A Note on Utility Maximization with Unbounded Random Endowment

Hi-Stat Discussion Paper

Research Unit for Statistical

and Empirical Analysis in Social Sciences (Hi-Stat)

Hi-Stat

Institute of Economic Research

Hitotsubashi University

2-1 Naka, Kunitatchi Tokyo, 186-8601 Japan

http://gcoe.ier.hit-u.ac.jp/index.html

Global COE Hi-Stat Discussion Paper Series

October 2009

A Note on Utility Maximization with Unbounded

Random Endowment

Keita Owari

ENDOWMENT

K

O

Graduate School of Economics, Hitotsubashi University 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan

This paper addresses the applicability of the convex duality method for utility maximiza-tion, in the presence of random endowment. When the price process is a locally bounded semimartingale, we show that the fundamental duality relation holds true, for a wide class of utility functions and unbounded random endowments. We show this duality by exploit-ing Rockafellar’s theorem on integral functionals, to a random utility function.

1. I

Maximization of expected utility has been a time-honored issue in the study of mathemat-ical finance. Especially, the following version of the problem withrandom endowmentis important in view of its application toutility indifference valuation:

(1.1) maximize EŒU. ST CB/; over all 2;

whereU is an utility function,Sis a semimartingale,is the set of admissible integrands (strategies), andBis a random variable expressing arandom endowmentor acontingent claim.

A sophisticated way of solving (1.1) is the convex duality method which pass (1.1) to a minimization over the set of local martingale measures forS, through the (formal)duality equality: (1.2) sup 2 EŒU.ST CB/D inf >0Qinf2ME V dQ dP CdQ dPB ;

whereV is the Fenchel-Legendre transform of the utility functionU, andMis a set of local martingale measures. The RHS of (1.2) is the optimal value of thedual problem. Note that the inequality “” is always true, while “” may not. This equality is shown by several authors in different settings, e.g., the case of no endowment.B 0) by Kramkov and Schachermayer [12] and Schachermayer [17], the case of boundedBby Bellini and Frittelli [2], and the case ofexponential utilitywithsuitably integrableBby Delbaen et al. [5], Kabanov and Stricker [11] and Becherer [1].

Then a natural question arises: to what degree of generality does the equality (1.2) hold true ? This is the theme of this note. Under the fundamental assumption thatS islocally bounded, we shall prove the duality for a wide class of endowmentsB. Our idea is based on a refinement of [2] from a slightly different point of view. Namely, we view the problem

E-mail address:[email protected], [email protected]. 2010Mathematics Subject Classification. 91G80, 60H30, 46N10 .

Key words and phrases. Utility maximization, Convex duality method, Martingale measures.

(1.1) as the maximization of expected utility functional associated to therandom utility function.!; x/_7! U.x_CB.!//. This allows us to take full advantage of Rockafellar’s theorem on convex integral functionals.

2. R

Suppose we are given a complete probability space.˝;F; P /equipped with a filtration F _WD .Ft/t2Œ0;T  satisfying the usual conditions of right-continuity and completeness,

whereT ₂.0;₁/is the fixed time horizon. We assumeF _DFT for notational simplicity.

LetSbe ad-dimensional càdlàglocally boundedsemimartingale on.˝;FT;F; P /, and

define

(2.1) bbWD f 2L.S /W 0D0; Sis uniformly bounded from belowg;

where L.S / _D L.S; P / denotes the set of d-dimensional predictable processes _D .1_{; :::;} d_{/which are.S; P /-integrable, andSD}R

0sdSsis the stochastic integral of

₂ L.S /w.r.t. S. For the precise definitions and basic properties of stochastic integrals and the setL.S /, we refer the reader to Jacod [9,10]. Any ₂bbis called anadmissible

strategy, and we explicitly include the condition0 D 0in the definition of admissibility

to avoid the contribution of the initial value0S0to the stochastic integral.

