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9
МАТЕМАТИКА MATHEMATICS
УДК 517.95 ББК 22.161.62 Ш 96
Shumafov Magomet Mishaustovich
Doctor of Physics and Mathematics, Professor of Department of Mathematical Analysis and Methodology of Teaching Mathematics, Head of Department, Adyghe State University, Maikop, ph. (8772) 593905, e-mail: ma-gomet_shumaf@mail. ru
Second order stochastic differential equations: Stability, dissipativity, periodicity. II. - A survey*
(Peer-reviewed)
Abstract. This paper is a continuation of the previous paper and presents the second 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, and their main properties are presented.
Keywords: probability space, random variable, stochastic process, Wiener process, martingale, stochastic integral, Ito integral, Stratonovich integral, isometry property.
Шумафов Магомет Мишаустович
Доктор физико-математических наук, профессор кафедры математического анализа и методики преподавания математики, заведующий кафедрой, Адыгейский государственный университет, Майкоп, тел. (8772) 593905, e-mail: [email protected]
Стохастические дифференциальные уравнения второго порядка: Устойчивость, диссипативность и периодичность. II. - Обзор**
Аннотация. Настоящая статья является продолжением предыдущей статьи и представляет собой вторую 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 дается конструкция стохастических интегралов Ито и Стратоновича и приводятся их основные свойства.
Ключевые слова: вероятностное пространство, случайная величина, стохастический процесс, вине-ровский процесс, мартингал, стохастический интеграл, интеграл Ито, интеграл Стратоновича, свойство изометрии .
The paper is a continuation of the previous paper [1]. We continue the section "2. Some Mathematical Preliminaries", where some basic notions and facts from probability theory and stochastic analysis are introduced. Here we recall the construction of Ito's and Stratonovich's stochastic integrals, on the basis of which a stochastic differential equation in a certain sense will be defined. More details can be found, for instance, in the books: [2, Ch. 2-4], [3, Ch. 1], [4, Ch. 4], [5, Ch. 4], [6, Ch. 8], [7, Ch. 5], [8, Ch. 12] (see, also original papers [9] and [10]). We keep the general numeration in the work and continue this here.
* 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.
2.15. Stochastic integrals: Ito's and Stratonovich's integrals
We shall define the stochastic integral for random processes j ft (w)] >0 with respect to a Wiener process (Brownian motion) (w)]>0 . Since for almost all 0eQ, the Wiener sample path % : t (w) is nowhere differentiable (therefore of infinite variation) the integral cannot be
defined in the ordinary way. However, the stochastic integral can be defined for a wide class of stochastic processes by making use of the stochastic nature of Wiener process. This integral was first defined by K. Ito in 1949. It is now known as Ito stochastic integral.
2.15.1. The Riemann-Stieltjes sums for Ito and Stratonovich integrals
Let %((,w) be a one-dimensional (i.e. Я-valued) Wiener process defined on some probability space (Q, У,Р). Here R denotes the set of all real numbers. Let f(t,a>), ie[0,oo), сое Q, be a real-valued function. Let 0 < a < b .
Our task is to define the stochastic integral
] f(t,W)%(t,W) . (2.1)
a
As noted above, since the paths tof Wiener process %(t,w) are of infinite variations, it is impossible to define the integral (2.1) in the Riemann-Stieltjes sense. It is reasonable to start with a definition for a simple class of functions f(t,w) and then extend it by some approximation procedure.
Let us first assume that f has the form
f (t,W) = W) for tk < t < tk+i (k = 0,1,...,m-l), (2.2)
where ja = 10 < t1 <... < tm = b] isa partition of \a,b], and 7]k(w) (0 < k < m -1) are real-valued random variables. For such functions (2.2) it is reasonable to define
) f (t,W%(t,W := • (k+1,W ,W) . (2.3)
a k=0
Note that (2.3) is a random variable. In general, it is natural to approximate a given function
f (t,W) by simple functions If (fk ,W) • X[tt ,tk+1)((), where X[h ,tk+1 )(t) is indicator function of
k=0
\tk, tk+1) (i e. Z\tk,tk+j (() = 1 if t e\tk, tk+1), and Z\tk (t) = 0 if t $\tk, tk+1)), the points tk belong to the intervals \tk, tk+1). Then we define the integral (2.1) as the limit (in some sense) of Riemann-Stieltjes sums
"if (t( ,w) •Ww) -%(tk ,W as m ^^ .
