Stochastic Leontieff type equations and mean derivatives of stochastic processes Текст научной статьи по специальности «Математика»

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

MSC 60H30, 60H10

STOCHASTIC LEONTIEFF TYPE EQUATIONS AND MEAN DERIVATIVES OF STOCHASTIC PROCESSES

Yu.E. Gliklikh, Voronezh State University, Voronezh, Russian Federation, [email protected], E. Yu. Mashkov, Kursk State University, Kursk, Russian Federation, [email protected]

We understand the Leontieff type stochastic differential equations as a special sort of Ito stochastic differential equations, in which the left-hand side contains a degenerate constant linear operator and the right-hand side has a non-degenerate constant linear operator. In the right-hand side there is also a summand with a term depending only on time. Its physical meaning is the incoming signal into the device described by the operators mentioned above. In the papers by A.L. Shestakov and G.A. Sviridyuk the dynamical distortion of signals is described by such equations. Transition to stochastic differential equations arise where it is necessary to take into account the interference (noise). Note that the investigation of solutions of such equations requires the use of derivatives of the incoming signal and the noise of any order. In this paper for differentiation of noise we apply the machinery of the so-called Nelson's mean derivatives of stochastic processes. This allows us to avoid using the machinery of the theory of generalized functions. We present a brief introduction to the theory of mean derivatives, investigate the transformation of the equations to canonical form and find formulae for solutions in terms of Nelson's mean derivatives of Wiener process.

Keywords: mean derivative, current velocity, Wiener process, Leontieff type equation.

Introduction

In papers [1, 2] a new approach to investigation on dynamically distorted signals is suggested that is based on Leontieff type differential equations. Further development of this approach requires taking interference (noise) into account that yields the transition to Stochastic Differential Equations. Here the correspondent stochastic differential equation takes the form

L((t) = MM f £(s)ds + f(t) + BW(t),

Jo

where L is a degenerate matrix n x n, M and B are non-degenerate matrices n x n, £(t) is an n-dimensional stochastic process, f (t) is a smooth n-dimensional vector-function and W(t) is a Wiener process in R™. The physical meaning of these objects is as follows: f (t) is the signal incoming into the device described by the matrices L and M, while BW(t) (where W(t) is «derivative> of Wiener process, i.e., white noise) describes the noise (interference).

The equations of such sort are called the Leontieff' type stochastic differential equations.

The features of Leontieff type equations require dealing with the derivatives of f (t) and w(t)

B

unit matrix and the equation has been reduced to canonical form, the so called current velocities (symmetric mean derivatives) of Wiener process are involved for describing the derivatives of Wiener process. As a result some physically reasonable analytical formulae for the solutions are obtained.

The notion of mean derivatives was introduced by E. Nelson [4, 5, 6] for the needs of the so-called Nelson’s stochastic mechanics (a version of quantum mechanics). Later a lot of applications of mean derivatives to some other branches of science were found. The investigation of Leontieff type stochastic differential equations is a new field of application of mean derivatives. Note that by general ideology of the theory of Nelson’s mean derivatives the current velocities are natural analogues of physical velocity of deterministic processes.

In this paper by the use of current velocities we investigate the general situation and do not suppose the equation to be already reduced to canonical form. Some constructions connected to reducing the equations to canonical form are announced in [7J.

An alternative approach to investigation of Leontieff'type stochastic equations, also based on the use of current velocities, is suggested in [8].

Besides the Introduction the paper contains three Sections. The first one is devoted to basic preliminary fact from the theory of mean derivatives necessary for the purpose of this article. In Section 2 we investigate the transition of Leontieff' type stochastic differential equations to canonical form. In Section 3 we find formulae for the solutions of equations under consideration.

Throughout the paper we use Einstein’s summation convention with respect to shared upper and lower indices.

We refer the reader to [9, 10J for details on the machinery of mean derivatives.

The research is supported in part by RFBR Grants 10-01-00143 and 12-01-00183.

1. Preliminaries on the mean derivatives

Consider a stochastic process {(t) in R™, t € [0, Z], given on a certain probability space (tt, F, P) and such that {(t) is Li-random variable for all t.

Every stochastic process {(t) in R™, t € [0, Z], determines three families of a-subalgebras of CT-algebra F:

(i) the «past> Pt generated % pre-images of Borel sets in R™ by all mappings {(s) : tt ^ R™ for 0 < s < t;

(ii) the «future> FI generated by pre-images of Bo rel sets in R™ by all mappings {(s) : tt ^ R™ for t < s < Z;

(iii) the «present> («now>) N generated % pre-images of Borel sets in R™ by the mapping {(t)

0

For convenience we denote the conditional expectation of {(t) with respect to N by E|(•). Ordinary («unconditional) expectation is denoted by E.

