SECOND-ORDER STOCHASTIC DIFFERENTIAL EQUATIONS: STABILITY, DISSIPATIVITY, PERIODICITY. III. - A SURVEY Текст научной статьи по специальности «Математика»

МАТЕМАТИКА MATHEMATICS

УДК 517.953

ББК 22.161.62

Ш 96

Shumafov M.M.

Adyghe State University, Maikop, Russia, [email protected]

Second-order stochastic differential equations: * stability, dissipativity, periodicity. III. - A survey*

(Peer-reviewed)

Abstract. This paper is a continuation of the previous paper and presents the third part of the author's work. The paper reviews results concerning qualitative properties of second-order stochastic differential equations and systems. In the first part we gave a short overview on stability of solutions of the second-order stochastic differential equations and systems by Lyapunov functions techniques and introduced some mathematical preliminaries from probability theory and stochastic processes. In the second part the construction of Ito's and Stratonovich's stochastic integrals are given. Here, in the third part, analog of the chain rule for stochastic differentials (Ito's formula), is presented. The stochastic differential equations in the sense of Ito and in the sense of Stratonovich are introduced and the relation between these two forms of equations is established. The existence and uniqueness theorem for solutions of stochastic differential equations is formulated.

Keywords: random variable, stochastic process, Wiener process, stochastic integral, stochastic differential, Ito formula, stochastic differential equation

Шумафов М.М.

Адыгейский государственный университет, Майкоп, Россия, [email protected]

Стохастические дифференциальные уравнения второго порядка: Устойчивость, диссипативность и периодичность. III. - Обзор**

Аннотация. Статья является продолжением предыдущей статьи и представляет собой третью часть работы автора. В работе делается обзор результатов исследований качественных свойств решений стохастических дифференциальных уравнений и систем второго порядка. В первой части работы был дан краткий обзор результатов работ по стохастической устойчивости решений дифференциальных уравнений и систем второго порядка с использованием аппарата функций Ляпунова. Были приведены некоторые предварительные сведения из теории вероятностей и теории случайных процессов. Во второй части дана конструкция стохастических интегралов Ито и Стратоновича. Здесь, в третьей части, вводится понятие стохастического дифференциала и приводится формула Ито дифференцирования сложной функции для стохастических дифференциалов. Дается определение стохастического дифференциального уравнения в форме Ито и в форме Стратоновича, устанавливается связь между ними. Формулируется теорема существования и единственности для решений стохастических дифференциальных уравнений.

Ключевые слова: случайная величина, стохастический процесс, винеровский процесс, стохастический интеграл, стохастический дифференциал, формула Ито, стохастическое дифференциальное уравнение

* This work represents the extended text of the plenary report of the Third International Scientific Conference "Autumn Mathematical Readings in Adygea" (AMRA - 3), October 15-20, 2019, Adyghe State University, Maikop, Republic of Adygea.

** Статья представляет собой расширенный текст пленарного доклада на Третьей международной научной конференции «Осенние математические чтения в Адыгее» (ОМЧА - 3), 15-20 октября 2019 г., АГУ, Майкоп, Республика Адыгея.

The paper is a continuation of the previous papers [1, 2]. We continue the section "2. Some Mathematical Preliminaries", where some basic notions and facts from probability theory and stochastic analysis are introduced. In [2] the construction of Ito's and Stratono-vich's stochastic integrals was presented. Here we shall introduce the notion of stochastic differential and present the chain rule, Ito's formula, for stochastic differentials. We shall give the definition of stochastic differential equations in the Ito and in the Stratonovich forms and formulate the existence and uniqueness theorem for solutions of stochastic differential equations. The details can be found, for instance, in the books: [3, Ch. 1, 2], [4, Ch. 3, 5, 6], [5, Ch. 4, 5], [6, Ch. 2-4], [7, Ch. 6-9, 11], [8, Ch. 5], [9, Ch. 8, §§ 2-4], [10, Ch. 5, § 5], [11, Ch. 12] (see, also the original papers [12, 13]).

2.16. Stochastic differentials. Ito's formula

Here we introduce stochastic differentials in the sense of Ito and in the sense of Stratonovich. The chain rule for stochastic differentials (Ito's formula) is presented.