k=0
However, there exist examples (see, for instance, [2, p. 23, 24], [4, p. 60]) which shows that (unlike the Riemann-Stieltjes integral) it do make a difference here what points t( we choice. The following two choices have turned out to be the most useful ones [2, p. 24]:
1) t( = tk (the left end point); then we get the Riemann-Stieltjes sum for Ito integral. It leads
to the Ito integral J f (t, w (t, w ) ;
a
2) t( = (tk + tk+0/ 2 (the mid point); then we get the Riemann-Stieltjes sum for Stratonovich integral. It leads to the Stratonovich integral, denoted by J f (t,w) ° d%(t,w) . Here the small circle
a
before the signifies the Stratonovich integral.
2.15.2. The space Ж1 ([a,6],i?). Definition of Ito's integral
Now we describe a class of functions for which the Ito integral will be defined. Let (Q, T, P) be a complete probability space with a filtration Let £ = {¿f,} be
a one-dimensional Wiener process (Brownian motion) defined on the probability space (Q, f , P) adapted to the filtration (i.e. Ff с F (o<s<t) and the function is
Tt - measurable for all t > 0).
Denote by У4(Й = F{ir:®Sf if) the sub- a -algebra of F, generated by Wiener process
is interpreted as the history of Wiener process up to (and including) time t). Clearly, £ Ts for all t > 0.
Also denote by Jil'(()= the sub-<x-algebra of T, generated by
t (УДЙ is interpreted as the future of the Wiener process beyond time t). Assume that [О is independent of Tt for all t > 0.
Definition 36. A family (F^}^ of a -algebras, =c У, is called nonanticipating with respect to if
1) iFjnB is a filtration (i.e. 3] с Ft с Г for all 0<s<t);
2) /s independent of Tt for all / >(). (The relation ГЙ1ft СЗ^ for all t> 0 holds.)
Definition 37. Let 0 < a < b < да. Denote by Жг = Ж1 {[af->\R) the class of all functions f it, a): [a, R such that:
1) the function (t,co)^f(t,co) is BXf measurable (3 denotes the Borel a-algebra on [a, b]);
2) f{t,a>) is Tt- adapted (i.e. f{t,a>) is Tt measurable for each time I > 0);
3) Ef t,a)2dt <да.
A real-valued stochastic process f it,a) satisfying the condition 2) of Definition 37 is called
nonanticipating with respect to the filtration It means that for each time t > 0, the ran-
dom variable f t(c°) "depends only the information available in the a -algebra Tt\ In turn, we may think of Tt as "containing all information available to us at time /". Notice that the condition
3) means that \baf i t,a
We can define in ЛГ" the norm
llfllab :=(е£/(t,a)2dt f. Then the set Ж2 becomes a normed space with norm
l|a,b
(we identify f and f in
if
f " f
a,b
= 0). This space is complete under the metric defined by the norm
For stochastic processes / e AT" we shall show how to define the Ito integral
if ](®)=}/(C^M,
a
where £((, c) is one - dimensional real-valued Wiener process.
b
a
The construction of Ito integral consists of two steps: first define the integral l\g ] for a class of simple processes g = git,®); then we show that each / g Ж- can be approximated by such simple processes g 's and we define the integral J fd% as the limit of J gd% as g ^ f. First introduce the concept of simple (or elementary) processes.
Definition 38. A real-valued stochastic process g = jg(t, w)] is called a simple (or elementary or step) process if there exist a partition a = tо < t1 <... < tm = b of [a, b], and bounded
random variables rjki®), 0 < к <m -1, such that rjk{со) is J7-., - measurable and
g Ы = iW w • X[tl,tl+1 )(() • (2.4)
m
m-1
k=0
Denote by the family = of all such simple processes. Clearly, M0 <= M2 .