Strictly speaking, almost surely (a.s.) the sample paths of {(t) are not differentiable for almost all t. Thus its «classical> derivatives exist only in the sense of generalized functions. To avoid

using the generalized functions, following Nelson (see, e.g., [4, 5, 6j) we give

Definition 1. (i) Forward mean derivative D{(t) of {(t) at time t is an Li-random variable of the form

D{(t)= lim Et ({(t + At) - {(t) (1)

sw At^+0 tK At J

where the limit is supposed to exists in Li(tt, F, P) and At ^ +0 means that At tends to 0 and At > 0.

(ii) Backward mean derivative D*{(t) of {(t) at t is an Li-random variable

D.Ut)= lim Et ({{t) - - At)) (2)

*sw At^+o tv At J w

where the conditions and the notation are the same as in (i).

Note that mainly D{(t) = D*{(t) but if, say, {(t) a.s. has smooth sample paths, these derivatives evidently coinside.

From the properties of conditional expectation (see [11] ) it follows that D{(t) and D*{(t) can be represented as compositions of {(t) and Borel measurable vector fields (regressions)

Y«(t,x) = Atim+o E(~ {(t) l«*> = x)

Y0(t,x) = Anm+o E({(t) — At — At) i{(t) = x) (3)

on R™. This means that D{(t) = Y0(t, {(t)) and D*{(t) = Y*°(t, {(t)).

The derivatives introduced in Definition 1, is a particular case of the objects defined as follows.

Let x(t) and y(t) be L^stochastic processes in R™, given on (tt, F, P). Introduce y-forward mean

derivative of x(t) by the formula

y x(t)

Dyxt = a>>J+o Ey(Xit) - A - ) V

where the limits must exist in Li(tt, F, P).

Recall that a process {(t) is called martingale (in our case - with respect its «past> Pt’), if for every time instants 0 < s < t < Z the relation E({(t) | pSt) = {(s) takes place.

Lemma 1. Let {(t) be a martingale with respect to its «past» P’. Then D{(t) = 0.

Proof. By the properties of conditional expectation E’(E(• | P’)) = E’(•). Then E’({(t + At) —

tm = Et (e&t + At)—m i Pt)) = Et m—«*» = 0 a

Definition 2. The derivative Ds = 2(D+D*) is called symmetric mean derivative. The derivative Da = i (D — D*) is called anti-symmetric mean derivative .

Consider the vector fields v’(t,x) = |(Y°(t,x) + Y*°(t,x)) and u’(t,x) = i(Y°(t,x) —

Y*°(t,x)).

Definition 3. v’(t) = v’(t, {(t)) = Ds{(t) is called current velocity of {(t); u’(t) = u’(t, {(t)) = Da{(t) is called osmotic velocity of {(t).

For stochastic processes the current velocity is a direct analogue of ordinary physical velocity of deterministic processes (see, e.g., [4, o, 6, 9, 10]). The osmotic velocity measures how fast the

«»

By w(t) we denote the Wiener process. Recall that w(t) is a Wiener process (in our case, with respect to its own «past» P™), if

t

2) w(t) is a square integrable martingale with respect to P™ such that w(0) = 0 and E((w(t) — w(s))2\P'W) = t — s for t > s.

Well-known Levi’s theorem says that in addition w(t) has stationary independent Gaussian increments and satisfies the equalities:

E (w(t) — w(s)) = 0, E ((w(t) — w(s))2) = t — s

for t > s. In the other words, the increment w(t) — w(s) for t > s is independent of Pf and has the same distribution as w(t — s). Note that the probabilistic density pw (t, x) of w(t) in R™ takes the form

1 x2

P™(t,x) = m n e 2t . (6)

(2nt) 2

Recall that the sample paths of w(t) are a.s. non-differentiable for almost all t and on every arbitrarily small time inervals they a.s. have infinite variation. Thus, the derivatives of w(t) in usual sense exists only as a generalized function.

Below we often deal with the processes of the form

{(t) = {o +f P(s)ds + w(t) (7)

o

where w(t) is a Wiener process. For such processes the above-mantioned «physical» properties

of current and osmotic velocities become clear from the following propositions.

Denote by p’(t, x) the density of process (7) with respect to Lebesgue measure A on [0, Z] x R™. This means that for every continuous inntegrable function f (t, x) on [0, Z] x R™ the following equality takes place:

[ f (t,x)p’(t,x)dA = I f (t, {(t))dPdt.

i[o,l]xR" ./nx[o,l]

Lemma 2. For porcess (7) in R™ the vector field u’(t,x) is represented in the form

u’(t,x) = ^gr&d Zog p’(t,x). (8)

Lemma 3. For process (7) in R™ the vector field v’(t,x) and the density pt(t,x) satisfy the equation of continuity

dp’ (t, x)

dt

The proofs of Lemmas 2 and 3 in the form convenient for us, can be found in [9, 10J.

For processes of more general type the above Lemmas can be generalized as follows.

