The Theory of Stochastic Processes III
- PDF / 7,485,461 Bytes
- 112 Pages / 481.89 x 691.65 pts Page_size
- 59 Downloads / 178 Views
Martingales and Stochastic Integrals
§ 1. Martingales and Their Generalizations
Survey of preceding results. We start by recalling and making more precise the definitions and previously obtained results pertaining to martingales and semimartingales (cf. Volume I, Chapter II, Section 2 and Chapter III, Section 4). Let {n, ~, P} be a prob ability space, let T be an arbitrary ordered set (in what follows only those cases where T is a subset of the extended real line [-00, +00] will be discussed) and let {iSt, tE T} be tI current of a-algebras (iSt C ~): if tl < t2 then iStl C iS t2' The symbol {g(t), iSt, tE T} or simply {g(t), iSt} denotes an object consisting of a current of a-algebras {iSt, tE T} on the measurable space {n, ~} and a random process g(t), tE T, adopted to {i1"t, tE T} (i.e., ger) if i1"t-measurable for each tE T). This object will also be referred to in what follows as a random process. A random process {g(t), iSt, tE T} is called an I5t-martingale (or martingale if there is no ambiguity concerning the current of a-algebras iSt under consideration) provided (1)
Elg(t)1 < 00
\itE T
and E{g(t) I iSs} = g(s)
for s < t, s, tE T;
it is called a supermartingale (submartingale ) if it satisfies condition (1) and moreover (2)
E{g(t)ll5s} ~ g(s),
s < t,
(E(g(t)ll5s}?g(s),
s < t).
s, tE T
Observe that the above definition differs from that presented in Volume I since we now require finiteness of the mathematical expectation of the quantity g(t) in all cases. Previously, in the case of the supermartingale, for example, only the finiteness of the expectation EC(t) was assumed. I. I. Gihman et al., The Theory of Stochastic Processes III © Springer-Verlag New York Inc. 1979
2
I. Martingales and Stochastic Integrals
The definition presented herein is equivalent to the following: U(t), 15" tE T} is a martingale (supermartingale ) if for any set B s E ~ sand for any sand t belonging to T such that s < t,
JB, g(t)dP = JB, g(s) dP Supermartingales and submartingales are also called semimartingales. In this section we shall consider mainly semimartingales of a continuous argument. The space of all real-valued functions on the interval [0, Tl which possess the left-hand limit for each tE (0, Tl and which are continuous from the right on [0, T) will be denoted by gg or by gg[O, Tl Analogous meaning is attached to the notation gg [0, T), gg [0,(0), and gg [0, 00 l. A number of inequalities and theorems concerning the existence of limits plays an important role in the martingale theory. The following relationships were established in Volume I, Chapter 11, Section 2: If g(t), tE T, is a separable submartingale, then
q=~ p>l, p-l'
(4)
(5)
E(g(t)-bf Ev [b) a, ~sup b ; leT -a
he re a + = a for a ;;,: 0 and a + = 0 for a < 0 and v[a, b) denotes the number of crossings down ward of the half-interval [a, b) by the sampie function of the process g(t) (a more precise definition is given in Volume I, Chapter 11, Section 2). We now recall the definition of a closure of a semimartingale. Let {g(t), 151, tE T} be
Data Loading...