For simple processes g we define the integral according to (2.3).
Definition 39. For a simple process g = git,®) with the form of (2.4) in define
Jg(i,0)d£(i,0) := I^(®) • [i,®) ~{(tk ,®)] . (2.5)
a k=0
The integral (2.5) is called the stochastic integral of g (t, ®) with respect to the Wiener process {((,®)} (or to Brownian motion). This integral is also called the Ito integral of g on the interval [a, b]. Note that the integral (2.5) is a random variable. It can show that the stochastic integral J*g(t)f£(t) belongs to
It is not hard to prove the following
Proposition 13. If g g Af^Qiij 6], Jt), then
EJ g(t,®]d%(t,®) = 0, e(j g( t,® )d£( t,®) J = EJ g( t,® )2dt.
a \ 'a
The latter relation expresses the isometry property of Ito integral for simple processes. It is easy to check the validity of the following assertion.
Proposition 14. Let gx,g^ g JUgQz, h\rR) and cx, cn be two real numbers. Then
and
ftl gl ((+ С2 g 2 ((, W Ш (f W)= С ¡"agi if, W)+ C2 tag 2 (t, .
The idea is now to extend the integral definition from simple processes to processes in For this purpose it should approximate an arbitrary process / g and
then pass to limit to define the Ito integral of f.
Proposition 15 ([3, p. 20]). For any f g Ai*" {\a,b\/i) there exists a sequence {gn{t\ ®)} of
simple processes such that lim Ej(( ( t,®)-gn ( t,®)Jdt = 0, i.e. gn ( t,®)^- f ( t,® as n ^œ
n^œ a
in the norm ||-|| b of the space Af".
Thus, by Proposition 15 the sequence {gn (t,®)} is a Cauchy sequence in the space
AT'([a,b]R). Therefore, Ej(n((,®) -gm((,®))2dt ^ 0 as n, m ^ œ. Making use of this fact,
a
by Propositions 13 and 14 we have:
e(( t,®) t t,®) t,®) ) = e(|[^„ ( t ,®)- gm ( t ,®)]( t ,®) ) =
= Ej(gn ( t,®)-g m ( ® )2 dt ^ 0,
a
as n, m ^œ.
Hence, the sequence of random variables {(gn (t,®)dg(t,®)} forms a Cauchy sequence in the space ¿^JQ. Therefore, in virtue of completeness of the space the limit of the
sequence {jhag„d%} exists as element of jC ~ (Dj jQ. This limit is taken as the stochastic integral.
Definition 40. (The Definition of Ito's integral.) Let f e Af3 (\a.h\R). The Ito integral of f = {f( t,®)} with respect to Wiener process {g(t,®)} is defined by
l|/](«):= ]f(t,co)dÇ(t,G)):= lim]gn{t,co)dÇ{t,co), limit in & ¡frflj
a a
where {gn (t, ® )} is a sequence of simple processes such that
lim Ej(f ( t,®)- gn ( t,®))2dt = 0
n^œ a
{i.e. limg„(t,<o) = f(t,a) in ATS ([a,è],i?)).
The above definition of the integral l[f ](®) is independent of the particular sequence
{gn (t,®)}.
2.15.3. Properties of the Ito integral
The Ito integral has many properties. We present some properties of them ([2, p. 29, 30],
([3, p. 22]).
Proposition 16. (The isometry property.) For all f e Jiff3 ([a, h\ R) the equality
e(( f ( t,®) dÇ( t,®))2 = Ejf ( t,®) 2dt,
holds, i.e. IITOGiOlltfttiuq = If^OaMJO-
Proposition 17. Let f, g g K3 ([a, h\ R), and let a,f3 be two real numbers. Then: 1 ) \ fit,® (t,®) is Th - measurable ;
a
2) Ej(f (t,®)£(t,®) = 0 ;
3) \ba[af + Pg ]t) = a|(fd£(t) +PtgdÇ(t) almost sure (a.s.);
4) J( r (® )f (t,® )d£ (t, ® ) = r (® ))( f (t, ® )d£ (t, ® ) almost sure (a.s.)
for any real-valued bounded Ta - measurable random variable //(&>), at that // • / g
5) Jbfd£(t,®) = J fd%(t,®) + ( fdÇ(t,®) (a.s.), if 0<a <c <b.
a c
The following proposition allows passage to the limit under the Ito's integral.