Lemma 4. [12] Let {(t) satisfies the Ito equation {(t) = a(s, {(s))ds + A(s, x)dw(s). Then

t 1 7TJ(a%jP’(t,x)) d

ut ^ = 2 - ~dx <10>

where (aij) is the matrix of operator AA* .

Proof. Let f be an arbitrary smooth function on R™ with compact support. Note that f ({(t)) is N’-measurable. Hence

E(f ({(t))Et ([ A(t, {(t))dw(t))) = E(f ({(t)) f A(t, {(t))dw(t)).

v Jt-At J v Jt-At J

Since f ({(t — At)) and J*_At A(t, {(t))dw(t) are independent and E J*_At A(t, {(t))dw(t) = 0, we E(f ({(t))( [ A(t, {(t))dw(t)) = E((f ({(t)) — f ({(t — At)))( [ A(t, {(t))dw(t))) . t-At t-At

E(f (Ш) [ A(t,£ (t))dw(t)\ = e( j (df • a(s,£(s)))A(s,£(s))dsdw(s)

\ .1+ A + / V/У- Av-

By Ito formula f ({(t)) — f ({(t — At)) = ft-At(df • a(s,{(s)))ds + 2ft-Attr f"({(s))ds + tt At (df • A(s,{(s)))dw(s) (by • we denote the coupling of 1-forms and vectors). Thus

A(t,{ (t))dw(t)\ = e(!

t-At t-At

+1 I trf"({(s))(A(s,{(s)),A(s,{(s)))dsdw(s)+[ (df • A(s,{(s)))A(s,{(s))ds

2 Jt-At Jt— At

The first two integrals in the right-hand side equal zero. Calculations in coordinates show that (df • A)A = df • (AA*).

On the other hand,

^ E{f(((t))u*(t,((t))) dt = -2 fQ Eyf (((t)) д1™_0 Et (J^ У—^ dt =

I E (df • AA * )dt = - 1 f df • AA * • p* dt л Л = \ I f • d(AA * • p* )dt л Л =

Jo 2 JRnx [0,T] 2 JRnx [0,T]

f fd—T^/fdt л Л = 1 Г Eh^—^yt =1- fT Eff&jlty

Jr-x[0,t]‘ p ’ 2 Jo V p* > 2 Jo У p* dx')

Since this is valid for an arbitrary f as above, this means that u’ = \ d<yAApii ^ 9x3 p P ^ dx?-

□

An alternative proof of Lemma 4 can be found in [12J.

Let A as above be constant and non-degenerate. Then the matrix (aij) = (aij)-i is well-posed and it can be considered as the matrix of new innner product in R™. In this case we obtain

Corollary 1.

u’ (t,x) = 1 Grad log p’ (t,x) = Grad log pt (t, x) (11)

where Grad denotes gradient with respect the inner product with matrix (aij).

Indeed, if A is constant, (aij) is constant as well, and formula (10) takes the form

1 dx(aijP’(t,x)) d 1 ijdxj(p’(t,x)) d

u*(t,x) = - dxJK J У ~d~ = Й ®ij —

2 p’ (t,x) dxi 2 p’ (t,x) dxi

1 Grad log p’ (t,x) = Grad log pt (t, x).

We are using formulae (8) and (11) below.

Now consider autonomous smooth field of non-degenerate linear operators A(x) : R™ ^ R™. x € R™ (i.e., (1,1)-tensor field on R™). Let {(t) be a diffusion process in which the integrand under

A({(t)) A(x)A (x)

symmetric positive definite matrices a(x) = (aij(x)) ((2, 0)-tensor field on R™). Since all these matrices are 11011-degenerate and smooth, there exist the smooth field of converse symmetric and positive definite matrices (aj). Hence this field can be used as a new Riemannian a(•, •) = aijdxi® dxj on R™. The volume form of this metric has the form Aa = ^det(aj,j)dxi A dx2 /••• A dx™.

2

Ю.Е. Гликлих, Е.Ю. Машков

Denote by p’(t, x) the probability density of random element {(t) with respect to the volume form dt A Aa = ^det(aj,j)dt A dxi A dx2 /• • •/ dx™ on [0,T] x R™, i.e., for every continuous bounded function f : [0,T] x R™ ^ R the relation

T

T

(12)

[ E(f (t,{(t)))dt = ! I f f (t,{(t))dP J dt = ! f (t,x)p’(t,x)dt A Aa.

o o \n J [o,T ] xRn

holds.

Lemma 5. For v’(t,x) and p’(t,x) the equation of continuity takes the form

dp dt ^ = —Div(v’(t,x)p’(t,x)),

where Div denotes the divergence with respect to Riemannian metric a(, •).

