Метод параметрикса и диффузионная аппроксимация стохастических моделей тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Биттер Илья Игоревич

• Биттер Илья Игоревич
• кандидат наук
• 2025

Оглавление диссертации кандидат наук Биттер Илья Игоревич

Contents

Introduction

1 Stability of Distributions of Diffusion Processes under Non-Uniform Perturbations

1.1 Assumptions and Main Result

1.2 McKean-Singer Parametrix Method

1.2.1 Parametrix Series for Transition Densities

1.2.2 Flow Associated with the Drift

1.2.3 Convergence of the Parametrix Series for Diffusions with Unbounded Drift

1.3 Stability of the Parametrix Series

1.3.1 ^-Stability Case

1.3.2 Proof of the Main Result in the ¿^-Perturbations Case

1.3.3 Numerical Example

1.4 Proofs

1.4.1 Proof of Lemma

1.4.2 Proof of Lemma

2 Markov Chains Converging to Diffusions and Asymptotic Version of the Parametrix Method

2.1 Example: Convergence of Sample Trajectories of Induced Order Statistics

2.1.1 List of Assumptions and Main Result

2.1.2 Convergence of Quantile Processes

2.1.3 Proof of the Main Result

2.2 Local Limit Theorem in the General Case

2.2.1 Assumptions and Main Result

2.2.2 Parametrix method in McKean-Singer form for Markov chains

2.2.3 Some properties of transport flows

2.2.4 Proof of the main result

3 ^-stability of transition densities for degenerate diffusions

3.1 Parametrix method for densities of degenerate diffusions

3.1.1 Approximating process

3.1.2 Auxiliary process

3.1.3 Convergence of the parametrix series

3.2 Stability of degenerate diffusions under non-uniform perturbations

3.3 Proofs

3.3.1 Proof of Lemma

3.3.2 Proof of Lemma

Conclusion

References

Введение диссертации (часть автореферата) на тему «Метод параметрикса и диффузионная аппроксимация стохастических моделей»

Introduction

For a fixed finite T > 0, consider the stochastic differential equation

dXt = b (t, Xt) dt + a (t, Xt) dWt, t G [0, T], (1)

where b : [0, T] x Rd ^ Rd is the drift of the diffusion process, and a : [0, T] x Rd ^ Rd 0 Rd is the diffusion coefficient. These coefficients are assumed to be measurable, given together with Ws, s > 0 - Revalued Brownian motion on some filtered probability space (Q, T, (Tt)t>o , P) satisfying the standard conditions [46]. The infinitesimal generator of the process (1) at time u for all functions <p G CO (Rd, R) , z G Rd, is defined as follows:

LuV(z) = 2Tr (aa*(u,z)D2z^(z)) + (b(u,z),Dz<p(z)) = (2)

1 ^ , .d z6(u,z) *d6(u,z)

= 2 2^ (u, ^ h(u,z)^~■ i,j=l J i=l

Solutions of equations of the type (1) are the most important examples of diffusion processes and arise in statistical physics, various areas of mechanics, financial mathematics, population models of biology, etc.

Since in most cases the solutions of stochastic differential equations cannot be written out explicitly, it turns out to be convenient to study their distributions specified by transition densities satisfying the Kolmogorov system of equations. In 1907, E. Levy proposed a method for constructing fundamental solutions of individual second-order partial differential equations. This method was called Levy parametrix method [37], and its application to the specified classes of equations allowed us to obtain upper bounds for their fundamental solutions [26, 29].

As it turned out later, the Levy parametrix method is not adapted for use in the discrete case, that is, for constructing transition Markov kernels. In 1984, Konakov and Molchanov [36] noted that an alternative method to the Levy parametrix can be applied to Markov chains, namely, the McKean-Singer parametrix method, constructed by McKean and Singer in 1967 for the needs of differential geometry [40].