Proposition 18. If /(i.ffljeK® ([«,¿1^) and f n(f,œ)&J&([a,b\É) for n = 1,2,..., and /„(¿,0)-» f(f®) as n —> 00 in
i.e.
Eb(fn(t,®)- f (t,®))2dt ^ 0 as n ^ro,
a
then
] f n{t,(o)d^{t,(o)^] f{t,co)d^{t,(o) as n^oo in £2(n,R).
a a
Next, an important property of the Ito integral is that it is a martingale. Definition 41. Let f g M1 ([o, T] R). Define
ft,®) :=jf(s,®)£(s,®) for t e[0, T],
0
where l(0) := J00 fd^ = 0. The l[f ]((,®) is called the indefinite Ito integral of f.
The integral l[f ](t,®) is depending on t. It turns out that the l[f ](t,®) can be chosen to depend continuously on t.
Proposition 19 ([2, p. 32], [3, p. 24]). Let f g JW2 ([o, Tj R). Then there exists a t-
continuous version of the indefinite Ito integral {l[ f ](t, ®)} , i.e. there exist a t -continuous
stochastic process {./(/, ®)} on (Q, T, P) {i.e. the function t^.j(t,co) is continuous for almost all ® e Q ) such that
P{® : J(t,®) = l(t,®)}= 1 for all t e[0,T]. When we refer to l[f ](t,®) we will mean a continuous version of it.
Proposition 20 ([3, p. 25], [4, p. 80]). Let f ([O, T] R). Then the indefinite integral l[/](/,®), t g [0, T], is a continuous martingale with respect to the filtration
Remark. Let f€Jlfi([B,6],fl) Suppose that the random function f(t,co) is continuous on t almost sure (i.e. for almost all cogQ), that is, fit,®) is continuous in probability one on t. Then it can be shown ([2, p. 39]) that the Ito integral can be defined as limit in of the
Riemann-Stieltjes integral sums (see 2.15.1)
b m-1
a m k=0
where a = 10 < t1 < ... < tk < tk+1 <... < tm = b is a partition of interval [a,b] into m subintervals
[tk, tk+1), 5m = max (tk+1 - tk). Here l.i.m. denotes mean-square or quadratic mean limit, that is
k
the limit on the metric of the space ¿^(il,^. The same is true ([7, ch. 5, §4]) if the condition of continuity in probability one on t of the function f(t,®) is replaced by the condition of continuity on t in mean-square (or quadratic mean) of a nonanticipating function f(t,®) .
2.15.4. Extension of the Ito integral
A. The Ito stochastic integral can be extended to the multi-dimensional case. Let {;t = )}t >0 be an n -dimensional Brownian motion defined on the complete
probability space (C,?,?} adapted to the filtration [i^J (i.e. ¿f, is ^-measurable).
Let denote the class of all (nxm) -matrix-valued measurable {^/-adapted
processes f = {flJ (t,®)} , i = 1,..., n, j = 1,..., m, such that
eJ| f (/I 2 dt
< ro.
(Here ||/|| denotes the Euclidean norm of the matrix f.) Then each entry f (t,0) satisfies the hypotheses 1)-3) of the Definition 37.
Definition 42 (multi-dimensional Ito integral). Let f € JtfaQfl, ij].,^'"-^), Then the multidimensional Ito integral
b b ffii (t) - fm №f det i
]/ ((,m)d^t = f : ;
a a vfn1 (() - fnm (t))
is defined to be the (n x l)-matrix, i.e. the n-column-vector-valued, whose i 'th component is the
m b
following sum of one-dimensional Ito integrals ^ f fij (t ,00dftJ .
j=
Similar to the scalar case (see Proposition 20) the indefinite integral ff(s,00df(s) is an
a
R"-valued continuous martingale with respect to the filtration £i*s}.
