SCIENTIFIC ABSTRACT SKOROKHOD, A.V. - SKOROKHOD, O.R.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001651110018-7
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: July 13, 2001
• Sequence Number: 18
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

22835
S/199/61/002/001/006/008
B112/B218
AUTHOR: Skorokhod, A. V.
TITLE: Existence and uniqueness of solutions of stochastic diffusion
equations
PERIODICAL: Sibirskiy- matematicheskiy zhurnal, v. 2, no, 1, 1961, 129-137
TEXT: The author proves an existence and uniqueness theorem for the dif-
ferential equation: d~(t) = a(t,%(t))dt + o (t,~(t))dw(t) (1), where
C2(t,x) denotes the diffusion coefficient, a(t,x) is the transport coeffi-
cient, and w(t) is a function that described the one-dimensional Brownian
movement. For such equations, existence and uniqueness theorems have been
derived by K. Ito, I. I. Gikhman, J. Doob, and Maruyma on the following
assumptions: There exists a number K and thus
ja(t 'x)J2 + la(t,x)l2---K (I +JX2 ); for every C> 0, there
exists a number L C and thus 10(t,x) - C(t.,Y) + ja(t,x) - a(t9y)j< LCIX - yl
for jx! 0, and
a
if there exist numbers a ~__-1/2 and L :>0 for every C~~-O so that for
C C a
jxj-_C, jyj-_-C one has Id (t,x) - a (t,y)j_-z:~LC(x - y) C, then the solution
S(t) is unique inasmuch as it is in agreement with any other solution with
the probability 1. For the proof of existence, the author uses a theorem
by A. N. Kolmogorov. There are 12 references: 7 Soviet-bloc and 5 non-
Soviet-bloc.
SUBMITTED: March 11, 1960
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Sko rokhod, A i V ~
25766
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C111/C222
TITLE- Stochastic eauations for diffusion processes in a bounded
region
PERIODICALt Teoriya veroyatnostey i yeye primeneniye, v. 6, no. 3, 1961,
297 - 298
TEXTs At the conference on probability theory and mathematical
statistics in Yerevan, 1958, it was reported about the results of the
paper.
From the results of Feller (Ref. 7: W. Feller, Diffusion processes in one
dimension, Trans. OMS, 77 (1954), 101) ; Ref. 8 : W. Feller, The general
diffusion operator and positively preserving semigroups in one di-
mension, Ann. Math., 60(1954), 417-435) it follows that in the case of
continuous trajectories of a diffusi6n process at the boundary there is
either absorption or instantaneous reflection or retarded reflection or
partial reflection. The author tries to construct the trajectories of
processes with boundaries being analogous to those considered by Feller
but which must not necessarily be homogeneous.
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Stochastic equations for diffusion ... C111YC222
The author considers only one-dimensional diffusion processes on the
semiline x>,,O . The single boundary point reads x = 0 . It is assumed
that the diffusion coefficients a(t,x) and 0'(t,x) are defined and
continuous for x >0, t Glt 0, T I and satisfy
I a(t,x) - a( t,y) !5~ KI x - Y I ,1 16 (t,ic) - (5r(t,y)l ~~ Kjx - yj (3)
The case of an instantaneous reflection is considered in detail. Let 1i M
be a process. The function '5 (t) is called a "reflection function" of
i~ (t) it '5(t) with a probability 1 is a continuous monotone function
the growth points of which can only be the zeros of r5(t). The process
with an instantaneous reflection is sought asta solution of
~w = -~(to) + j a(s,'~(s))ds + 6(s,q(s))dw(s) +r,(t) (5)
0 0
where 15(t) is the reflection function of (t), (to) = 0 and 5(t);?,,O
-or all t. Since the set of t - values for which 0 has the
N
Lebesgue measure 0 then it is unessential how a(t,x) and ~,(t,x) are
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Sjtcchat~t~-: equations for diffusion C111/C222
defined in x = 0 ; w(t) is the Brownian motions
It is proved that (5) has a unique solution for a single possible
function ~;(t), where has the property
lim e(t, + (6)
At~o
for almost all points ; *0 (x) 0 for x>0 and 0(x) 1 for x = 0.
The integral form for (6) reads i
s + 0) E; fds
t) =
-F-aultt 0
0
so that the equation of a process with an instantaneous reflection can
also be written in the form
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Stochastic equations for diffusion C111/C222
t t
0 ) + a(s,~ (s))ds + (~(s,~(s))dw(s) +
0 0
t
(5'(s, + 0) (s)) ~ ds (7)
8 t~ 0
0
Since it is proved that (5) may have a solution only for a single -5'(t),
for the proof of existence it must be found at least one 15 (t) for which
(5) has a solution. For this aim the author considers differential
equations. Let t 0< t1z- t2 < ... 4-1 ti, = T ; tk+I -tk =At k 7W(t k+1 )-W(t k)
= Wk . Let h,, h2, h,3 hn be a sequence of positive random magnitudes;
let qo be a random magnitude not depending on w(t). Let the sequence
Olk 9 k n be defined by
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Stochatitic; equations for diffusion ... C1 11Y0222
4 t +
'k k- k- I lk- 1 )'k-. 1tk- 1 tk- 1
(1-2)
+ t )h
o k-! k-l''k-O'k-1 (tk-1'-k-1)Atk-1 k
where (x) = i for x:!~O and \P (x) = 0 for x >0. Let the random
process v?(t) be defined by : I(t) = ?k for tG Itk'tk+1) I
k = n - 1~ It is shown that for max dtk-0 converges
to a solution of (5)-
A continuation of the paper is announced. The author mentions S.N.
Bernshteyn, I.I: Gikhman and A.D. Ventsell. There are 6 Soviet-bloc and
7 non-Soviet-bloc references. The references to the four English-
language publications read as follows : K. Ito, On stochastic differentials
equations, Mem,Am,11ath.Soc-4(1951),1-51 ; J.Doob, Martingales and one-
dimensional diffusion,Tr=AITS,78(1955),168-208 ; W.Feller, Diffusion pro-
cesses in one dimension, Trans. OMS, 77(1954), 101) ; W.Feller, The general
diffusion operator and positively preserving semigroups in one dimension,
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Ann. Math,, 60 (1954), 417 - 435-
SUBMITTED~ September 15. 1959
25766
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... Clll/C222
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SKOROKHOD, A.V.
Integrodifferential equations associated with solutions of stochastic
equations. Dop.M1 URSR no-7:854-858 161. (KRA 14:8)
1. Kiyevskiy gosudarstvennyy universitet. Predstavleno, akademikom
AN USSR B.V.Gnedenko [Hniedenko, B,V,j
(Integrodifferentlal. uquations)