For a given parameter e > 0, we introduce a perturbed analogue of the process (1) with the dynamics

dXH = b£ (t, XH) dt + a£ (t, XH) dWt, t G [0, T]. (3)

The coefficients b£, a£ satisfy at least the same assumptions as b, a, and are also in some sense close to b, a if the parameter e is small. When both coefficients b, a are bounded and Holder continuous in space, and the diffusion matrix a(t,x) := aa*(t,x) is assumed to be uniformly elliptic, it is well known that there exists a unique weak solution of the equation (1) that admits a transition density p(0,x,t,y) [26, 29, 35], and the application of the parametrix method in

combination with the chain method [6] leads to a two-sided Gaussian estimate [4, 5]

C-19x-i (t,x - y) < p(0,x,t,y) < Cgx(t,x - y),

where

gx(t,x) = t~d/2 exp^-^^ , A G (0,1], t> 0.

Such methods have been successfully improved for application to more general cases, namely to operators satisfying strong Hormander conditions and Kolmogorov operators with linear drift [25, 34, 41]. When the drift coefficient is unbounded and nonlinear, less is known. To obtain upper bounds, it is necessary to control the terms of the parametrix series, which becomes a subtle problem in the case of unbounded drift. For drift with sublinear growth, that is, \b(t,x)\ < C + l) G (0,1), a generalization of the parametrix method was obtained in [23], but this result turned out not to be applicable in the case of linear growth. In this sense, linearly increasing drift is precisely the boundary case, starting from which it is necessary to introduce a forward flow corresponding to the transfer of the initial condition, or, equivalently, a backward flow corresponding to the transfer of the final condition [24, 34, 39]. For an unbounded at most linearly increasing drift, an upper bound for the transition density can be obtained using the truncation method introduced in [24], see also [39]. This method consists in studying the logarithmic Fleming transform (see [24, 43]) of the transition density as a cost function of a suitable stochastic control problem [16], where the desired density estimates are obtained by choosing the appropriate controls. In the case of unbounded drift, the truncation method allows one to obtain an upper bound on the transition density, but to study densities under perturbations, one must work with the full parametrix series without truncation. Note also that upper bounds for the total variation, entropy, and Kantorovich metric between two solutions (including stationary ones) of the direct Kolmogorov equations with different diffusion matrices and drifts on Rd x [0,T] with fixed T > 0 were obtained in [13, 14]. Chapter 1 of this dissertation is devoted to the study of how a perturbation of the coefficients of the process (1) affects its transition density p. A similar problem was considered by the authors in the paper [35] under the assumption of boundedness of the drifts (1) and (3), as well as uniform closeness of the pairs of coefficients b,bs and a,as. We extend the results of [35] to the case of unbounded drift with at most linear growth, and, moreover, the condition L^ - closeness of the coefficients is replaced by a weaker one, namely, closeness in the sense of the metric L1. For this purpose, an expansion of the transition densities in a complete parametrix series will be constructed, which, unlike [23], will converge absolutely and uniformly in the case of linear drift growth.

For every n > n0 > 1, consider a non-homogeneous Markov chain X^ defined on the lattice tk = k/n with k =1 ■ ■ ■ n and taking values in Rd. The dynamics of the chain Xn is as follows:

1 bn(U,x) + -1 n Jn

K+1 = X^ + -bn(U, x) + —=Cr+i, i =1 ■ ■ ■ n - 1, (4)

where the function bn : [0,1] x Rd ^ Rd, and the family of errors satisfies the following Markov condition:

£ = xi, •••) = ^ (•). (5)

The probability distributions corresponding to the densities q™ x(•) have zero mean for any values of n,U,Xi, and the corresponding conditional covariance matrices are defined by the relation

/ ^iZjq™x(z)dz = a™ (t,x).