Proof. Here by Ae we denote the form dxi A • • •A dx™. So, A = ^det(aj,j )Ae■

Recall that Div(p’v’) = *-id((p’v’) J A) where J is the interrior product of vector (p’v’) and n-form A. But (p’ v’) J A = ^det(aj,j )Y^ 'i=i(pt v’ )idxi A^ • •A dxi-i A dxi+i A^ • •A dx™ and

“ mvp v’) = ^ + mgt.

Specify a smooth function f (t, x) with compact support. By df we denote the differential with respect to spatial coordinates: df = dXdxi. Note that by coordinate calculations we get

f (df • (p’v’(r, {(t)))) dr A A =

J [s,t] x Rn

[s,t] x Rn

'[s,t]xRn

f (r, x)

df • (p*v*(r,((r)))^det(aij)) dr Л Ae =

det(aij)

d (p* v* )i dxi

+ (p* v* )i

дл/ det(aij)

dx1

dr Л Ae =

[s,t]xRn

f (r, x)

d (p* v* )i (p* v* )i det(aij)

+

dxi y/det(aij) dx

\Jdet(ai,j) j dr Л Ae =

[s,t]xRn

f (t,x)

d (p* v* )i (p* v* )i d^Jdet(aij)

+

dr Л A =

dxi y/det(aij) dx{

— I (f (r,x)Div(ptv’)} dr A A.

J[s,t]xRn

By It.o formula

E(f (t,{(t)) — f (s,{(s))) = E(Js dTdT + J df • y°(t,{(t))dr + \js tr f(A,A)dr)

and by backward It.o formula

E(f(t,{(t)) — f (s,{(s))) = E(Js dT^ + J df • Y*(r,{(r))dT — \js trf”(A,A)dr).

Hence,

E{f(t,((t)) - f(s,((s))) = E(^ dTdr + J df'v*(r’€(r))dr)-

But

e(/ dTdr +f df • v* (r,£(r))dr^ =! n(lfp* + ^ ' (p*v* (r,£(r)))]) dr Л Л =

Js J s J [s,t] x Rn

i d ((f (r,x)p*) dr Л л - f f (r,x)dr Л л

■J[s,t]xRn dr V J J[s,t]xRn' dr J

(f (r, x)Div(p*v*)} dr Л Л =

E(f (t,£(t)) - f(s,Z(s))) -f (f(r,x)dr Л Л -f (f(r,x)Div(p*v*))dr Л Л.

V ' J[s,t]xRn ' dr J J[s,t]xRn J

' [s,t] x Rn

) dp’'

’[s,t]xRn dr J[s,t]xRn

ThuS J[s,t]xR^f (T x) dT) dr A A + J[s t]xRn(f (r, x)Div(p’v’) j dr A A = 0. Since this is valid for an arbitrary f (t, x) as above, this means that = —Div(p’v’).

□

An alternative proof of can be found in [6].

Since w(t) is a martingale, Dw(t) = 0, t € [0,Z) (see above).

Lemma 6. [See, e.g., [9, 10JJ For t € (0,Z] the equality D*w(t) = holds.

Proof. From the definition of osmotic velocity uw(t,w(t)) it follows that D*w(t) =

—2uw(t, w(t)). Since pw(t,X) is given by formula (6), from formula (8) it follows that uw(t,x) =

— 2 • j. Thus, D*w(t) = ^t—.

□

Corollary 2. DSw(t) = w^.

w(t)

system of notation from [9, 10], we look for the k derivative as Dw, Df or Df (see (4) and (5)) of the (k — 1)-th derivatives. This notation emphasizes that we always use the a-algebra «present» of w(t).

Lemma 7. [See, e.g., [9, 10JJ (i) DwW= — wjt fort € (0,Z).

(ii) DfW^ =0 far t € (0,Z],

(in) DwWt- = — W2- for t € (0,Z].

Proof. Indeed,

DW w(t)=( d + Dw(t) = - w(t)

t dt t t t2

and

'-f- = (d1 wt)+tD*w(t) = — '-f + af = o.

Assertion (iii) follow from the last two formulae.

□

Lemma 8. (i) Dw() = -kWrr; (ii)DW (wtt)) = -(k - 1) W+ (m) DWW(wf) = -2-1WB-

Proof.

(i) Dw(wjk)) = dttkw(t) + ttDw(t) = -kW+1 + 0 = -kW+

(ii) DW() = dtitw(t') + tkD*w(t) = -kp+T + tkW^tt) = -(k - 1)tk+r',

(iii) From the last two formulae we obtain that DW (wikr) = - W+r-

Lemma 9. 1 For integer k > 2

□

h-l

n(2i - ^ (t)

DS w(t) = (-1)h-1 • --------tr •

This formula is proved by induction starting from the assertions of Corollary 2, Lemma 7

(iii) and Lemma 8 (iii).