B. Stochastic Ito integral can be extended to a larger class of stochastic processes. This extension consists of weakening to the condition 3) of Definition 37 to
b
3') f f ((,0)2dt <w (a.s.), i.e.
a
P-J0 :\f ((,0)2dt <wj = 1. Denote this class of the stochastic processes by ([«,&],.
The construction of Ito integral for random functions (stochastic processes) from :?*£[«*].<! is carried out according to the same scheme as for functions from the class ¿M (Details can be found in [4, p. 64-71], [6, ch. 8, §1].) Let {^(t,co)}mi be an one-dimensional Wiener process. If / € one can show
that there exist step functions (simple processes) {gn(i,o)}G?s([c( such that
b
f(t ((,0) - f ((,0))2 dt ^ 0
a
in probability, i.e. in measure with respect to P. For such a sequence one has that the sequence of
b
integrals fgn(t,00df(t,0 converges in probability to some random variable, and the limit only
a
depends on f, not on the sequence {gn}. Thus we may define the Ito stochastic integral for random functions from the class ^([«^t^fl).
Definition 43. (The Definition of extension of Ito's integral.) Let f S P ■ ([e, c],R), i.e. the co^s 1) and 2) o^nor, 37, and h p{|/(},0)2 * <„} = 1
The Ito integral of f with respect to Wiener process {f (t, 0)} is defined by
bb
ff (t,00df(t,0) : = lim fgn(t,0 №, 0) (limit in probability),
J n^w J
where {gn it, a) is a sequence of simple functions (step functions) such that
b
lim [itit,a) - f it,a))2dt = 0 (limit in probability).
n—J
a
As before there exists a t-continuous version of the corresponding indefinite integral
t
[ f (s,a)d^(s,a) .
a
2.15.5. Definition of Stratonovich's integral
Let be an one-dimensional Wiener process, and f it,a) a real-valued random func-
tion on [a, b]x Q. As is pointed out in 2.15.1 the limiting value (in certain sense) of the approximating Riemann-Stieltjes sums
m-1 , ,
2 f ((>М[+i,a),a)],
k=0
where a = 10 < t1 < ... < tm = b, t* e [tk, tk+1 ), depends on the choice of intermediate points t*.
A. Special case
As an example, consider the special case f (t, co) = ^(t, co). It can be proved (see [4, p. 60], [5, p. 61]) that for integral sum
m-1 , .
^4*) [J-c((k)] №=%(t,®C),
k=0
where t* = (l - A) k + Atk+1, 0 <A< 1, we have
lim ;g4*b((k+i)^('J = ^ 22 +f a- ±> - a) . (2.6)
I
Here 5m = max(tk+1 - tk). In particular, if we choose A = 0 in (2.6), that is t* = tk, we ob-
k
tain the Ito integral:
(I) J f(()df(() ; 2 - ;<g) 2 - ^. (2.7)
a
If we choose A = 1/2 in (2.6), that is, t* = (tk + tk+1 )/2, then we obtain the stochastic integral in the sense of Stratonovich:
(S)J;((,©);((,©) =:J;((,c)°d;((,c) =^(b) -^(a) (2.8)
aa
(Here, in (2.8), the small circle before the d; signifies the Stratonovich integral.)
Thus, in our special case f ((, co) = ;(t, co), where ;(t) = ;(t, co) is Wiener process, the Stratonovich integral is the following:
b ' \ + tk+r
J;(() ° d;((): = n.m.J]d ^ [+1)- )] .
a k=0 V y
Also, it is valid the equality ([4, p. 61], [5, p. 120])
J ;(() ° d;(t) = lln0g ^(tk) +/(tk^ [J-^((k)] . (2.9)
a m n0 k=0 2
Note that among integrals of the type (2.6), Ito's integral is characterized by the fact that, as a
i
function of the upper limit, i.e. the indefinite integral f^(s)d^(s) is a martingale [4, p. 61]. How-
a
ever the formula (2.7) does not coincide with the value obtained by formal application of the classical rules. This disadvantage is removed by the choice A = 1/2 in (2.6), which leads to the Stratonovich integral, but this entail other disadvantages (for instance, Stratonovich integrals are not martingales [2, p. 37]).