JRd

Under the conditions considered and the uniform convergence of bn ^ b, an ^ a as n ^ *x>, it follows from, for example, Theorem 1 in [44] that the chain (4) converges in distribution to (1). Weak convergence of distributions of discrete-time Markov processes to diffusions has been widely studied. The classical literature in this area includes the results of Skorokhod [45], Struck and Varadhan [46]. These results were obtained by probabilistic methods. However, recently it has become quite common to use an analytical approach to considering the convergence of transition densities of Markov chains to diffusions, see [32, 35]. Namely, the application of the parametrix method to parabolic PDEs and a modification of this method to discrete-time Markov chains allowed us to quantify the weak convergence mentioned above. This approach can also be used to prove local limit theorems for stochastic approximation algorithms known as Robbins-Monroe procedures [31]. An important typical example of the (4) model arises when considering the concept of induced order statistics, which was first introduced independently by David [20] and Bhattacharyya [11].

Let (X1,Y1) , (X2,Y2),... be independent identically distributed random vectors with distribution (X,Y), where X G R and Y G Rd. We assume that the first components of the vectors have continuous distribution function F with quantile function F-1, so with probability 1 for the k-th order statistic Xnk obtained from X1,... ,Xn, we have Xn1 < ... < Xnn. We define the induced order statistics Yn1,..., Ynn as Ynk = Yj if Xnk = Xj. Let m(x) be the conditional expectation, and a2(x) be the conditional covariance matrix of Y given X = x, and let ^(t) = /-co1(i) a2(x)dF(x), 0 < t < 1.

The main result [11] concerns the limit behavior in the case of d =1 sample trajectories

j^n, k = (Ynj - m (Xnj)) ,k = 1,..., raj .

The first part of Chapter 2 studies the convergence of the Markov chain

Zn,k = 2 (1)Sn,k, (6)

Jn

where Sn,k is the multidimensional perturbed analogue of Sn,k:

= ^ (Ynj - m (Xnj) - -—raf- 2 (1)HXnj ^ ,k = 1,..., n J

to diffusion with nonzero trend

dXt = ^-1(1)0(F-l(t))dt + ^-1(1)a (F-1(t)) (7)

The (6) process has dynamics

^n,fc+i = + 1 ^-1(1)0(Xra,fc+i) + —=1 (1)a (F-1 (ifc+i)) 4+i n \/n

with tk = k/n, and the errors 5k+1 are of the form

4+i = °-1 (F-1 (tk+1)) (Yn,k+1 - m (Xn,k+1)). (8)

The asymptotic theory of induced order statistics is discussed in [20, 21, 22, 42, 48].

In [31] the authors compute the rate of convergence of the discrete scheme (4) to the diffusion (1) under the condition bn = b and an = a for all natural n, and assuming that the drift coefficient is bounded. In the second part of Chapter 2, a generalization of the above result is given to the case when the pairs of coefficients (b,a) and (bn,an) coincide only asymptotically, and b, bn are unbounded and allow no more than linear growth.

Finally, in Chapter 3, an analysis of the stability of transition densities of degenerate Kolmogorov-type diffusions is performed. The most general model of this type has the form

dX\ = Fi (t, X]X?) dt + a (t,Xi,..., X?) dWt, (9)

dx2 = Fz (t,xi,...,x?) dt,

dX3 = F3 (t,X? ,...,Xn) dt,

dX'n = Fn (t,Xn-1,Xn) dt

with Ws, s > 0 - Revalued Brownian motion on some filtered probability space (Q, T, (Ft)t>o , P. In the case when all coefficients of the equation (9) are smooth, the existence of a transition density of the vector (X 1,Xz, ...Xn) is ensured by Hormander's theorem [27]. In [24] the authors consider a uniformly Lipschitz family (Fi) and a uniformly elliptic Holder continuous diffusion coefficient a with exponent rq G (1/2,1], for which the existence of a density of the vector (X 1,Xz,...Xn) is proved. Finally, for the case rq G (0,1/2] the existence of a density under the same assumptions is shown in [38].