2. Leontieff type stochastic equations and their canonical form

As it is mentioned in the Introduction, the stochastic differential equation of Leontieff type is a stochastic differential equation in R™ of the form L^(t) = MM J0 £(s)ds + J0 f (s))ds + Buj(t), where £(t) is a random and f (t) is a deterministic n-dimensional vectors, L, M and B are n x n matrices, where L is degenerate (has zero determin ant) while M and B are non-degenerate and w(t) is a Wiener process. Their physical meaning is the following: f (t) is an incoming signal into the device described by operators L and M, Bw where w(t) is white noise, is interference, and £(t) is outgoing signal. The vector-function f (t) is supposed to be smooth.

If the sheaf M + XL is regular, one can apply the Kronecker-Weierstrass transformation and reduce the matrices L and M to the quasi-diagonal form (see [13]). This transformation is described by a pair of linear non-degenerate operators (matrices) that we denote by A = (aj) and Ar. The conjugate to A operator is denoted by A*. In the quasi-diagonal form, under appropriate numeration of basis vectors, in the matrix L = ALAr first along diagonal there are Jordan boxes with zeros on diagonal, and the last matrix along diagonal is the unit one. In M = AMI Ar in the lines corresponding to Jordan boxes, there is the unit matrix and the last block along diagonal is a certain non-degenerate matrix. In the next section, for the sake of convenience, we present LM

Denote by (•, •) the standard inner product (Euclidean metric) in R™. Recall that the Wiener process w(t) is Gaussian with mean value 0 and covariation matrix tl, where I is the unit matrix,

i.e, with density (6) with respect to the volume form of Euclidean metric (•, •).

Introduce the matrix C = AB. Since the matrices A and B are non-degenerate, C is nondegenerate as well and such is also CC* = ABB*A*. Hence the inverse matrix (CC*)-1 = C*-lC-l is well-posed. Thus (see [14]), Cw(t) is also Gaussian with mean value 0 and covariation tCC

pCW(t, x) = ((2nt)-n/2A-l/2)exp(-((CC^t lx,x')) (13)

lNote the misprint in this formula in [3j where mistakenly 2k 1 instead of 2k is set in the denominator.

with respect to the same volume form, where A is determinant of CC*.

Introduce the new inner product (Eucliden metric) {•, •} in R™ by formula {X, Y} = ((CC*)-lX, Y).

Theorem 1. (i) For every vectors X and Y in Rn the identity {CX,CY} = (X,Y) holds, (ii)

The process w(t) = Cw(t) is a Wiener process in Rn with Euclidean metric {•, •}.

Proof. Recall that (CC*)-1 = C*-lC-l. Then

{CX, CY} = (C*-lC-lCX,CY) = (C-lCX, C-lCY) = (X, Y)•

The volume form of metric {•, •} differs from that of (•, •) % the coefficient A-l/2, i.e., the density of Cw(t) with respect to the volume form of {•, •} takes the form

((2nt)-n/2)exp(-((CC2-x,x)) = ((2nt)-n/2)exp()• (14)

Obviously the other properties of Wiener process are satisfied for Cw(t) in R™ with metric {•, •}.

□

Let et, •••,en be a natural orthonormal basis in R™ with (•, •).

Corollary 3. Cet, • • •, Ce™ is an orthonormal basis in R™ with {•, •}.

Corollary 4. Introduce n(t) = ARx^(t). In R™ with {•, •} the Leontieff type stochastic equation

takes the form Lrj(t) = Mn(s)ds + Af (s)ds + w(t).

Taking into account formula (11), we see that the expression of current velocity for w(t) contains Grad(C-lx,C-lx) where Grad is the gradient with respect to inner product {•, •}.

Lemma 10. d{x,x} = d(C-lx,C-lx) = 2C*-lC-lx, where d is exterior differential.

Lemma 11. Grad{x,x} = Grad(C-lx,C-lx) = 2x.

The proof follows from the formula of lifting the indices

Grad(C-lx,C-lx) = CC *d(C-lx,C-lx)

and from Lemma 10.

Hence, in R™ with {•, •} formulae for current velocity and higher symmetric derivatives of Wiener process w(t) have usual form as in Lemmas 6-9.

3. Solutions of Leontieff type stochastic equations

So (see Corollary 4), if the sheaf M + XL is regular, after the Kronecker-Weierstrass trans-

R™ {• , •}

Ln(t) = [ Mn(r)dr + ( Af(t)dr + w(t), (15)

Jo Jo

where n(t) = Ar £(t),

L = ALAr =

/ 0 1 0 0 0 0 0 0 1 0 0 0

0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

V 0 0 0 0 0 0

0 0 0 ... 0 \

0 0 0 ... 0

0 0 0 0 0 0 1 0 0 0 1 0

0 0 0 0 1 0 0 0 1

0 0 0

.1

(16)

and

M = am ar =

/ 1 0 0 0 0 0 0

0 1 0 0 0 0 0

0 0 1 0 0 0 0

0 0 0 1 0 0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0

0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0 0 0 0

^0000000 0

0 0 .. 0

0 0 .. 0

0 0 .. 0

0 0 .. 0

0 0 .. 0

0 0 .. 0

0 0 .. 0

1 1 3 3 ,, ^ ^ 1 1 n— q an—q+1 . an—q+1 an—q+1 . n— q an q n—q+1 an

n n—q an.. an—q+1 an an

(IT)

/

Everywhere below we deal with equation (15) in Rn with {■, ■}.

