EXISTENCE AND UNIQUENESS OF SOLUTIONS TO STOCHASTIC FRACTIONAL DIFFERENTIAL EQUATIONS IN MULTIPLE TIME SCALES Текст научной статьи по специальности «Математика»

TOM 28, № 141 2023

SCIENTIFIC ARTICLES © A. Ponosov, 2023

DOI 10.20310/2686-9667-2023-28-141-51-59

Existence and uniqueness of solutions to stochastic fractional differential equations in multiple time scales

Arcady PONOSOV

Norwegian University of Life Sciences №-1432 As 5003, Dr0bakveien 31, Norway

Abstract. A novel class of nonlinear stochastic fractional differential equations with delay and the Jumarie and Ito differentials is introduced in the paper. The aim of the study is to prove existence and uniqueness of solutions to these equations. The main results of the paper generalise some previous findings made for the non-delay and three-scale equations under additional restrictions on the fractional order of the Jumarie differentials, which are removed in our analysis. The techniques used in the paper are based on the properties of the singular integral operators in specially designed spaces of stochastic processes, the representation of delay equations as functional differential equations as well as Picard's iterative method.

Keywords: Jumarie derivative, Brownian motion, multi-time scales

Mathematics Subject Classification: 60H20, 34A8, 34K50.

Acknowledgements: The author gratefully acknowledges the financial support from internal funding scheme at Norwegian University of Life Sciences (project no. 1211130114), which financed the international stay at Scuola Normale Superiore in Italy.

For citation: Ponosov A. Existence and uniqueness of solutions to stochastic fractional differential equations in multiple time scales. Vestnik rossiyskikh universitetov. Matematika = Russian Universities Reports. Mathematics, 2023, vol. 28, no. 141, pp. 51-59. DOI 10.20310/2686-96672023-28-141-51-59.

НАУЧНАЯ СТАТЬЯ © Поносов А.В., 2023 DOI 10.20310/2686-9667-2023-28-141-51-59 УДК 519.216.2

Существование и единственность решений стохастических дробных дифференциальных уравнений в нескольких временных шкалах

Аркадий Владимирович ПОНОСОВ

Норвежский университет естественных наук 1432, Норвегия, г. Ос, Дрёбаквайен, 31

Аннотация. В статье вводится новый класс нелинейных стохастических дифференциальных уравнений дробного порядка с запаздыванием и дифференциалами Жюмари и Ито. Цель исследования — доказать существование и единственность решений этих уравнений. Основные результаты статьи обобщают некоторые предыдущие выводы, сделанные для уравнений без запаздывания с тремя временными шкалами и при дополнительных ограничениях на дробный порядок дифференциалов Жюмари, которые снимаются в нашем анализе. Методы, использованные в статье, основаны на свойствах сингулярных интегральных операторов в специально сконструированных пространствах случайных процессов, представлении уравнений с запаздыванием в виде функционально-дифференциальных уравнений, а также на итерационном методе Пикара.

Ключевые слова: производная Жюмари, броуновское движение, мультивременные шкалы

Благодарности: Автор выражает благодарность за финансовую поддержку работы в рамках внутренней программы Норвежского университета естественных наук (проект № 1211130114), которая финансировала международный визит в Высшую нормальную школу в Италии.

Для цитирования: Поносов А.В. Существование и единственность решений стохастических дробных дифференциальных уравнений в нескольких временных шкалах // Вестник российских университетов. Математика. 2023. Т. 28. № 141. С. 51-59. DOI 10.20310/26869667-2023-28-141-51-59. (In Engl., Abstr. in Russian)

1. Introduction.

Processes operating in a multi-time scale modus arise in a number of fields including finance, science and engineering. In [1] it was suggested to use the fractional Jumarie derivative introduced in [2] v = (f (0 < a < 1) and the classical white noise g = wdf to model the deterministic and the stochastic parts of the multi-time scale processes, respectively. In the integral form this reads as

f (t) - f (0) = ai (t - s)a-1v(s)ds and g(t) - g(0) = i w(s)dB(s). J 0 J 0

Adopting this approach we study the following fractional stochastic delay differential equation in multiple time scales:

m

dx(t) = £ (fj(t, (Hijx)(t))(dt)aj + gj(t, (Hjx)(t))dBj(t)). (1.1)

j=i

Here fj (t,v) and gj(t,v) are random functions and H1j-, H2j- are linear delay operators, (dt)aj are the fractional Jumarie differentials and dBj (t) are the Ito differentials generated by the standard scalar Wiener processes (Brownian motions) Bj. The initial condition for (1.1) is

x(s) = p(s) (s < 0), (1.2)

where ^(u,s) is some random function (not necessarily continuous).