2.16.1. Definition of stochastic differentials

Let (Q,F,P) be a complete probability space, where Q = {c} is a set of elementary events c, F is a a -algebra on Q, P is a probability measure on F . Let [Ft}t>0 be a filtration, i.e. {Ft}t>0 is a family of increasing sub-a-algebras of F: F^F^F for all 0 < s < t < ro .

Let £(t) = £(t,c), £(t) = ( (t),...,£m (t)) T, denote an m -dimensional ^"-valued.

Wiener process defined on the (Q,F,P) adapted to the filtration {Ft}t>0, i.e. for every t > 0 the vector random variable £(t,c) is Ft-measurable (the random variables

£ (t),...,£m (t) are assumed to be mutually independent). Assume that for every t > 0 the increments £r (t + h) -£r (t) (r = 1 ,...,m), h > 0, are independent from the a algebra Fu i.e. for every event At Ft. More precisely, we assume that for every t > 0 the a -algebra Fm,h>t= a{^(t + h) -£(t) , 0 < h < ro}, generated by the process (t + h) -£(t) ,0 < h < a>}, is independent of the a-algebra Ft In this case, the family {Ft} is said to be non-anticipating with respect to the £ (t) .

Let b (t,c) and ax (t,c) ,..., am (t,c), t e[a, b], ceQ , 0 < a < b, be ^"-valued column-vector random functions which are measurable in (t, c) e [a,b]xQ. Assume that the vector-functions b (t,c) and a (t,c) ,..., am (t,c) are Ft-measurable for all t e[a, b], i.e. they are adapted to the filtration {Ft} (they are said to be nonanticipating with respect to the family {Ft}).

Let x(t,c) , t e[a,b], c eQ, be an ^"-valued stochastic process that has continuous sample functions (sample paths) for almost every c eQ (i.e. with probability 1), and that is Ft-measurable (and hence non anticipating).

Definition 47. We shall say that the stochastic process x (t) = x (t, c) has the Ito stochastic differential (on t e [a, b] e

m

dx(t) = b(t,c) dt + Y,ar (t,c)d£ (t) (2.13)

r=1

or (in another notation)

dx (t) = b (t,®) dt + a( t,®) dg(t), (2.13')

where cr(t,®) = (< (t,®) ,...,<m (t,®)) is an nx m-matrix, d£(t) :=(d£1 (t),...,d£m (t))T if:

a) ||b(t,®)e L([a,b], < (t,®) eL([a,b],i.e.

b b JI|b (t, ® )) ) dt < m, J||a-r (t, ®)) ) 2 dt < m

a a

for almost every ® eQ, that is, with probability 1 (| |-|| is the Euclidean norm in ^n);

b) for all a < s < t < b the relation

t m t

x (t,®) = x (s,®) + J b (r,®) dr + ^J<r (r,®) d%r (r,®) (a.s.) (2.14)

s r=1 s

holds.

Remark. The two integral conditions in a) may be replaced by (more strict conditions):

b b eJ||b(t,®)) ) dt <M and Ej|o-r (t,®)) ) 2dt <m.

a a

It should be pointed out that the stochastic differentials are simply a compact symbolic notation for relationships of the form (2.14): strictly speaking the symbols "dx(t)", "dt",

"d%r (t)" and "d%(t)" in expressions (2.13), (2.13') have no meaning alone.

A stochastic process of the form (2.14) is usually called Ito process. Note that the differential (2.13) in vector form is assumed to be equivalent to the n "scalar" stochastic differentials

m

dx, (t) = b (t,®)dt + £ < (t,®)d£r (t),

r=1

where x(t) = x(t,®) = ( (t,®) ,...,xn (t,®) )T, b(t,®) = (b (t,®) ,...,bn (t,®) )T,

< r (t,®) = (< (t,®) ,...,^rn (t,®) ) .