B. General case
Let f E Ji'Qcjb],^. Suppose the function t^ f{i,co) is continuous almost sure (or continuous in quadratic mean). The similarly to the Ito integral (see 2.15.3, Remark) it can be defined the Stratonovich integral of f by [2, p. 39]
b m-1 f t i t A
ff M °dZ(t,m):= Li.in.Xf ,0 [1,0)-£((,0)],
a Sm10 k=0 V 2 J
whenever the quadratic mean limit (i.e. in Z:(0,JJ)) exists. Here {a = 10 < t1 < ... < tk < tk+1 <... < tm = b} isapartition of the interval [a,b] into m subintervals
[tk, tk+i] (0 ^ k ^ m -1 , S m = max(tk+i - tk).
k
We generalize the formula (2.9).
Let F (x, t) be a real-valued nonrandom function that is continuous with respect to t , that has first-order continuous derivative F'x with respect to x € R (for instance, F e C1 (R X |ar&](jQ), and that satisfies the condition
b
Ej(F(¿¡(,0),t))2dt . (2.10)
a
Then one can define Stratonovich integral of the random composite function F (¿¿((,0), t).
Definition 44. (The definition of Stratonovich integral of the composite function in special case.) Let ¿((,0) be a Wiener process and the condition (2.10) holds. Then the Stratonovich stochastic integral of the composite function F (¿(t, 0), t) is defined as follows:
]f (,0), t) ° d£M := U.m.g F [¿¿tk ,0Mtk+1,o), h W +1,o) ,0],
a m k=0
where a = 10 < t1 < ... < tm = b, Sm = max(tk+1 - tk ).
k
In the right-hand side it may be also written F |f^(tk >°) + %itk+1,o),tk + t+1 j, however it
does not change the definition 44 [10, p. 363].
It can be shown that limit "l.i.m." in Definition 44 exists [4, p. 168].
Definition 45. (The definition of the Stratonovich integral of the composite function in general case.) Let ¿((,0) be a Wiener process and x(t,o) be a real-valued stochastic process. Let
F{x,t) be a real-valued nonrandom function: F\ R Y. [a, &] —i R, that is continuous in t e \a,b\ and continuously differentiable in s <E R . Then the Stratonovich stochastic integral of the composite function F(((,0), t) is defined by
fF (x(( ,0), t) ° d^) := llm.g F f x(tk ,0Mtk+1,0, h jt ^,0) ,0)],
provided that this limit (in Z2{Urii)) exists.
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'x{tk,©) + x{tk+l,o) tk + tk
As above, in the right-hand side it may be also written FI
4+1
2 2 J
without loss of generality of meaning of the Definition 45.
Similarly to one-dimensional case one can define the Stratonovich integral for multidimensional case.
Definition 46. (The definition of the Stratonovich integral in the multi-dimensional case.) Let <^(t) = ^(t,a( be an m -dimensional Wiener process and let F(x, t) be a (n x m matrix)-valued
nonrandom function such that F-. Rfl >: [otfc] —* £fi:>:>\ that is continuous on t, and that has first-order continuous partial derivatives SF/dxj with respect to the n components xj of the vector x. Let x{t,co) be a stochastic process with values in Then the Stratonovich integral of F(x(t, (), t) is defined in a similar manner as in the Definition 45 by the formula
b m-1 , s
J F (x(t,(), t) o d^(t( := Li.in.X F (x*, tk )[ J-£(tj ] ,
a sn -o k=0
where x * = (x(tk, () + x(tk+1, ())/ 2, provided that this limit exists for all sequences of partitions of
the interval [a, b], with 5n = max(tk+1 - tk) i 0.
k
In [4, p. 169] sufficient conditions for existence of the limits in definitions 45 and 46 are presented.