It is clear (cf. (7)), that here for simplicity the initial value in (15) is supposed to be £(0) = 0. Note that the solutions that we construct below, cannot satisfy this condition since they are t=0

condition but become solutions only after a certain, a priori given and arbitrarily small positive time instant t0 > 0 (see below).

Remark 1. Rewrite (15) in the form Lri(t) — M n(s)ds — A f (s)ds = w(t). We see that «present> to the pro cess Ln(t) — M n(s)ds — A f* f (s)ds coincides with the «present> for w(t). Thus we use the latter a-algebra for calculation of mean derivatives< i.e., we apply to (15) the derivatives Dw, Df or D'W- Note that the solutions found below, are measurable with respect to the «present> of w(t) for every t.

Taking into account the structure of matrices (16) and (17), it is clear that (15) is decomposed into several independent systems of equations. The one «at the bottom> corresponds to the unit LM by K, and by ((t) the vector of dimension q + 1 constructed from the last q + 1 coordinates of

a

a

a

n(t). Then ((t) is described by the equation

Z(t) = K f z(s)ds Jo

+

Af (t )dr + w(t)

(18)

in R9+1. Here w(t) is a q+1-dimensional Wiener process constructed from the last q+1 coordinates of w(t) in R™ and Af (t) is a q + 1-dimensional vector constructed from the last q + 1 coordinates of Af (t). For (18) there is a well-known analytical formula of solutions: Z(t) = fg eK(t-T)Af (t)dr + to eK(t-T\Iw(t).

The other systems correspond to the Jordan boxes in L and unit matrices, constructed from the lines and columns in M. As an example, we consider (p + 1) x (p + 1) matrix (Jordan box) N in the left upper corner of (16)

N =

and the corresponding unit matrix from (17). The other systems are quite analogous.

Denote by (Af )(p+1) the (p+1)-dimensional vector constructed from the first p+1 coordinates of Af, by n(p+i)(t) ~ the (p + 1)-dimensional vector with coordinates (n1(t),...,np+1(t)) constructed from the first (p+1) coordinates of n(t) and by W(p+1) (t) - the vector with coordinates (w1(t),...,wp+1(t)) constructed from the first p + 1 coordinates of w(t). It is clear that the coordinates of Af have the form (Af) = E™=1 a)fj- Then n(p+1)(t) is described by the equation

Nn(p+1)(t) = f (n(p+1)(s) + (Af)(P+1)(s)) ds + w(p+1)(t).

Jo

Witten via coordinates, this system takes the form

/

0 1 0 0 . .0

0 0 1 0 . . 0

0 0 0 0 . . 1

0 0 0 0 . .0

/ 0 0

0 V 0

\ ( n1(t) \

n2(t)

t)

/ V nj+1(t)

1/

Jo(n2(s) + E ?=1 a2jf j )ds

fo(vp(s) + H n=iapf j )ds

V/o(nP+1 ^П=1 ajj+1 fj)ds J

( w1(t) \

w to t

+

wj(t)

\ wj+1(t) )

(19)

From the last equation of (19) we obtain

p t p t ™

Jo nP+1(s)ds = -J §2 ap+1fj )ds

w

p+1

(t).

(20)

j=1

Since the current velocity (symmetric mean derivative) corresponds to the physical velocity, from this equation we find np+1(t) by applying the derivative DSW to both sides of the equality (see Remark 1). Obviously application of the mean derivatives Dw and Df (and so D'W) to the integrals both in the left and the right-hand sides yields the same results: np+1(t) and J2™j=1 aP+1fj, respectively. Thus we obtain that

nj+1(t) = - V aj+1fj - DWwj+1(t) = - V aj+1 fj

j

j=1

j

j=1

fj -

w

j+1

(t)

2t

(21)

t

u

n

n

From the last but one equation we obtain

r t ™

np+1(t) = J (np(s) + Y, ap fj)ds + wp(t).

(22)

j=1

Applying the arguments analogous to the above ones, we derive

Vp(t) = Df np+1(t) - £ apfj - Df wp(t).

3=1

Substituting the expression for np+1(t) from (21) into the latter equality and using Lemma 7, we obtain

np(t) = -£

ap+1df j ^ ap f j + wp+1 (t) wp(t)

a3 d -2s a3f + j=1 j=1

4t2

2t

(23)

By complete analogy, for 1 < i < p we obtain the recurrent formula

n*(t) = Df ni+1(t) ^ aj fj - Df wi(t).