By analogy with definition of Ito's stochastic differential one can define stochastic dif-

n

ferential in the sense of Stratonovich. We shall say that the ^ -valued stochastic process x (t) = x (t,®) has the stochastic differential in the sense of Stratonovich

m

dx(t) = b(t,®) dt + (t,®) °d£r (t)

r=1

or

dx(t) = b(t,®)dt + <(t,®) °(t) , where £ (t) = £ (t,®)e^, b (t,®) , < (t,®)e (r = 1 ,..., m ); £(t) = ( (t),..., ^ (t)) , <r(t,®) = (< (t),...,<jm (t) ) is nxm-matrix, if for all a < s < t < b the relation holds

t m t

x (t,®) = x(s,®) + J b (r,®) dr + ^J<r (r,®) ° d%r (r,®) (a.s.) (215)

s r=1 s

provided that the integrals in (2.15) exist. (Here, in (2.15) the small circle " ° " before the d%r signifies the Stratonovich integral).

Example. Suppose n = m = 1, s := 0, 0 < t <b . Asis known (see [7, p. 61],

[8, p. 63]) for one-dimensional Wiener process &( t) = & (t, c) we have

(I) J &(s) d£(s) = £(sf/2 -1/2, (S) J &(s) o d&(s) = £(sf/2.

0 0

There fore the Ito stochastic differential of the process x (t) = &(t ,c)2 is of the form

d ((t)) 2 = dt + 2£(t) d£(t), (2.16)

and the Stratonovich stochastic differential is d(& (t))2 =£(t) o d& (t). The latter two equalities are only the differential notations of integrals above.

If we construct the increment A&(t)2) of &(t)2 formally, using Taylor's formula

A(x2) = 2xdx + (dx)2, we obtain formally (" At ^ 0")

d ((t)2) «A&(t)2) = 2&(t) d&(t) + (d&(t) )2.

(Another motivation: let :=&(tk,c). Then

A ) = & - & = (+! - £) 2 + 24 (+! - &) = ( A&)2 + 2& A& . Replacing a(&2) , A& by d (&2), d&, formally, we obtain

d (£) = 2&d& + (d&)2. Comparison with (2.16) shows, that in the case of the stochastic differential of &(t)2 we must replace (d& (t))2 with dt, that is, (d& (t))2 = dt. 2.16.2. Ito's formula

The formula of Ito [12] implies (in the language of stochastic differentials) that smooth functions of stochastic processes defined by (2.14) are themselves stochastic processes of this type.

Ito's theorem [12]. Let u e d ([a, b]x^n, i.e. u = u(t, x) be a continuous

nonrandom function defined on [ a, b t e[ a, b], x = (,..., xn) T with values in ^

and with the continuous partial derivatives up to first order in t and to second order in x:

du du d 2u

dt' 5x/ dxidxj

(i, j = 1 n).

If the valued stochastic process x (t) = x (t, c) on t e[a, b],

x(t) = ( (t),..., xn (t)) , has an Ito differential (2.13)

m

dx(t) = b(t,c)dt + Y,ar (t,c)d&r (t)

r=1

(i.e. the process x(t, c) is defined on [a, b] by the stochastic differential (2.13)), then the stochastic process

y (t) = y (t ,c) = u (t, x (t ,c)) defined on [a, b] with initial value y (a, c ) = u (a, x (a, c )) also possesses an Ito stochastic

differential given by du

dy (t) =

dt (t. *(t))+>> f-(t.x (t)) ь(®+2>> ^(, x(t)) [>>, (t (®

i 1 i i 5 .1 1 i 1

+

;V r=1

( " Я?/ ^

> (t,x(t)К (t,®) d£ (t) . (a.s.)

r=1 V i=1 ЯХг'

dt +

(2.17)

The formula (2.17) is called Ito'sformula. This formula can be rewritten as

dy (t) =

Яи (Яи 1 , Л 1 -А Я2и / , ч , чт\ — + 1 —, b(t,®) l + —> -(к(t,®)er(t,®) )

Яt V^x V ') 2 j Яхг Ях;Л V ;

Яи

dt + >V^X, К (,®)Jd£r (,

„ , ..... du du (du

where the argument of the partial derivatives —, — =

dt dx

Яи

Ях1 Яхп)

Я2 и йх,.Ях1

is

(t,x(t)) ; <(t,®) = (< (t,®) ,...,<m (t,®) is nxm -matrix with column-vectors

< (t,®) ,...,<m (t,®) e []&", (<(t,®)<j(t,®)Tj denotes an entry (element) of the matrix

^ 'ij

product <(t,®) <(t,®) T, (*,*) denotes scalar product.