A solution of the initial value problem (1.1), (1.2) is a stochastic process x satisfying (1.2) for s < 0 and the integral equation

m //• t n t

x(t) - ^(0) = E(JQ aj(t - s)aj-1fj(s, (Hijx)(s))ds gj(s, (Hjx)(s))dB3(s)

The main result of the paper is a generalization of the existence and uniqueness theorem from [1] to the case of Eq. (1.1) and its operator counterpart.

2. Preliminaries

We keep fixed a stochastic basis (Q, F, (F)teR, P) satisfying the standard conditions [3] assuming, in addition, that Ft = F0 for all t < 0. All stochastic processes in this paper are supposed to be progressively measurable w.r.t. this stochastic basis or parts of it [3]. The following notation is used throughout the paper:

• R = to), R+ = [0, to), R- = (-to, 0).

• ^ is the Lebesgue measure defined on R or its subintervals.

• E is the expectation corresponding to the probability measure P.

• Bj(t) (t E R+, j = 1, ...,m) are the standard scalar Wiener processes.

• The space Lq(J, Rl) (1 < q < to, J C R is a subinterval), contains all progressively

measurable l -dimensional stochastic processes x(t) (t E J) such that J E|x(t)|qdt < to.

j

• The space Mp(J, Rl) (1 < p < to ) consists of all progressively measurable, l -dimensional stochastic processes x(t) (t E J) such that

sup E|x(t)|p < to. teJ

• The space kn consists of all n-dimensional, F0 -measurable random variables, and k = k1 is a commutative ring of all scalar F0 -measurable random variables.

• The space kn = : £ G kn, E|£|p < w} (1 < p < w) is a linear subspace of kn.

The spaces Lq(J, R1), Mp(J, R1) and kn are supposed to be equipped with the natural norms. Clearly also that for q < p and finite intervals J we have Mp(J, R1) C Lq(J, R1), also in the topological sense (the topology of the larger space is weaker). We will also assume that Mp(J, R1) C Mp(J',Rl) if J C J1 by letting the processes on J to be 0 outside J.

3. Properties of some delay operators

Consider the delay operator

(Hx)(t) = x(h(t)), (3.1)

Theorem 3.1. Let J = [0,T] and 1 < q < w. Assume that h(t) (t G J) is a Borel function such that h(t) < t ^ -almost everywhere on J. Then the operator (3.1) is a linear bounded operator from Mq(R- U J, Rn) to Mq(J, Rn).

Proof. Evidently, H is linear and maps progressively measurable processes defined on

R- U J to the ones defined on J. In addition, sup E|x(h(t))|q < sup E|x(t)|q, which proves

teJ t<T

boundedness of H from Mq (R- U J, Rn) to Mq (J, Rn). □

Next, consider the distributed delay operator

(Hx)(t) = i dsR(t,s)x(s). (3.2)

J—x

Theorem 3.2. Let J = [0,T] and 1 < q < w. Assume that the values of R(t,s) (t G J, — w < s < t) are l x n -matrices and R satisfies the following conditions:

1. R is Borel measurable on its domain;

2. sup Var—xR(t, ■) < w. teJ

Then the operator (3.2) is a linear bounded operator from Mr(R— U J) to Mq(J, R1).

Proof. Using the componentwise description of the operator (3.2) we may assume, without loss of generality, that l = n =1, so that (Hx)(t) = f^ x(s)dsR(t, s). Evidently, the operator H maps progressively measurable processes defined on R— U J to the ones defined on J. Putting Var—x[R(t,-)](s) = R(t, s) we get

supE| i x(s)dsR(t, s)|q teJ J—x

< sup (E f |x(s)|qdsR(t, s) x f f dsR(t,s) teJ y J—x \J—x

< sup( f E|x(s)|qdsR(t, s)) x sup ( I dsR(t, s)) teJ VJ—x / teJ VJ—x /

< sup E|x(s)|q sup I / dsR(t, s)) x sup I / dsR(t, s)

sR x sR

seR-uJ teJ \J—x / teJ VJ—x

< sup E |x(t)|q sup f i dsR(t,s)) < ('sup Var—xR(t,•)) sup E |x(t)|q

teR-uJ teJ VJ—x / \teJ / teR-uJ

which proves boundedness of H from Mq(R— U J) to Mq(J, R1). □

Remark 3.1. The delay operator (3.1) can be regarded as a particular case of the delay operator (3.2) if one puts R(t, s) = diag[xh,...., Xh] to be the n x n diagonal matrix containing the indicator Xh of the set {(t, s) : s < h(t)}. Moreover, if we define R(t, s) to be the (rn) x n -matrix of the form

