I. Martingale characterization of processes with independent increments (B. ... that the solutions of the optimal stopping problem for these processes (the value of ...
THEORY OF STOCHASTIC PROCESSES R. Kud~ma and V. Mackevi6ius
UDC 519.21
The development of the theory of stochastic processes at Vilnius University and generally in Lithuania is connected with the name of B. Grigelionis, Professor at Vilnius University, head of a sector of the Institute of Mathematics and Cybernetics, corresponding member of the Academy of Sciences of the Lithuanian SSR. We shall indicate the basic directions of the studies carried out by him and his students. I. Convergence of sums of piecewise-stochastic R. Banis).
processes
(B. Grigelionis,
If. Optimal stopping and optimal control of stochastic processes Kudzma, V. Mackevicius, G. Pragarauskas, R. Eidukyavichyus). III.
I. Sapagovas,
(B. Grigelionis,
R.
Theory of semimartingales.
I.
Martingale characterization of processes with independent increments
2.
Pointwise stochastic measures
3.
Absolute continuity of measures
4.
Structure of functionals of stochastic processes
5.
Equations of nonlinear filtrations
6.
Processes in a half space-(B. Grigelionis,
(B. Grigelionis).
(B. Grigelionis). (B. Grigelionis, M. Radavichyus). (B. Grigelionis). p
7. Weak convergence of measures Kubilyus). IV.
(D. Surgailis, B. Grigelionis,
E. Tinfavichyus).
R. Mikulyavichyus).
(B. Grigelionis,
R. Morkvenas, V. Mackevicius,
K.
Other problems.
i.
Ergodicity of Markov chains
(Z. Navitskas).
2.
Approximation of solutions of stochastic equations
3.
Sheaves and stochastic processes
(V. Mackevicius).
(R. Kudzma).
Later we consider in more detail the work carried out at Vilnius University. If.
Optimal Stopping and Optimal Control of Stochastic Processes
Let X = (xt, ~ t , Px) be a homogeneous Markov process with continuous time t ~ 0 in the semicompactum (E, ~), 9 ~ b e the class of all (finite) stopping times relative to ~ t ) , g be some (sufficiently " g o o d " ) function on E. We write s (x) = sup E~ g (x~), x ~ E.
If there exists a stopping time-.0~.~=')~ for which, for all x ~ E, s(x) = Exg(Xzo) , then zo is called an optimal stopping time. The basic problem of optimal stopping is to find To and hence the value of the game s, since under general hypotheses To is the time of first landing in the set D = {x:s(x) =g(x)). Grigelionis and Shiryaev [i], Grigelionis [2, 3] established sufficient conditions under which the value of the game s together with, in general, the unknown set D c E = R n is a solution of the so-called generalized Stefan problem (the condition of "smooth gluing" is indicated for a diffusion process)
aCs(x) =0, xeE\D, s(x)=g(x), xeD,
(1)
V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 20, No. 3, pp. 107-116, July-September, 1980. Original article submitted September 4, 1979.
0363-1672/80/2003-0255507.50
9 1981 Plenum Publishing Corporation
255
Os (x) _ Of(x) Oxi Ox~ '
x ~OD
(ag is the characteristic operator of the process X). Sufficient conditions are also obtained under which the solution s of (i) is in fact the value of the game. Such a reduction of an optimal stopping problem to a Stefan problem was used earlier in a series of examples without rigorous justification by V. S. Mikhalevich, H. Chernoff, J. A. Bather, etc. Among later studies in connection with this problem, one should note particularly the papers of Krylov on the theory of optimal control (cf., e.g., [4]). For piecewise-Markov processes, finding the value of the game cannot be reduced to a Stefan problem. Grigelionis proved [5] that the optimal stopping problem for a piecewiseMarkov process with transition function P(t, x, F) is equivalent with the corresponding optimal stopping problem for a Markov chain with transition functions
II(x, 17) , II(x, P)=lim--1 P(t, x, r \ ( x } ) .
