¯s]given by

Γ1V(s, x) = Rt−s,tφ(x) + Zt

t−s

Rt−s,r B(r, ·),∇>V(t−r, ·)(x)dr, 0< s ≤¯s, x ∈RN,(12)

for every V∈Λγ

1(0,¯s]. Notice that such an application is well deﬁned and with values in Λγ

1(0,¯s], thanks to

the properties of Rdiscussed above, the dominated convergence theorem and the next computations based

on (10):

sup

x∈RNZt

t−s

∂xjRt−s,r B(r, ·),∇>V(t−r, ·)(x)dr≤N C kBk0,T kVkΛγ

1(0,¯s]Zt

t−s

(r−(t−s))γ(t−r)γ

≤4γ

1−γNC kBk0,T kVkΛγ

1(0,¯s]s1−2γ,0< s ≤¯s, j = 1, . . . , N. (13)

Here C=C(α, A, Q)>0is the same constant as in (10), and the last inequality is obtained using the

bound

t−s

(r−(t−s))γ(t−r)γ=(Zt−s

t−s

+Zt

t−s

2)dr

(r−(t−s))γ(t−r)γ= 2 Zt−s

t−s

(r−(t−s))γ(t−r)γ

≤2

1−γ2

sγs

21−γ=4γ

1−γs1−2γ,(14)

where for the second equality we perform the substitution u= 2t−s−r. Estimates similar to those in (13)

allow to write, for every V1, V2∈Λγ

1(0,¯s],

sup

x∈RN|(Γ1V1−Γ1V2) (s, x)|+ sup

x∈RN∂xj(Γ1V1−Γ1V2) (s, x)

≤4γ

1−γNkBk0,T s1−γ+Cs1−2γkV1−V2kΛγ

1(0,¯s],0< s ≤¯s, j = 1, . . . , N.

Hence we obtain

kΓ1V1−Γ1V2kΛγ

1(0,¯s]≤4γ

1−γNkBk0,T ¯s+C¯s1−γkV1−V2kΛγ

1(0,¯s].(15)

This shows that, for ¯ssuﬃciently small, the map Γ1is a contraction in Λγ

1(0,¯s]: we denote by V1its unique

ﬁxed point. Now deﬁne

uφ

s(t, x) = Rs,tφ(x) + Zt

Rs,r B(r, ·),∇>V1(t−r, ·)(x)dr, t −¯s≤s≤t, x ∈RN,(16)

and notice that uφ

t−s(t, x) = V1(s, x),0< s ≤¯s, x ∈RN.Therefore uφ

(t, ·)is the unique, local solution of

(11) (in the strip [t−¯s, t]×RN) such that uφ

t− (t, ·)∈Λγ

1(0,¯s].

At this point, we can repeat the same procedure to construct the solution of (11) in the interval

[t−2¯s, t −¯s], because the relation among constants in (15) –which is necessary to get a contraction– does

not depend on the initial condition. Speciﬁcally, we take φ1=uφ

t−¯s(t, ·)∈C1

bRNand deﬁne the map

Γ2V(s, x) = Rt−s,t−¯sφ1(x) + Zt−¯s

t−s

Rt−s,r B(r, ·),∇>V(t−r, ·)(x)dr, ¯s≤s≤2¯s, x ∈RN,

for every V∈Λγ

1[¯s, 2¯s]. Computations analogous to the ones in the previous step show that Γ2: Λγ

1[¯s, 2¯s]→

Λγ

1[¯s, 2¯s]is a contraction: its unique ﬁxed point is denoted by V2. Then we call

uφ1

s(t−¯s, x) = Rs,t−¯sφ1(x) + Zt−¯s

Rs,r B(r, ·),∇>V2(t−r, ·)(x)dr, t −2¯s≤s≤t−¯s, x ∈RN;

notice that uφ1

t−s(t−¯s, x) = V2(s, x),¯s≤s≤2¯s, x ∈RN, and that by the deﬁnition of φ1, one has

uφ1

t−¯s(t−¯s, ·) = uφ

t−¯s(t, ·). Now we extend the function uφ

s(t, x)in (16) assigning

uφ

s(t, x) = (uφ

s(t, x), t −¯s≤s≤t

uφ1

s(t−¯s, x), t −2¯s≤s≤t−¯s, x ∈RN.

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Probability computation for highdimensional semilinear SDEs driven by isotropic -stable processes via mild Kolmogorov equations

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