2 8687
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D274/D304
AUTHOR: Skorokhod, A-V.
TITLE: On the existence of solutions to stochastic equations
PERIODICAL: Akademiya nauk UkrSSR. Dopovidi, no. 9, 1961,
1119-1121
TEXT: Ito's stochastic equation
t t
9 (t) -~(to) +t~ cL(s, F, (s))ds +o-(s,~(s))dw(s) +
0 t0
t
+ S (s), u)q(dsXdu)
to
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On the existence ... D274/D304
is considered (Ref. 1: Matematika sb. perevodov, 1:1, 78, 1957).
Ito established that (1) has a single solution if a K can be found
so that
12+ Id(t,x) zr(t,Y,12 +
(CL(t,x) - OL(t,y)
W du Y12
+ lf(t,x,u) - f(t,y,u 2
u
The existence of the solution to (1) can be proved under less ri-
gorous conditions. Theorem: (1) has a solution if: 1) a(t,x) and
Cf(t,x) are continuous for xERI , tF- Etc) 12) for all t1ELto 2 TI
x.FR 1 the condition
lima lf(t,x,u) - f(t x ~u )12 du = o
1 u2
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On the existence ... D274/D304
is satisfied; 3) a number K can be found so that
,GL(t,x) 1 2+1 a( t,x) 12+ If( t,x,u) 12 du
U2 (l+x2)
4) f(t,x,u) is bounded in each bounded domain of variation of x
and u and for each to 9 x0 f(t,x,u) continuous in t,x at to X0
for nearly all u. a(t,x), 01(t,x), -P(t,x,u) - are functions deter-
J'
mined for tt Ito,T] , XER1, uER1, w(t) - describes the Brownian
movement, q(A)=p(A)-Mp(A) ). The prcof of the theorem involves
three lemmas. The third lemma is: Let the sequence of stochastic
continuous processes L,(t) (possibly vectorial processes) satisfy
the conditions
lim 1M SUPP > c"', = 0
0 n -o~ t
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D274 Y
On the existence ... D304
and b) for each positive F_:
lim lim sup P~J~ (t 0
n-.*O n-- It,-t 2 1-,o

S/052/63/ooa/001/004/6C5
A limit theorem for... B112/B186
Q. -~ p~91 < Ir < g, k,=O,
n n
will converge towards
Q - P f9, (1) < E (1) < g(1); 0 < ' < I).
The following theorem is derivedo if a I, (II d" are bounded and con-
x x xx
tinuous, if U(X-) >0# and if
+ z (t (s)) ds (s)) dw (s). K
t-;hare w(t) is ind ependent of then there will exist such a number L
that 0
n - ql,---L log n/-fi~.
-,'j3;il I TT &D iOctober 26, 1960
Card 2/2