We consider a degenerate diffusion process of the form

dXt = b (t, Xt, Yt) dt + a (t, Xt ,Yt) dWt,

, (10)

dYt = Xtdt,t G [0,T],

and its perturbed analogue

dXl = b£ (t, X^, Yt£) dt + a£ (t, X£, Yt£) dWt,

dYt£ = X£dt,t G [0,T].

whose coefficients certain regularity conditions are satisfied, ensuring the existence of transition densities. The main goal of Chapter 3 is to extend the result of [30] on the closeness of the transition densities of the processes (10) and (11) for close values of the pairs of coefficients (b, a) and (bs, a£) to the case of unbounded drift with at most linear growth. Moreover, similar to the models (1) and (3) of Chapter 1, the condition of uniform closeness of the coefficients in the original work [35] is replaced by a weaker smallness of the distance in the L1 metric.

The purpose and objectives of the study

The main objective of this study is to expand the class of diffusion models for which results on the stability of distributions can be obtained for sufficiently small perturbations of the coefficients using the classical parametrix method in the McKean-Singer form, as well as to develop an asymptotic version of the parametrix method when the diffusion matrices and drift coefficients of the pre-limit objects only asymptotically coincide with the diffusion matrices and drifts of the diffusion limits. The solution of the problems posed is divided into the following stages:

1. Consider the stability of the distribution of the diffusion process (1) under perturbation of the coefficients by estimating the distance between the transition densities of the specified diffusion and the perturbed analog (3) in some integral metric. Generalize the well-known result [35] obtained under the assumption of closeness of the pair b, b£ in the uniform metric and their boundedness to the case when the drifts grow at most linearly and are close in the sense of the metric L1.

2. Construct an asymptotic version of the parametrix method for inhomogeneous Markov chains converging to diffusions. Unlike the result [31], we only assume an asymptotic coincidence of the corresponding coefficients of the limit process (1) and the Markov chain (4). In addition, the local limit is proved under the assumption of unboundedness of the drift b, allowing at most linear growth. As a model example, an estimate for the rate of weak convergence of sample trajectories of induced order statistics to the limit diffusion process is derived using the classical limit theorem.

3. Study the stability of Kolmogorov-degenerate diffusion processes under non-uniform perturbations of the coefficients and obtain generalizations of the work [30] similar to item 1 of this list.

Scientific Novelty

All the main results of the dissertation are new. Thus, in Chapter 1, estimates of the stability of non-degenerate inhomogeneous diffusion processes are obtained, including for previously un-

considered non-uniform perturbations of the coefficients. Moreover, the fundamental difference from the cases known in the previous literature is the assumption of unboundedness of the drift coefficients of the processes under study. In Chapter 3, a similar generalization is given for Kolmogorov-degenerate diffusions. Chapter 2 is devoted to the proof of a local limit theorem for inhomogeneous Markov chains converging to diffusions. This theorem is a justification for the so-called asymptotic version of the parametrix method, that is, a modification of the main result of [32] for the case when the corresponding coefficients of the Markov chain and the limit diffusion process coincide only asymptotically. Note that in this problem, unlike [32], the unboundedness of trends is also assumed.

Theoretical and practical significance of the results

The stability estimates of non-degenerate diffusions of type (1) in Chapter 1 and degenerate Kolmogorov processes (10) from Chapter 3 can be applied in many applied problems and contribute to the theory of estimation. Thus, an important application can be found, for example, in financial mathematics. It is often very useful to know how a change in the volatility a affects the density, and thus the price of the corresponding option [3, 8, 15]. In problems of unknown parameter estimation, having estimates (bs, a£) of the true parameters (b, a) and some control over the differences \b — 6£| , |a — a£| can help to estimate the difference p — p£ of the densities corresponding to the dynamics of the true model and the model with estimated parameters. Another important application involves the case of smoothing by convolution. This special kind of perturbation is useful for studying the error between the densities of non-degenerate diffusion of type (1) with Holder coefficients (or piecewise smooth bounded drift) and its Euler-Maruyama scheme. For this problem, some convergence results can be found in [7] and [33]. Thus, the stability results are useful in any application where the diffusion and drift coefficients may be misspecified. The asymptotic version of the parametrix method from Chapter 2 can be very useful in the theory of stochastic approximation. Examples include the asymptotic theory of induced statistics [20, 21, 22, 42, 48], Robbins-Monroe-type stochastic approximation algorithms [9, 31], and discretization problems for solutions of stochastic differential equations

[33].