The Ito's formula (2.17) can be presented in another form. Let us introduce formally the following multiplication rules

dt• dt = 0, d£r • dt = dt• d£r = 0, d£r • d£r = dt, d£r • d£p = 0 (p *r). (2.18) Using the rules (2.18) and coordinate representation

m

dxt (t) = bt (t,®)dt + X< (t,®)d^r (t), i = 1 ,...,n, (2.19)

r=1

of the vector relation (2.13), we have

m

dxt (t) • dxj (t) = X < (t, ® ) • < (t, ® ) . (2.20)

r=1

Taking into account the latter (formal) equality (2.20) and applying relations (2.19), the Ito formula (2.17) can be represented as

dy (0=§-(x (0 ) dt+X f (x (0 ) ^ (o+2 X ffU- (x w) ^ w • dxj w, (221)

or in symbolic form

dy (t) = —dt + [ dx (t), —| и +1 (dx (t), —| и ,

Яt V Я^х) 2 V Ях)

Я

where dx(t) = (¿гх1 (t),...,dxn (t)) , —

Ях

Я

Я

Ях/ ' Ях;

. (In (2.21) the product dxt (t) • dx} (t)

n )

is calculated by using the formal rules (2.18).)

We single out the special case n = m = 1.

Corollary (Ito's theorem for n = m = 1 [7, p. 92]). Let u = u (t, x) be a scalar continuous nonrandom function defined on [a,b]x^ with continuous partial derivatives -du-,

du d2u dx ' dx2

If x (t) = x (t, c ) is a process defined on [a, b ] with stochastic differential

dx (t) = b (t,c) dt + a( t ,c) d&( t), where b(t,c) , a(t,c), &(t) = &(t,c) are scalar functions, then the process y (t) = u (t, x (t, c ) ) possesses on [a, b] the stochastic differential

dy (t) =

¥ (t' x(t))+äX (t' x(t) ) b (t ^+2 (t' x(t) ))a(t ^

dt +

(t, x (t) )a(t,c) d&(t). (a.s.) (2.22)

The formula (2.22) is called the one-dimensional Ito formula ([5, p. 44; 6, p. 32]). It can be rewritten as

dy (o=|u (t, x (o ) dt+|x (t, x (o ) dx (o+2 fu (t, x (o) (dx (o)

where (dx (t)) 2 = (dx (t)) -(dx (t) ) is computed according to the rules

dt - dt = dt - d& = d& - dt = 0, d&-d& = dt.

Note that the Ito's formulas (2.17), (2.22) are sometimes are called Ito's chain rule. Let us formulate Ito's theorem in its most general form.

Ito's theorem: general case [12]. Let u e C]'l ([a, b]x i.e. the nonrandom

function u = u (t, x) = (u1 (t, x),..., uk (t, x)) be a C)'2X map from [ a, b ]x ^ into ( k > 1). Let the ^-valued stochastic process x (t) = x (t,c) be defined on [a, b] by the stochastic differential

m

dx(t) = b(t,c)dt + ^ar (t,c)d& (t)

r=1

(as above in the case k = 1).

Then the k -dimensional stochastic process

y (t) = u(t,x(t,©)) = (u (t,x(t,c)),...,uk (t,x(t,c)))

defined on [a, b] with initial value y (a) = u (a, x(a,c)) also possesses an Ito stochastic differential dy (t) given by (cf., (2.21))

Du n du 1 n d2u

dyq (t) = —-dt + Y —-dx, (t) + - T-—dx, (t) dx, (t) ,

dt t! dx, iW 2, Dx, dx, ,w ;W

where yq (t) = uq(t,x(t,c)), q = 1 ,...,k, is qth component of the process y(t): y (t) = (y1 (t),..., yk (t) ) , the argument of the partial derivatives of uq is (t, x (t) ) , differentials dx, (t) are defined by (2.19), and the product dx, (t) - dxj (t) is calculated by the rules (2.18).

Now we illustrate the Ito's formula by two examples.

Example 1. Let u (t, x) = u (x) = xp, x e l[R, and x(t) = 4(t) = &(t,c). Then

dx(t) = d&(t) and therefore b(t,c) = 0, a(t,c) = 1. Hence for y(t) = 4(t)p the Ito's formula (2.22) yields

d (t (0 p)=pt (0 p-j dt (t)+2 p (p -1) t (t) p2 dt.