(24)

j=1

Taking into account Lemma 9 we derive from (24) the explicit expression for every ni(t) 1 < i < p in the form:

p ™ k-i+1 f j\ ™

ni № = ^ EaJ+1 ^ aj fj

k=i Vj=1 I j=1

k-i

+

p+1

p

k=i+1

(-1)

k-i+1 j=1

n (2j - 1) k, .

wk(t)

2k-i+1 tk-i+1

\

wi(t)

(25)

/

Let us turn back to the question on zero initial values for solutions of system (19). From the definition of symmetric mean derivatives it clearly follows that they are well-posed only on open time-intervals since their construction involves both forward and backward time increments. Taking into account formula (25), one can easily see that the solutions constructed above, have the form of sums where some summands contain multipliers of , k > 1, type. So, the solutions tend to zero as t ^ 0, i.e., at t = 0 the values do not exist.

A version of solving this problem is as follows. Specify an arbitrary small time instant to € (0, l) and consider the function t0(t) by the formula

to(t) =

t0 if 0 < t < 10; t t < t.

In formulae (21), (23) and (24) replace the elements by Processes wiU

take zero value at t = 0 but only for t > t0 they will be the solutions of (15). Note that for two different time instants t(1) and t(2), for t > max(t(1),t(2)) the values of corresponding solutions coincide a.s.

n

References

1. Shestakov A.L., Sviridyuk G.A. A New Approach to Measurement of Dynamically Perturbed Signals. Bulletin of the South Ural State University. Series «Mathematical Modelling, Programming & Computer Software», 2010, no. 16 (192), issue 5, pp. 116-120. (in Russian)

2. Shestakov A.L., Sviridyuk G.A. Optimal Measurement of Dynamically Distorted Signals. Bulletin of the South Ural State University. Series «Mathematical Modelling, Programming & Computer Software», 2011, no. 17 (234), issue 8, pp. 70-75.

3. Gliklikh Yu.E. Investigation of Leontieff Type Equations with White Noise by the Method of Mean Derivatives of Stochastic Processes. Bulletin of the South Ural State University. Series «Mathematical Modelling, Programming & Computer Software», 2012, no. 27 (286), issue 13, pp. 24-34. (in Russian)

4. Nelson E. Derivation of the Schrodinger Equation from Newtonian Mechanics. Phys. Reviews,

1966, vol. 150, no. 4, pp. 1079-1085.

5. Nelson E. Dynamical Theory of Brownian Motion. Princeton, Princeton University Press,

1967. 142 p.

6. Nelson E. Quantum Fluctuations. Princeton, Princeton University Press, 1985. 147 p.

7. Gliklikh Yu.E., Mashkov E.Yu. On Reduction of Leontieff' Type Stochastic Equations to Canonical Form. Izmereniya: sostoyanie, perspektivy razvitiya: tez. dokl. mezhdunar. nauch.-prakt. konf., Chelyabinsk 25-27 sentyabrya 2012 g. [Measurements: the State of the Problem and the Prospects of Developmets. Abstracts of Communications of the International Scientific-Practical Conference. Chelyabinsk 25-27 September 2012. Vol. lj. Chelyabinsk, Publ. Center of the South Ural State University, 2012, pp. 73-75. (in Russian)

8. Shestakov A.L., Sviridyuk G.A. On the Measurement of the «White Noise». Bulletin of the South Ural State University. Series «Mathematical Modelling, Programming & Computer

»

9. Gliklikh Yu.E. Global and Stochastic Analysis in the Problems of Mathematical Physics. Moscow, KorriKniga, 2005. 416 p. (in Russian)

10. Gliklikh Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics. London, Springer-Verlag, 2011. 460 p.

11. Parthasarathy, K.R. Introduction to Probability and Measure. N.Y., Springer-Verlag, 1978. 343 p.

12. Cresson ,J., Darses S. Stochastic Embedding of Dynamical Systems. Journal of Mathematical Physics, 2007, vol. 48, pp. 072703-1 - 072303-54. [DOI: 10.1063/1.2736519J.

13. Gantmakher F.R. Theory of Matrices. Moscow, Fizmatlit, 1967. 575 p. (in Russian)

14. Gikhman I.I., Skorokhod A.V. Introduction to the Theory of Stochastic Processes. Moscow, Nauka, 1977. 567 p. (in Russian)

