On the uniform integrability of continuous parameter stochastic processes

J. h e a n rnh Soe.

Vol. 19, No. 2, 1983

ON THE UNIFORM INTEGRABILITY OF CONTINUOUB PARAMETER BTOCFIASTIC PROGJWSES

L.

J. L. Daub (1975) [6] introduced a notion of optionaIly separable processs which generi- both the separable pr- and weE-measurable processes.

G. Johnson and L. L. Helms [a] showed that right continuous supermar- bingale (X,) is of class (D) if and only if lim E(X,) --E{X-) for every increasing sequence (TJ of stopping times &verging to

+

see [IO, p. 1021). In this paper we w i l l extend G. J o h n and L L.

Helms' result to the optionally separable pr-

Let

(a,

A process (X,) is adapted if X, i5 3#-meaaurable for each c. Unless Borne

other convention is mted explicitly, p r o m (X,) ^a stmhastic pmms

(X,),,,,, adapted to (S*). A function T : n + & U {+W) k a stopping time for (3,) iff ( T g t ] for aU t ~ & . It is h w n that S f T = { F ~ 8 : F n { T l t ) ^E& for d ER,) is a @-algebra. In order that XT is gr

measurable, it m t f f i a s to a m m e that the promm (XJ is progrewive i. e.,

for all t E R+ the map CO, t ] xSL-*R d d e d ^{by (S,} m) +X,(w) is measurable

with raped to

[a

J. L. h b introduced optiodly separable ^{p m [6].}

Definition [G]. If (X,) is a p r o m , a sequence (SJ of finite stopping

times is called ^an optional separability set for (X,) _{if for} each U , the ^set

(S, (a) : n E N ) contains 0 and is dense in CO, 0 0 ) and the graph of the sample function c+&(QI) is in the closure of the graph r e a t r i d to the countable dense ^set (&(m) : n E N).

Note that the ^set (&/\h : ^R, k>l) is also an o f i d y separability set

whose stopping timea are bounded. If the stopping times S, can be chosen constant, then the process is separable. J. L. Doob [6] has shown that every well measurable process is (indistinguishable from) an optiondly separable p m

M v e d 24, 1982

Thim research iu auppmled in part by the Korean Traders Scholarship Foundation.

Definition [g]. h (Xt)p,,,, be ^a progressive pm-. (X#) is called regular supermartingale if fox any stopping tiina S md Y' with S s T, we have E (XT-) <m and E (XT I ³³⁾ 5 X*

It is known that a right continuous supmartingale is regular supermar- tingale CIO, p. W.

We begin with a amaximaln lemma which is known for separable (or

*right continuous'') supermartingale [7, p. 3533.

LEMMA 1. ik (X,),a,-l be a progressive, optionally sepmtle regular supermartingda Then for each 230,

~ ~ s ~ P X , ( ~ I > ~ I 61 ~SQPX.>A X--EX-+EXO

osrs- osts-

Pmf. Let (S*) be an o p t i o d y separability set such that the S, _are bounded. For fixed ⁿ let (Sl', SBf, ^{. a * ,} Ss') be the rearrangement of (Sl, Ssl

.m-, Sa) in increasing order, then (Sr', SBt, --m, S/) are also stopping times

(see Cl, 611.

Let AI-- I& ( d < J l ₁

Then AI, ^m*., 11, A are disjoint, A r ~ 3 s w , _k LE^, and AI U U A, U A,= ( min Xp (a) <RI

l=+-.*- J

-

Uing the supermartingale inequality and the fact Xp Iw) S R on AI, we find that

5AZ P(AL) =JP min XBJ<R))

k J=t--.ra

As a+

+

{ i d X,, (m) <A)

-

j=l,..rm OS#<-

set. By taking n-tm, we get

J-

oils- a<t<-

(b) Aa in (a) let Srl, ^..m, S'= be a rearrangement of SI, S2, ^a*., S, in increasing order. Let &=o and S:=

+

X X,, . W . , X,+, X,. ) is a supermartingale.

l a W

On the uniform integrability of continuous parameter stochastic processes 113

Let M_n= { max X s'. >il}. Define

j=O'l... J

( ) {minIS; :Xs'. >il} if wEMn

T w = J .

00 If w$Mn

Then ^T is a stopping time for ([Js', [Js', "', as' , as') ._o By the optional

1 n 00

sampling theorem, we have E(Xs~)"2.E(Xr ) , so that E(Xs') "2.E(X^r)=f Xr+ f x.,

o Mn Ja'-Mn

"2. ilP(Mn )+E (X.,) - f X.,

Mn

Thus we have

ilP(Mn ) s:;;f_MX., - E (X.,)+E(Xo)

n

Since M_n increase to {sup Xs'.>il} = {sup Xt(w»il}, by taking limit as n

l:5::"j:S;oo J O::5:"t$OO

~OO, we obtain

ilP( {_0,;;/,;;.,sup Xt>ill) s:;;f_{I sup} x/>.<lX.,-E(X,,,) +E(Xo).

O:s;;t::s;oo

THEOREM 2. Let (Xt)o,;;t,;;oo be a progressive, optionally separable process such that for any stopping time S there exists a stopping time T"2.S wz"th EZr -

<

satisfying XtS:;;Zt for all t .. Furthermore (Zt)o,;;t,;;., is progressive, optionally separable process and Zr=ess sup E(Xsl[Jr) for any stopping time T.