R(t,s) = (diag[Xhi ,....,Xhi ],..., diag[Xhr ,....,Xhr]), then we get the multiple delay operator x(t) M (x(h^t)), ....,x(hr(t)).

4. Main results

Let us first consider the following fractional functional differential equation:

m

dy(t) = E ((Fj(y))(t)(dt)aj + (Gj(y)x)(t)dBj(t)), t e [0,T] (4.1)

j=i

equipped with the initial condition

y(0) = yo e kpn. (4.2)

The solution of Eq. (4.1) is understood in the following sense:

*jj\t - s)aj-1 Fj(y)(s)ds + J' Gj(y)(s)dBj(s^ , t e [0,T]. (4.3)

Definition 4.1. Let X, Y be two separable metric spaces and a : Q x X ^ Y be a random map. The operator x(-) ^ a(-,x(-)) is called the superposition operator generated by a.

D e f i n i t i o n 4.2.

• A continuous map V : X ^ Y, where X, Y are two separable metric spaces of functions defined on an interval J C R is called Volterra if

xi(s) = X2(s) ^ (Vx)(s) = (Vy)(s)

for all x1,x2 e X, any t e J and almost all s < t, s e J.

• A map V : Q x X ^ Y is called a random Volterra map if V (w, ■) is Volterra for almost all w e Q and V(-,x) is F-measurable for all x e X.

• The superposition operator generated by a random Volterra map is defined by x(-) MV (-,x(-)).

• A random Volterra map V : Q x X M Y, such that V4(-,x) is Ft-measurable for all t e J will be called non-anticipating.

Evidently, any Volterra map V gives rise to a family of Volterra maps V* : Xt M Yt (t e J), where Xt and Yt consist of the restrictions of the functions from X and Y, respectively, to (—to, t] fl J. It is also easy to check that the superposition operators generated by random Volterra maps are continuous in probability and if V is non-anticipating, then the superposition operator generated by V preserves progressive measurability of stochastic processes.

m

y(t) — y(0) =

In the proofs below we use the following inequalities:

E

f (s)dB (s)

q / t \ q/2

< cqE n |f(s)|2ds

(t G R+, q > 2),

(4.4)

where f (t) is an arbitrary scalar, progressive measurable stochastic process on R+, B(t) is the standard scalar Brownian motion and cq is a certain constant, which is independent of f ;

(t - s)a-1g(s)ds

< dq tqa-1 / |g(s)|q ds (t G R+, q > a-1)

1-1/q

(4.5)

where g : R+ ^ R is a Lebesgue measurable function and dq = ^qq--^T^)

Inequality (4.4) follows from the estimates proved in e.g. [4], while (4.5) is a direct consequence of Holder's inequality.

Theorem 4.1. Let J = [0,T] and assume that

1. 0 < « < 1, p > 2, «-1 < Pj < p (1 < j < m).

2. The superposition operators generated by the non-anticipating operators Fj,Gj (1 < j < m) map the space Mp(J, Rn) into the spaces Lpj (J, Rn) and L2(J, Rn), respectively, and satisfy the Lipschitz condition

||Fjyi - Fjy2||Lp. ( J,Rn) < t|y - ^2 ||Mp( J,Rn), ||Gjyi - Gjy2||L2(J,Rn) < ^||yi - y2||Mp(J,Rn)

for some constant t and the sub-linear growth condition

||Fj£||lPj(j,Rn) < b||e||fc?, ||Gje||L2(j,Rn) < b||em

for some constant b and any £ G kp1.

(4.6)

(4.7)

Then the initial value problem (4.1), (4.2) has a unique (up to the natural equivalence of indistinguishable processes) solution y(-,y0) G Mp(J, Rn).

If the constants t and b are independent of J, then the solution y(t,y0) is defined for all t G R+.

