文档介绍：MEMOIRS
O F T H i-
AMERICAN MATHEMATICAL SOCIETY
NL'MBKR 4
ON STOCHASTIC LUA'UONS
KFYOSl 1TO
PUBLISHED BY THh
AMERICAN MATHEMATFCAL SCXJF1T
531 West 116th St., New York City
ON STOCHASTIC DIFFERENTIAL EQUATIONS
By
KIYOSI ITO
Let be a Markoff with a continuous and
Xj. simple process parameter t, F(t,"$;s,E)
be the transition probability law of the process:
-
(D F(t,| ,-s,E) Prfx^E/X.- 3},
where the side means the of x E under the condition:
right probability a x^. f
Hie differential of x. at t * s is given by the transition probability law of x^
in an infinitesimal neighborhood of t s:
(2) FCs-A^jjs^E).
W. Feller^) has discussed the case in which it has the following form:
=
F(s-A , )G(s-A
(3) 2 JJS+A^E) (1-p(s,I) (/yA2 2 ,j js+A^E)
+ +
(yA2 )p(s,j)P(s,3 ,E) o(/yA2 ),
where is a distribution as a function of E and
G(s-Ag,5 ;s+A,j,E) probability
satisfies
(5) T-^T; f
1**2 J (^-j)
h-jl<f
(6) (l-J)G(s-A ->b(t,J),
"^2 J, 2 ,J js^^dn)
for A *A > and p(s, J ) > and P(s, J ,E) is a probability distribution in E. The
M 11 '
special case of p(s, J ) O has already been treated by A, Kolmogoroff and
S. Bernstein. 3/
We shall introduce a somewhat general definition of the differential of the
process x. (Cf. 85). Let P A denote the conditional probability law:
L 8,5 ,^,^2
3 }, A A > 0.
Mx^-V^* E^- V 2
If the ]- times^) convolution of P tends to a probability law L
[1/^*A fl A ^
with regard to Levy's law-distance as A +A > 0, then L is called the
I d S,J
stochastic differential coefficient at s. L is clearly an infinitely divisible
law. In the above Feller's case the logarithmic characteristic function 5)
Received by the editors March 29,
KIYOSI I TO
of is
V (*,L ) given by
S>$>$ L^ f
03 iu2
= - 6)
(7) (z,L ) ib(s,j )z a(s,j )z^p(s, 5 ) f (e -1)P(s, J ,du(+) J ).
8 >j 7
^ -00
A problem of stochastic differential equations is to construct a Markoff p