Methodology and methods of the study

This study makes significant use of the analytical approach to the analysis of distributions of solutions to stochastic differential equations. The main tool for quantitatively assessing the stability of diffusion processes under perturbations of the coefficients is the parametrix method in the McKean-Singer form [40], which, unlike the classical parametrix method in the Levy form [37], can be adapted to the discrete case, allowing one to obtain results on the rate of

convergence of inhomogeneous Markov chains to the diffusion limit.

Publications based on the research results

The main results of the dissertation were published in the following articles:

1. Bitter, I. and Konakov, V. L1 and L^ stability of transition densities of perturbed diffusions. Random Oper. Stoch. Eq. 2021, 29 (4), 287 - 308.

2. Bitter I.I. L1-stability of transition densities of perturbed degenerate diffusions // Control of large systems. - 2022. - Issue. 100. - P.6-35.

3. Bitter I.I. Local limit theorem for perturbed sample trajectories of induced order statistics // Control of large systems. - 2025. - Issue. 113. - P.6-20.

Approbation of research results

The results of the dissertation research were reported by the author at the following conferences

and scientific seminars:

1. XII International Conference

"Application of Multivariate Statistical Methods in Economics and Quality Assessment named after S.A. Aivazyang, 09/21/2022

2. Joint probabilistic online seminar «Probability and mathematical statistics (three-city seminar)g, 04/12/2022

3. Research seminar «Stochastic Analysis and Applicationsg, HSE

Propositions submitted for defense

1. In Chapter 1, using the parametrix method in the McKean-Singer form, a local limit theorem is proved that characterizes the stability of the distribution of the diffusion process (1) under non-uniform perturbations of the drift and diffusion coefficients. This is a direct generalization of the result of [35], which considered perturbations in a uniform metric. Moreover, in our case, no more than a linear growth of the trend is allowed.

2. Chapter 2 contains a proof of a local limit theorem that gives a quantitative estimate of the rate of convergence of the inhomogeneous Markov chain (4) to the limit diffusion of the form (1). This statement generalizes the estimate from [32] to the case when the Markov chain coefficients only asymptotically coincide with the corresponding limit coefficients, and the drift coefficients allow no more than linear growth. As an example

of the model under consideration, the convergence of perturbed sample trajectories of induced order statistics to the diffusion limit is studied.

3. Chapter 3 solves a problem similar to that considered in Chapter 1 for Kolmogorov-degenerate diffusion processes of the form (10).

Author's Personal Contribution

The candidate's contribution was decisive in the results presented in Chapters 2 and 3 of the dissertation. The main results listed in these chapters reflect the candidate's personal contribution to the published works. The statements obtained in Chapter 1 are the results of the joint work of the candidate and the co-author of the work [12] (in equal shares).

Structure and volume of work

This dissertation consists of an introduction, three main chapters, a conclusion and a list of references. The total volume of the work is 83 pages, including 6 figures and 48 titles in the list of references.

Заключение диссертации по теме «Другие cпециальности», Биттер Илья Игоревич

Заключение

Таким образом, данное исследование содержит следующие результаты:

1, В Главе 1 обобщен известный результат об устойчивости распределений диффузионных процессов при возмущениях коэффициентов на случай, когда сносы растут не более чем линейно, а соответствующие коэффициенты данной диффузии и ее возмущенного аналога близки в смысле метрики Ь1.

2, В Главе 2 построен асимптотический вариант метода нараметрикс дня неоднородных цепей Маркова, сходящихся к диффузиям, причем соответствующие продольные и допредельные коэффициенты сноса допускают не более чем линейный рост, В качестве модельного примера получена оценка скорости сходимости выборочных траекторий индуцированных порядковых статистик к продольному диффузионному процессу,

3, В Главе 3 получены аналогичные пункту 1 настоящего списка обобщения результатов об устойчивости распределений вырожденных диффузионных процессов типа Колмогорова при возмущениях коэффициентов.

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