Proof. We prove this theorem for the equivalent integral equation (4.3). Notice that due to the Volterra property of the operators Fj and Gj we have

(4.8)

11 Fjyi - Fj||LP. ([0,t],Rn) < t||yi - 1 |Mp([0,t],fi"), ||Gjyi - Gjy2||£2([0,i],Rn) < t||yi - y2 ||MP([0,i],Rn)

for any t G J. Now, the proof becomes a standard application of Picard's iterations. Put

y(v)(t) = yo + £ («^(t - s)aj-1Fj(y(v-1))(s)ds + jf Gj(y(v-1))(s)dBj(s)) (t G J, v G N)

t

q

t

t

and y(0) = y0. Using (4.6)-(4.8) and inequalities (4.4), (4.5) with q = pj we obtain

E|y(v+1)(t) - y(v)(t)|p < K f E|y(v)(s) - y(v-1)(s)|p ds (t G J, v G N) (4.9)

0

and | |

E |y(1)(t) - y(0)(t)|p < Kot||yo||kn (t G J). (4.10)

Iterating (4.9) and using (4.10) yield

K v tv

E |y(v+1)(t) - y(v)(t)|P < Ko— (t G J, v G N),

which ensures convergence of the sequence {y(v)} to some y in the space Mp(J, Rn). The stochastic process y(t) satisfies then Eq. (4.3) due to continuity of the operators

Fj : Mp(J, Rn) ^ Cp.(J, Rn) and Gj : MP(J, Rn) ^ A(J, Rn)

and boundedness of the linear operators

(I1jy)(t) = T(t - s)a-1y(s)ds and (I2jy)(t) = f y(s)dB(s) 00

acting from Lpj(J, Rn) to Mp(J, Rn) and from L2(J, Rn) to Mp(J, Rn), respectively (see estimates (4.4), (4.5)).

Assume y1(t) and y2(t) to be two solutions of Eq. (4.3). Then we have, exactly as in (4.9), that

E |y 1 (t) - y2(t)|p < K f E |y1(s) - y2(s)|p ds (t G J),

0

and the property of uniqueness follows from Gronwall's lemma. □

To prove the existence and uniqueness theorem for (1.1) we represent it as Eq. 4.1. This is a standard procedure in the deterministic theory of functional differential equations [5]. To this end, we assume given two stochastic processes y G Mp( J, Rn) and ^ G Mp(R- U {0}), put

y (t) = i y(t) (t G J) and „ (t) = I 0 (t G J)

y+(t) = \ 0 (t g R-) and (t) = \ p(t) (t G R-)

and define

Fj(y) = fj(■, H1jy+ + H1jGj(y) = gj(■, H2jy+ + H2j(4.11)

which yields Eq. (4.1).

The result below connects Eq. (1.1) and (4.1).

Proposition 4.1. Let J = [0,T] and assume that the k -linear operators Hj Mp(R- U J, Rn) ^ Lp(J, Rl) are bounded for all i = 1, 2,j = 1,...,m. Then the stochastic process

( ) \ P(i) (t e R-)

is the solution of the initial value problem (1.1), (1.2) on the interval J if and only if y is the solution of the initial value problem (4.1), (4.2) on the same interval.

Proof. Let y be a solution of the problem (4.1), (4.2). Then (4.12) can be rewritten as x(t) = y+ (t,<(0)) + (t) (t E R— U J), and for all t E J we obtain x(t) = y(t) and Hjy+ + Hj= Hjx due to linearity of Hj. Hence x(t) satisfies Eq. (1.1). In addition, x(t) = <(t) for t < 0.

Assume now that x is a solution of the problem (1.1), (1.2) and put y = xJ. Then x(t) = y+(t) + (t) (t E R— U J), so that Hjx = Hjy+ + Hj, which means that y(t) satisfies Eq. (4.1) if Fj and Gj are defined as in (4.11). By construction, y(0) = <(0), and the result follows. □

Example 4.1. The representation (4.1) of Eq. (1.1) with the distributed delay operators Hij given by

(Hijx)(t) = I dsRjj(t, s)x(s), J—x

where Rij(t, s) are n x l -matrix valued, Borel measurable functions defined on {(t, s) : t E J, < s < t}, reads as

dy(t) = J2 (f(t^QdsRij(t,s)y(s) + Uj(t))(dt)a + gj(t, jf^R;(t,s)y(s) + U2j(t))dBj(t)

where uij(t) = (Hij)(t) = /(—x0) dsRij(t,s)<(s) In particular, Eq. (1.1) with time-dependent delays given by