2.15.6. The connection between Ito and Stratonovich integrals
We give here a connection between the two stochastic integrals understood in the sense of Ito and in the sense of Stratonovich.
Let F (x, t) be a real-valued nonrandom function defined in Definition 45. Let ^(t) = <f(t,() be an one-dimensional Wiener process and x(t,() be a real-valued stochastic process.
Remember that Ito's integral of composite function F (x(t, (), t) can be computed, as it is usually done, this way:
b m-1
J f (x(tt)d^(t,()=n.in.;£ f (x(tk),() -[(tk+1() ()].
^k=0
The following proposition establishes the relationship between Ito's and Stratonovich's stochastic integrals in scalar case (n = m = l) .
Proposition 21. (The conversion formula: scalar case [4, p. 170], [10].) Let the functions F, x(t, () and g(t, (( be defined as in the Definition 45. Then the relation
(S)J F (x(t, (), t) o d^) = (I )Jf (x(t, ((, t)d^( + 2 J MXX^M f (x(t, (), t)dt (2.11)
a a a
holds provided that the stochastic integrals exist.
As seen from this, in particular, that the stochastic integrals of Ito and Stratonovich coincide when F (x, t) = F (() is independent of x.
In the special multi-dimensional case (n > 1, m = 1) the relationship between Ito and Stratonovich integrals expressed by the formula, which formally coincides with (2.11), where the functions F, x(t,() and ^((,() are defined as in the Definition 46: F is an n-vector-valued function: ß№ X [et,&] £№, i^/ojefi" is an n-vector stochastic process, %{t,(o) is one-dimensional Wiener process, and dF/ dx = (öF/dx.)^.=1 is Jacobi matrix of the vector function F .
In the general multi-dimensional case (n > 1, m > l) the relationship between Ito's and Stra-tonovich's integrals is given by the following
Proposition 22. (The conversion formula: multi-dimensional case: n > 1, m > 1 [4, p. 170], [10].) Let the functions F, x(,o) and ¿;(t,a) be defined as in the Definition 46. Then the relation
(S)JF(x(t,41)о d£(() = (/){F(xM, fa® + 2±J F {X10Fr (xM,t)dt, (2.12)
a a r-1 a
holds, where F\ Д" X [я,,b] —> 4®)ей", = Fr is the r-th column
of the matrix F, dFr / dx - (cFir/ Эх.). is Jacobi matrix ofthe vector function Fr (r - 1,..., m) .
Now we present some aspects of the definitions of Ito's and Stratonovich's stochastic integrals when comparing these ones.
2.15.7. A comparison of Ito and Stratonovich integrals
It can be shown ([10], [2, p. 37], [4, p. 171]) that the Stratonovich integral (on the basis of which the Stratonovich differential is defined) satisfies all the formal rules of an ordinary integral (or differential (integration by parts, change of variable, chain rule for stochastic differentials)). This makes the Stratonovich integral natural to use it in the stochastic calculus; since it can be more convenient and easier than Ito integral (or differential), when calculating.
However, Stratonovich integrals are not martingales, and Ito integrals are. This give the Ito integral an important advantage. But the conversion formulas (2.11) and (2.12), that we have given above in 2.15.6, enable at all times to move from one type of integral to the other.
We conclude this section by summarizing the advantages of each definition of the stochastic integrals by Ito and Stratonovich:
Advantages of Ito's integral
1. Simple formulas:
b Г ь Л2 b
eJf((,©)£(()- 0, eijf((,©)£(() I - ejfM2dt.
a V a I a
t
2. if ]((, ®) - Jf (s, ®)d%(s) is a martingale.
0
Advantages of Stratonovich's integral
1. The formal rules of an ordinary integral (as integration by part, chain rule, etc.) hold.
2. In consideration physical problems it is often natural to regard the stochastic differential equations (which are defined by integral equations) as the equations in the sense of Stratonovich.
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