УДК 517.9 + 519.216.2

СТОХАСТИЧЕСКИЕ УРАВНЕНИЯ ЛЕОНТЬЕВСКОГО ТИПА И ПРОИЗВОДНЫЕ В СРЕДНЕМ СЛУЧАЙНЫХ ПРОЦЕССОВ

Ю.Е. Гликлих, Е.Ю. Машков

Ю.Е. Гликлих, Е.Ю. Машков

Стохастические дифференциальные уравнения леонтьевского типа мы понимаем как специальный класс стохастических дифференциальных уравнений в форме Ито, у которых в левой части имеется вырожденный постоянный линейный оператор, а в правой части - невырожденный постоянный линейный оператор. Также в правой части имеется слагаемое, зависящее только от времени. Его физический смысл - входящий сигнал в устройство, описываемое указанными выше операторами. В статьях А.Л. Шестакова и Г.А. Свиридюка подобные уравнения использованы для описания динамически искаженных сигналов. Переход к стохастическим дифференциальным уравнениям возникает при необходимости учета помех. Отметим, что для исследования решений таких уравнений необходимо использовать производные произвольного порядка от сигнала и от помех. В этой статье для дифференцирования помех мы применяем аппарат так называемых производных в среднем по Нельсону от случайных процессов. Это позволяет при исследовании не использовать аппарат теории обобщенных функций. Мы даем краткое введение в теорию производных в среднем, исследуем преобразование уравнений к каноническому виду и находим формулы для решений в терминах производных в среднем винеровского процесса.

Ключевые слова: производная в среднем, текущая скорость, винеровский процесс, уравнение леонтьевского типа.

Литература

1. Шестаков, А.Л. Новый подход к измерению динамически искаженных сигналов / А.Л. Шестаков, Г.А. Свиридток // Вестник ЮУрГУ. Серия: Математическое моделирование и программирование. - 2010. - №16 (192), вып. 5. - С. 116-20.

2. Shestakov, A.L Optimal Measurement of Dynamically Distorted Signals / A.L. Shestakov, G.A. Sviridyuk // Вестник ЮУрГУ Серия: Математическое моделирование pi программирование. - 2011. - №17 (234), вып. 8. - С. 70-75.

3. Гликлих, Ю.Е. Изучение уравнений леонтьевского типа с белым шумом методами производных в среднем случайных процессов / Ю.Е. Гликлих / / Вестник ЮУрГУ Серия: Математическое моделирование pi программртроватше. - 2012. - №27(286), вып. 13. -С. 24-34.

4. Nelson, Е. Derivation of the Schrodinger Equation from Newtonian Mechanics / E. Nelson // Pliys. Reviews. - 1966. - V. 150, №4. - P. 1079-1085

5. Nelson, E. Dynamical Theory of Brownian Motion / E. Nelson. - Princeton: Princeton University Press, 1967. - 142 p.

6. Nelson, E. Quantum Fluctuations / E. Nelson.- Princeton: Princeton University Press, 1985.

- 147 p.

7. Глр1клр1х, Ю.Е. О прршедепрш стохастртческртх уравпетшй леонтьевского тртпа к канотш-ческому вртду / Ю.Е. Глртклртх, Е.Ю. Машков // Измеретшя: состоятше, перспектртвы разврттрш: тез. докл. междупар. пауч.-практ. копф., Челябршск 25-27 сентября 2012 г. -Челябршск: Издательскрш центр ЮУрГУ, 2012. - Т. 1.- С. 73-75.

8. Shestakov, A.L. On the Measurement of the «White Noise> / A.L. Shestakov, G.A. Sviridyuk // BecTiiPiK ЮУрГУ Ceppra: Математртческое моделртроватше pi программртроватше. -2012. - №27 (286), вып. 13. - P. 99-108.

9. Глртклртх, Ю.Е. Глобальный pi стохастртческрш апалртз в задачах математртческой Фртзрткрт / Ю.Е. Гликлих. - М.: УРСС, 2005. - 416 с.

10. Gliklikh, Yu.E. Global and Stochastic Analysis with Applications to Mathematical Physics / Yu.E. Gliklikh. - London: Springer-Verlag, 2011. - 460 p.

11. Parthasarathy, K.R. Introduction to Probability and Measure. N.Y., Springer-Verlag, 1978. 343 p.

12. Cresson, ,J. Stochastic Embedding of Dynamical Systems / ,J. Cresson, S. Darses j j ,J. of Mathematical Physics. - 2007. - V. 48. - P. 072703-1 - 072303-54. [DOI: 10.1063/1.2736519J.

13. Гатттмахер, Ф.Р. Теоррія матррщ / Ф.Р. Гатітмахер.- М.: Фрізматлріт, 1967. - 575 с.

14. Гихматт, И.И. Введение в теорию случайных процессов / И.И Гихматт, А.В. Скороход. -М.: Наука, 1977. - 567 с.

Юрий Евгеньевич Гликлих, доктор физико-математических паук, профессор, кафедра алгебры и топологических методов анализа, Воронежский государственный университет (г. Воронеж, Российская Федерация), [email protected].

Евгений Юрьевич Машков, аспирант, кафедра математического анализа рі прріклад-тіорі математрікрі, Курскрій государственный утіріверсрітет (г. Курск, Россрійская Федератція), [email protected].

Поступила в редакцию 20 февраля 2018 г.