=(x, r)-
n(x,e)
~ xvO
t
In many cases it is hard to get explicit solutions for optimal stopping problems. In connection with this, there arises a natural problem: what conditions should be imposed on the convergent (in some sense) sequence of Harkov processes Xn and payoff functions gn, so that the solutions of the optimal stopping problem for these processes (the value of the game and the e-optimal stopping times) should converge to solutions of the limit process. Mackevicius [6] got general sufficient conditions in terms of the convergence of the corresponding a) subgroups of operators; b) infinitesimal operators; c) measures in spaces of functions without discontinuities of the second kind. The general results are applicable to the study of the asymptotic behavior of the boundary of the domain of optimal stopping in certain concrete problems. Kudzma [7] in problems of optimal stopping developed a general invariance principle, analogous to the invariance principle in mathematical statistics. He applied this idea to the optimal stopping of semistable (in the sense of J. Lamperti) Markov processes, whose transition functions satisfy the condition P(at, x, B ) = P g ,
for some ~ > O.
a - ~ x , a - ~ B ) , a > O , t > O , xc[O, ~), B ~ [ O ,
~),
Considering the value of the game s(x, y ) - s u p E ~ ~ ~me iY~)v '
he found conditions under which the optimal stopping times have the form
~v--~inf{t~ O : x , ~ c ( y + t ) ~ } or
~=inf{t~O:xt>(y+t)
~
or x,=O).
In the diffusion case he got equations for c. The powerful analytic methods of the theory of Markov processes are directly applicable for solving optimal stopping problems for general (nonmarkov) processes, in terms of which one can formulate many concrete problems, e.g., various statistical hypotheses, the problem of "dissonance," etc. Grigelionis [8, 9] introduced and studied the concept of sufficient statistics and sufficient Markov statistics, whose use allows one to reduce many problems of optimal stopping of nonmarkov processes to the Markov case. Here the basic difficulty turns out to be the verification of the Markov property. In connection with this Grigelionis [i0, ii, 3], Surgailis [12] studied conditions under which a stochastic process satisfies the stochastic equation of K. Ito and thus (in the case of uniqueness of the solution) has the Markovian property. These studies led Grigelionis to new fruitful ideas in the theory of stochastic processes, especially in the theory of semimartingales. III.
Theory of Semimartingales
Problems connected with statistics of stochastic processes, stochastic controls, the theory of information, etc. led to the study of nonmarkovian processes. In time it became clear that a basic role in these studies is played by the apparatus of stochastic integration
256
with respect to martingales and point measures. B. Grigelionis [13-15] introduced the concept of locally infinitely divisible (l.i.d.) process, as a process (xt, 3vt) for which the conditional distribution of infinitesimal increments x ( t + d t ) -- x(t) f o r ~ t is infinitely divisible with deflection vector ~(t) dt, diffusion matrix A(t) dt and Levy measure T(t, P)dt (for the exact definition, cf. below). The papers of J. Jacod on the structure of semimartingales (in the sense of P. A. Meyer and K. Doleans) determined the place of l.i.d, processes in the general theory of stochastic processes and allowed B. Grigelionis, J. Jacod, A. N. Shiryaev, etc. to get new results in describing properties of a semimartingale in terms of its characteristics. Let X = {x t = (x[, .... x~), o~t, t .>-0} be an m-dimensional semimartingale, p((s, t] x F) be the jump measure of the process X on the o-algebra of Borel subsets of the space (0, ~) x (Rm\ {0}). It is known (cf. [16, 17]) that any semimartingale can be uniquely represented in the form
/
x :xo+a,+x~+ " 0
w h e r e ~o = 0 ,
~ is a predictable
process
f xq(ds, d x ) + f ;x[~l
0
having
'
f xp{ds, dx),
(2)
[xi>l
on e a c h f i n i t e
interval
finite
variation
a.e.,
%-%_=
f xrl({t},
dx)
t>O,
Ixi~