In particular case p = 2 we have the formula

d (t(t)2) = 2t(t) dt(t) + dt,

which we have established in 2.16.1 above.

Example 2. Let u(t,x) = tx, t,x e[, and x(t) = t(t,®) . Then dx(t) = dt(t) . Therefore b(t,®) = 0, c(t,®) = 1. Hence for y(t) = tt(t) the Ito's formula (2.22) yields

d (tt( t)) = t( t)dt + tdt(t),

which means that

t t tt(t) = jt(r) dr + Jrdt(r)

0 0

or

t t JVdt(r) = tt(t) -jt(r) dr. 0 0 The latter is analogous to classical integration-by-parts formula.

Example 3. Let u (t, x) = u (x1, x2) = x1 x2. Assume that the stochastic processes

x1 (t) = x1 (t,®) and x2 (t) = x2 (t,®) possess the stochastic differentials

dx1 (t) = b1 (t,® dt + ct1 (t,® dt (t),

dx2 (t) = b2 (t,® dt + a2 (tdt (t) .

Here k = 1, n = 2, m = 1. Then, taking into account the rules (2.18) the formula (2.21) yields the following result:

d ( (t) • x2 (t) ) = x2 (t)dxx (t) + x1 (t)dx2 (t) + dx1 (t) • dx2 (t) = = [x2 (t) bj (t,®) + xj (t) b2 (t,®) + o"j (t,®) a2 (t,®) ]dt + [x2 (t)Cj (t,®) + xj (t)o2 (t,®) )dt (t).

This means that the integral relation holds

t t t

x1 (t) x2 (t) = x1 (t0) x2 (t0) +1 x2 (r) dxx (r) +1 xj (r) dx2 (r) + (r, ®) c2 (r,®) dr

¿0 ¿0 ¿0 which is the rule for integration of stochastic integrals by parts. (Here dxj (t)• dx2 (t) = c (t,®) •c2 (t,®) by (2.20).)

2.17. Stochastic differential equations

Let (Q,F,P) be a complete probability space with a filtration [Ft}i>0 satisfying the usual conditions (see, 2.16.1). Let t(t) = (t (t),..., tm (t))T, t ^ 0, be an m -dimensional

([m-valued) Wiener process defined on the space (Q,F,P) adapted to the filtration {Ft}t>0. As in the subsection 2.16.1 the family {Ft} is assumed to be nonanticipating with respect to

the Wiener process t(t). Denote dt(t) :=(dtj (t),...,dtm (t) ) .

Let 0< t0 <T<^ . Let b(• , •): [t0,T]x and , •): [t0,T]x [n^[nxm

n

be Borel measurable [ -vector-valued and (n x m matrix)-valued functions defined on

[t0, T]x respectively. They are assumed be nonrandom ones, that is, for fixed (t, x) the values b(t,x) and u(t,x) are independent of aeQ, i.e. the random parameter a appears only indirectly as b (t, x (t,a)) and a(t, x (t,a)) if we substitute a stochastic process

x (t) = x (t,a) in place of variable x.

/ \ " Assume that x0 = x0 (a) is a ^-measurable [ -valued random variable such that

E||x0 (a)| |2 or x0 (a) is constant with probability 1.

Definition 48. An equation of an Ito's stochastic differential form dx (t) = b (t, x (t ,a)) dt + ct( t, x (t,a)) d%( t)

(x(t0) = x0, x0 = x0 (a), t0 < t <T<rc), (2.23)

or in the integral form

t t

x (t) = x0 + J b (r, x (r,a)) dr +Ja(r, x (r,a) ) d% (r) (2.24)

'0 h

interpreted as the defining equations for an unknown stochastic process x (t) = x (t, a) with given initial value x (t0) = x0, is called an Ito's stochastic differential equation.

Notice that, (2.23) together with the initial value is only a symbolic way of writing the stochastic integral equation (2.24) (cf., (2.13') and (2.14)).