S:.T

(Zt) O,;;t,;;., is called the Snell envelope of (Xt)O,;;t,;;.,.

Proof. The proof of this theorem is similar to the proof in the case of well-measurable progress [9J and is omitted.

THEOREM 3. Let (Xt)^tEEO,00] be a progressive, optionally separable process.

Assume that supI_{E(Xr )} 1<+00, where supremum is taken over the set of all

T

extended stopping times. If Hm E(Xrn) =E(X.,) for every increasing sequence (T_n) of extended stopping times converging to 00, then (Xt) is of class ^(D), i. e., (Xr ) is uniformly integrable over the set of all extended stopping times.

REMARK. G. ]ohnson and 1.L. Helms [8J showed that a non-negative right continuous supermartingale (Xt ) tEEO,") is of class (D) if Hm E (Xrn)=

E (X.,) for every increasing sequence (Tn) of extended stopping times converging to 00. So Theorem 3 is an extension of this result.

Proof. Let (Zt)tEEO,"] be the Snell envelope of (Xt )tEW,⁰⁰⁾ in the Theorem 2. Define

The (Rn) is an increasing sequence of extended stopping times converging to 00, since sup Zt<^{+ 0 0} a. e. by Lemma 1. We will break the remainder

t

of the proof into three steps.

Step 1. We will show that E(ZRn)-E(X",,). Suppose that lim E(ZR»EXoo

+e for some e>O. Since E(ZR)=sup E(XT) >E(Xoo)+e for each n, there

T;;,R"

exists an stopping time Sn~Rn such that E(Xs) >E(Xoo) +e. It remains to show that we can replace the sequence (Sn) by an increasing sequence.

Define

T~(w)=min{Sk(w) ISk(w)~Rn^{(w), X}Sk(w)~E(X_SnIas..)}

Then R n-::;' T:-::;'Sm XT:(w)~E(XsnlaT:) and T: is an extended simple stopping time, because

{w; Sk(W) ~Rm XSk~E(XsnlaSk)}Eask. Let Tn=max T;.

l,;;j';;n

Then (T_{n )} is an increasing sequence of extended simple stopping times converging to 00. Now we will show that_{E(XTn )}~E(XT:) ~E(Xs) ~E(Xoo)

+c:, which contradicts the hypothesis of theorem. For fixed n, let T₁"=

T'm T_i+1"=T_i VT/, then Tn'= T/-::;'T/'··· -::;'Tn-/'-::;'Tn"= Tn • We assert that _E(XT';)-::;'E(XT;:~) for all i, thus we have E(XT:) -::;'E(XTn ).

E(XT;') =S_T;"=_T;+l"XT;~ldP+S" ,XT~'dP_{T; <T;}

-::;,S·" "XT;~ldP+S" ,E(Xs;laT)dP T; =T;+l T; <T

=f"_{T; =T}"XTi~ldP+J" ^,XsidP

H1 . T; <T;

=S "_ "XT;;ldP+f" ,E(Xs;laT;)dP T; -T;+l T; <T;

-::;,J" "

=J

T; -TH1 T; -Tj+1

The first and second inequalities followed from the definition of the stopping time T/.

Step 2. We will prove that (Zt) tECO,ooJ is of class CD). Let T be an arbitrary stopping time and T' stopping time defined by

T'(w)= {T(W) if ZT~W) ~n

+^{00 otherwIse}

Then we have R n-::;' T' and consequently E(ZRn) ~E(ZT') ~E(Zoo).

By step 1 we obtain that

On the uniform integrability of continuous parameter stochastic processes 115

J

IZT",nl IZT<nl

On the other hand, {supZtS;;n} c {ZT::S;;n} for any stopping time T. Since

0';;1';;'"

supZ/<+oo a.e. by Lemma 1, P({supZt:5:n})-+l as n- oo . Therefore

0-:;:1$"00 Os:ts:oo

P ({ZTS;;n} )-1 as n-00 uniformly on T. Thus

J

uniformly in T, which implies (ZT+)rEr is uniformly integrable where

r

Step 3. We will show that (XT)rEr is uniformly integrable. Since _{X t :5:Zt} for all t and (ZT+)rEr is uniformly integrable, it follows that _(XT+)rEr is uniformly integrable. In order to prove that (XT-)rEr is uniformly integrable, consider process (-Xt)tE[O, "'] which satisfies all the conditions of the theorem. Using _(-Xr)+=XT- and step 2 we obtain that (Xr-)rEr is uniformly integrable. Thus (XT)rEr is uniformly integrable.

COROLLARY4. Let (Xt) tER+ be progressive, optionally separable process with supIE(XT) 1<+00. If E(XT)n-E(XT) for every increasing sequence (T_{n )} of finite stopping times, which converges to any finite stopping time T, then

(X/AT) tER+ is of class (D) for any finite stopping time T.

Proof. For any finite stopping time T, let Yt=XtAT for tE[0, 00J, then

(Y_{t )}tE[O, "'] is progressive, optionally separable process. Applying Theorem 3

to the process (Y_{t ),} we obtain that (Y_{t )} is of class (D).

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Kook Min University Seoul 132, Korea

Korea Advanced Institude of Science and Technology Seoul 131, Korea

Kyungpook National University Daegu 635, Korea