SYOROKHODJ A.V. (Kiye-,r)
,.,a -1 a--e 7.~ r t nzaleS -
t'
.-0 -
Homogeneous cOntinucus F'ar-'v 7 P-rocesses 117,
J ee prim. S f6-3. I , - t ii::,~ , . 1 )
Teor. veroiat.

fami`e5 of andom stochast'c
71
N '64. (MIRA 17:12)
Ln~vert'l let. Predstavieno akademikom

kA0_0P_-_6_3 wk(i)
ACCESSION NR.--AP4045050 s-/0052/64/009/003/0492/0497
AUTHOR:. Skorokhod, A. V. (Kiev)
TITLE: Branching diffusion processes
SOURCE: Teoriya veroyatnostey i yeye primenenlya, v, 9, no. 38 1964, 492-497
TOPIC TAGS: branching difftision. -process :Afarkbv branch' ces diffusio
ing pro s
process
'-ABSTRACT: The- author obtains -equations for the transitiom probabilities of
branching diffusion processes for:one single type o~ partiele,nimilar proces ses
for several types of particles were previously investigated by B. A. SevasVyanov
(Degeneration Conditions for branching processes with diffusion Aeoriya veroyatm
1 eye primen. VI, 3 (1961), 276-286). Orig. art. has: 7. equations
ASSOCIATION: None
SUBMITTED: 19Dec63 ENCL: 00 SUB CODE: N?
NR REF SOV: 003 OTHER: 000
Cardl,1

L 4580-65 EWT(d) IJP(a)
sk,
Acussiox xR A003736 BOOK EXPLOITATION
-3korokhod AnatoTix Vladimirovich~,
Processes with independent random varialb4 (Siuchaynyye prot essay a
nezavIsimymf-]pr1rasH0Sea17am1 _Izd-vo, offauka" 64*
Moscow, 19
_._.278_p._biblio,, 9,000,eopies printed. Series note: Teoriya
veroyatnostey f-ma-tematichookata-statiatika.---
TAGS: mathematics, random process theory, independent-random
variable, Brownian movement
PURPOSE AND COVERAGE: This book is devoted to the.theory.of random
processes with independent variables--one of the most important
parts of random proceso theory. In-this book, for the first time,
are collected the many Important results obtained in the study of
.random'processes with independent variables. These results had boen
.scattered among various articles. The book is of Interest for
apecialiats in probability theory,working on random processes and
.for those vho study random process-theory and are ooncerned with
its applIcation in various branches of acience.
Co,d

f. 4,;808-6c
ACCESSION BR.AK4043736
TABLE OF CONTENTS fabridgedli
Foreword 6
Ch, 1. Independent random values 9
Ch. II. Processes with independent Variables. Definition and
properties of trajectory -- 36
Ch. III. Analysis of stochastical17 continuous processes with
independent variables -- 5
Ch. IIII. General properties of processes with independent
variables -- 90
Ch. V. Homogeneous processes with independent variables 126
Ch. VI. The Brownian motion process -- 169
Ch. VII. The convergence of random processes with independent
variables -- 198
Ch. VIII. Limit theorems for funotionale of random processes with
independent variables -- 216
~Cho IX. Degreas.corresponding to processes with independent
variables 239
:Bibliography 275
2/3
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ACC NR1 AF6007535 SCUM GODE1 UR/04()6/65/001/002/0101/01()7
AU111CR: Skorokho-11 Aevo 3,F
OAG: nonp
TITLE;t Quantity of information encoied by a nonlinear channel with internal noise
S--URCE: Froblemy pereiachi informatsil, v. 1, no. 2, 1965j 101-107
TOPIC TAGO'i encoiling 11hoory, Gaussian iistribution
ABSTRACTt The author consilors a random signal x(t) which paoses through a channel
yealling a signal y(t) relatel to x(t) by a liff9rential nquation
X-M''
which lopnnils on the InIlornal nols-o 'of -thp nystfn, and is assumed to be a Gaussian
process * Let IT(x,y) r3enote the (unntity of information In the processa Tho author
introrlules functions v(t) an'l ~t) F~nrj ,Ierives th.e equatlon
i r (o) 3
Ir (r, ) do.
LJ3t x(t) b-i a GuussJan process where Y- U)'~- a (t), Mx (t) x1? (1, a) +
Then, v (i) / t (j) d (t, t) where ~(t,s) for o