Definition 49. A stochastic process x(t) = x(t, a), t e[t0, T], is called a solution of

the equation (2.23) or (2.24) on the interval [t0, T] if it has the following properties:

a) x (t ,a) is continuous in t e[t0, T] for almost all aeQ and F-adapted, i.e. Ft-measurable (hence nonanticipating for t e[t0, T]).

b) x (t, a) is measurable as a function of two variables (t, a) from [t0, T]xQ to (or more weak condition, x (t, a) is progressively measurable with respect to Ft).

c) The random ["-valued B(t,a) := b(t, x(t,a) ) and ["xm-valued S (t,a) :=o(t, x (t,a) ) functions are such that, they are measurable in (t,a) (or progressively measurable) and

T T

eJ IB (r, a)) ! dr < rc, eJ ||S (r, a)) ! 2 dr t0 t0 (or, more weak conditions: with probability 1

T T

J||B (r,a) dr<rc, J||s (r,a)f dr<rc).

t0 t0

(iBii=a 12, iisi=£№■

1 1, j

d) Equation (2.24) holds for every t e [t0, T] with probability 1. Remarks

1. If x(t,a) is a solution of (2.23), then every process stochastically equivalent to x(t,a) is also a solution (see [7, p. 103; 9, p. 469]).

2. For every solution of (1) there exist a stochastically equivalent solution with almost surely (i.e. with probability 1) continuous sample functions. It follows from the fact that substitution of a solution into the right-hand side of (2.24) yields a process which is a continuous function of t with probability 1 since both integrals in (2.24) are continuous functions of the upper limit for almost all a (the first integral of B(t,a) is absolutely continuous for almost all a, the second integral of S(t,a) has a continuous version). Since the right-hand side of (2.24) is almost surely equal to the left-hand side (for fixed t), i.e. stochastically equivalent to the left-hand side, then by virtue of the above Remark 1 it is a solution with almost surely continuous sample functions.

Let us now find the conditions that guarantee the existence and uniqueness of the solution to equation (2.23).

Theorem 2.1. (Existence and uniqueness theorem for stochastic differential equations [7, p. 105]). Suppose that we have an Ito's stochastic differential equation (2.23). Let

E| x0 (a)) |2 . Suppose that -valued function b (t, x) and the ["xm-valued matrix function c(t, x) are defined and measurable in (t, x) on [t0, T]x and have the following properties: there exist constants L and K such that

PI " "

a) (Global Lipschitz condition) for all t e[t0, T], x e [ , y e [ ,

||b (t, x) - b (t, y)) ) + \\c(t, x) -c(t, y) 11 < L ||x - y||;

b) (Linear growth condition) for all t e[t0, T], x e[",

lb (t, x) ! j + ||c( t, x)| |< K (1 + || x||).

Then the stochastic equation (2.23) has on [t0, T] a unique ["-valued solution x (t ,a) , continuous in t for almost all aeQ (i.e. with probability 1), that satisfies the

2 T 2 initial condition x (t0,a) = x0 (a) . In addition, E|| x (t, a)) ) and eJ| |x (t, a))) dt

t0

Here uniqueness means that if x(t,a) and y(t,a) are solutions of (2.23), continuous in t (withprobability 1) with the same initial value x0 = x0 (a), then

P{a : x (t,a) ^ y (t,a) for all t e[t0, T]} = 0.

Remarks

1. From hypothesis a) it follows that the functions b (t, x) and c(t, x) are uniformly continuous in the variable x for all t e[t0, T].

2. If the functions b (t, x) and c(t, x) are continuous with respect to t then the

hypothesis b) follows from a).

3. The theorem 2.1 remains valid if we replace the Lipschitz condition with the more general local Lipschitz condition: for every N > 0, there exists a positive constant LN such

that, for all t e[t0, T], ||x|| < N and ||y|| < N, the inequality from condition a) is satisfied

when L is replaced by LN .

4. For the Lipschitz condition to be satisfied, it is sufficient that the functions b (t, x) and c (t, x) have continuous partial derivatives of first order with respect to the components

of x for every t e [t0, T] and that these be bounded on [t0, T]x ^ (or, in the case of the local Lipschitz condition, on [t0, T]x{|| x|| < N ]).