(Hij x)(t) = x(hij (t)),

where hij (t) < t are Borel measurable functions (i = 1, 2, j = 1,...,m), has the following representation:

m

dy(t) = E (fj(t, (Sijy)(s) + uij(t))(dt)aj + gj(t, (Sjy)(s) + U2j(t))d£,(t)), j=i

where Sj, known as inner superposition operators (see e.g. [5]), are defined as

(Sj y)(t)

and

y(hij (t)) (t G J) y)(t)~ï 0 (t G R-),

«ij(t) = (Hij(t)) (t G R-).

Now we are ready to prove the existence and uniqueness result for Eq. (1.1). Theorem 4.2. Let J = [0,T] and assume that

1. o < a < i, pj > 2, a-1 < pj < p (i < j < m

2. For all j = 1,...,m the random functions fj , gj : ^ x R+ x R1 ^ Rn are such that fj(■,-,v) and gj(■,-,v) are progressively measurable for any v G R1 and fj(w,t, ■) and gj(w,t, ■) are continuous for P ® p.-almost all (w,t), satisfy the Lipschitz condition

|fj(w,t,xi) - fj(w,t,X2)| < t|xi - X2I, |gj(w,t,xi) - gj(w,t,X2)| < t|xi - X2I a. s.

for some constant t and all x1,x2 G R1, t G J and the sub-linear growth condition

|fj(^,t,x)| < b|xL |gj(^,t,x)| < b|x|.

t

t

3. The k-linear operators Hj : Mp(R- U J, Rn) ^ Lp(J, Rl) are bounded for all i = 1, 2, j = 1,...,m.

Then for any p G Mp(R- U {0}, Rn) the initial value problem (1.1), (1.2) has a unique (up to the natural equivalence of indistinguishable processes) solution x(-,p) G Mp(J, Rn).

If the constant t is independent of J, then the solution x(t, p) is defined for all t G R.

Proof. The proof is based on Theorem 4.1. Define Fj and Gj using the formulas (4.11). It is easy to see that the superposition operators generated by the non-anticipating operators Fj, Gj (1 < j < m) map the space Mp(J, Rn) into the space Lp(J, Rn), which contains both Lpj (J, Rn) and L2(J, Rn), because p > max{2,pj : j = 1,...,m}. These operators satisfy the Lipschitz condition (4.6) and the sub-linear growth condition (4.7) as well. Therefore, Eq. (4.1) with Fj, Gj so constructed has a unique solution y G Mp(J) satisfying the initial condition y(0) = p(0). Applying Proposition 4.1 completes the proof. □

Remark 4.1. As Mp([0,T],Rn) C Lp([0, T], Rn), the delay operators (3.1) and (3.2) satisfy condition (3) of Theorem 4.2.

References

[1] J.-C. Pedjeu, G. S. Ladde, "Stochastic fractional differential equations: Modeling, method and analysis", Chaos, Solitons & Fractals, 45 (2012), 279-293.

[2] G. Jumarie, "Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results", Computational Mathematics and Applications, 51:910 (2006), 1367-1376.

[3] B. 0ksendal, Stochastic Differential Equations. An Introduction with Applications, Springer, 2014.

[4] I. Neveu, Discrete Parameter Martingales, North-Holland, Amsterdam, 1975.

[5] N. V. Azbelev, V. P. Maksimov, L.F. Rakhmatulina, Introduction to the Theory of Functional Differential Equations. Methods and Applications, Hindawi, New York, 2007.

Information about the author

Arcady Ponosov, Doctor of Natural Sciences, Professor of the Institute of Mathematics. Norwegian University of Life Sciences, As, Norway. E-mail: arkadi@nmbu.no ORCID: https://orcid.org/0000-0001-5018-6577

Received 25.01.2023 Reviewed 28.02.2023 Accepted for press 10.03.2023

Информация об авторе

Поносов Аркадий Владимирович, доктор естественных наук, профессор Института Математики. Норвежский университет естественных наук, г. Ос, Норвегия. E-mail: arkadi@nmbu.no ORCID: https://orcid.org/0000-0001-5018-6577

Поступила в редакцию 25.01.2023 г. Поступила после рецензирования 28.02.2023 г. Принята к публикации 10.03.2023 г.