5. If the functions b (t, x) and o(t, x) are defined on [t0, ro)x and if the assumptions of the Theorem 2.1 hold for every finite subinterval [t0, T) of [t0, ro), then the equation (2.23) has a unique solution defined on the entire half-line [t0,ro) (herewith, Lipschitz constant L depends, in general, on T; or on T and N, if we assume local Lipschitz condition). Such solution is called a global solution.

6. In the case of autonomous stochastic differential equation (2.23), where b (t, x) = b (x) and a(t, x) ^ct(x) , this equation has a unique continuous global solution

x(t,o) on the entire interval [t0,ro) such that x(t0, o) = x0 (o) provided only that the following Lipschitz condition is satisfied: there exists a constant L > 0 such that, for all

n

x, y e^

Ilb (x) " b (y) |HK x)-a( y)| |< L||x - y||.

Herewith the restriction on the linear growth condition b) in the Theorem 2.1 follows from global Lipschitz condition.

Above we have dealt with stochastic differential equations in the sense of Ito. Similarly one can define a stochastic differential equation in the sense of Stratonovich in terms of the Stratonovich's integral, a namely, as an equation of a Stratonovich's differential form

dx (t) = b (t, x (t) ) dt + a[t, x (t) )o d£(t), x (t0) = x0 = x0 (o), (2.25)

interpreted as the integral equation

t t

x (t) = x0 +J b (r, x (r,o)) dr + Ja(r, x (r,o) ) o d£(r), t e[t0, T] (2.26)

t0

with respect to a process x(t) = x(t,o). Here the second integral on the right-hand side of

the integral equation is to be understood in the sense of Stratonovich.

We again note that (2.25) is only a symbolical notation of the integral equation (2.26). A stochastic process x(t) = x(t, o) , t e[t0, T], is called a solution of the equation

(2.25) or (2.26) if it satisfied to the equation (2.26) for every t e[t0, T] with probability 1

provided that the integrals on the right exist.

Since the conversion formula ([7, p. 170]) for the two Ito's and Stratonovich's integrals yields the equality

ja(r,x(r)) od£(r) = {a(r,x(r)) d£(r) + -(,x(r))ar (,x(r)) dt,

2 r=11 cx

r0 r0 r0

then the stochastic differential equation (2.26) in the sense of Stratonovich is equivalent to the Ito equation

1 m Qq.

b (t, X (t)) + - X-^x- (t, Ж (t, x(t)) dt + a(t, x(t)) d£(t),

2 r=i CX

dx (t) =

where ul (t,x),...,am (t,x) are column-vectors of the (nxm)-matrix t,x),

dajdx = (d&rjCx;) ^ is Jacobi matrix of the vector function ar (r = 1,...,m),

dt(t) :=( (t),...,dtm (t) )T .

Remark. If the matrix function c(t, x) is independent of x, c(t, x) = c(t), then the stochastic equations in the sense of Ito and in the sense of Stratonovich coincide.

References:

1. Shumafov M.M. Second-order stochastic differential equations: stability, dissipativity, periodicity. I. - A survey // The Bulletin of the Adyghe State University. Ser.: Natural-Mathematical and Technical Sciences. 2019. Iss. 4 (251). P. 11-27. URL: http://vestnik.adygnet.ru

2. Shumafov M.M. Second-Order Stochastic Differential Equations: Stability, Dissipativity, Periodicity. II. - A Survey// The Bulletin of the Adyghe State University. Ser.: Natural-Mathematical and Technical Sciences. 2020. Iss. 4 (271). 2020. P. 11-25. URL: http://vestnik.adygnet.ru

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8. Evans L.C. Introduction to Stochastic Differential Equations. Version 1.2. Berkeley: University of California, 2002. 139 p.

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2. Shumafov M.M. Second-Order Stochastic Differential Equations: Stability, Dissipativity, Periodicity. II. - A Survey// The Bulletin of the Adyghe State University. Ser.: Natural-Mathematical and Technical Sciences. 2020. Iss. 4 (271). 2020. P. 11-25. URL: http://vestnik.adygnet.ru

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7. Arnold L. Stochastic Differential Equations: Theory and Applications. New York: John Wiley and Sons, 1974. 228 p.

8. Evans L.C. Introduction to Stochastic Differential Equations. Version 1.2. Berkeley: University of California, 2002. 139 p.

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