In this paper, we consider only a class of utility functions defined on thewhole real line. More precisely, we assume:

(A1) U _WR_!Ris a continuously differentiable, increasing, and strictly concave function satisfying the so-called Inada condition:

(2.2) lim

x! 1U

0_{.x/D C1 and lim}

x!C1U

0_.x/D0:

For a given utility functionU, the Fenchel-Legendre transform ofU is defined by

V .y/_WDsup

x2R

.U.x/ xy/; y ₂R:

In the language of convex analysis, V is the convex conjugate of the convex function

˚.x/ _D U. x/. Under (A1),V is also differentiable withV0.y/ _D .U0/ 1.y/, and has the explicit representation: V .y/ _D U..U0_/ 1_{.y// y.U}0_/ 1_{.y/ify > 0,V .0/ D}

U._C1/_WDlimx!C1U.X /, andV .y/D C1ify < 0. Furthermore, we have

lim

y#0V

0_{.y/D 1 and lim}

y!1V

0_{.y/D C1:}

(2.3)

Note in particular thatV is bounded from below. For utility functions, we assume also the condition ofreasonable asymptotic elasticities:

(A2) AE ₁.U /_WDlim inf

x& 1

xU0.x/

U.x/ > 1; AEC1.U /WDlim supx%C1

xU0.x/ U.x/ < 1:

This condition is introduced by Kramkov and Schachermayer [12] and Schachermayer [17] as a necessary and sufficient condition for the existence of optimal investment strategy. Also, (A2) is equivalent to (see [6]): for any closed intervalŒa; b.0;₁/, there exists

C1; C2> 0such that

V .y/C1V .y/CC2.yC1/; 8y > 0; 2Œa; b:

(2.4)

A probability measure Q P under whichS is a local martingale is called an ab-solutely continuous local martingale measure for S, and the set of all such measures is

denoted byMloc. For the domain of the dual problem, we introduce the following subset

ofMloc:

MV WD fQ2MlocW EŒV .dQ=dP / <1g:

Note that, by the consequence (2.4) of (A2), we have for allQP,

EŒV .dQ=dP / <_{1 ,} EŒV .dQ=dP / <₁; ₈ > 0:

Generically, for any set Q of positive measuresQ P, we denote by Qe the set of

Q₂QwithQP. We assume a version ofno-arbitragecondition: (A3) Me_{V ¤ ;}.

Finally, letBbe aFT-measurable random variable such that:

(A4) There exists some" > 0for which,

EŒU. .1_C"/B / > ₁;

(2.5)

EŒU. "BC/ > ₁:

(2.6)

2.2. MAINTHEOREM ANDRELATEDRESULTS

We are now in the position to state the main theorem. The proof will be given in Section3.

Theorem 2.1. Under (A1) – (A4), the duality equality holds, i.e., (2.7) sup 2bb EŒU.ST CB/D inf >0Q2infMV E V dQ dP CdQ dPB ;

and the infimum in the RHS is attained by some.;O Q/y ₂.0;₁/Me_V.

From a practical point of view, it is also important to ask whether the optimal expected utility can be approximated bybounded stochastic integrals, i.e., by admissible strategies such thatSis bounded not only from below, but also from above. If the utility function is bounded from above, the answer is positive. Let

(2.8) bD f 2L.S /W 0D0; Sis uniformly boundedg:

Corollary 2.2. If, in addition to (A1) – (A4),U is bounded from above, then we have (2.9) sup 2b EŒU.ST CB/D inf >0Q2infMV E V dQ dP CdQ dPB :

Finally, as pointed out by [5] in the case of exponential utility, the duality equality is quite robust in the choice of admissible class. Let

(2.10) V WD f 2L.S /W 0D0; Sis a supermartingale under8Q2MVg:

Corollary 2.3. Suppose (A1) – (A4), and letL.S /be sandwiched bybb(resp.b

ifU.₁/ <₁) andV, i.e.,bb V (resp.b V). Then (2.7) remains

true withbbreplaced by.

We conclude this section with a brief review of related literature. Generally speaking, our result is an intermediate one among duality results of the type (1.2), in that, we require

Sto be locally bounded, but give a duality of theclassical-type(i.e., exclude the unpleasant intervention ofbizarresingular term, see below) for a wide class ofU andB.

Bounded Endowment.To our best knowledge, a duality result as our Theorem2.1appears first in [2]. Their argument (from our view point) is based on the analysis of the functional

X _7! EŒU.X / onL1_{, and its conjugate defined on ba ' .L}1_{/ (Banach space of}

finitely additive signed measures), giving the duality for the caseB0. Then the case of bounded endowment follows by translation of the domain inL1.

Exponential Utility.The “Six-Author Paper” [5] and its refinement [11] develop a general duality theory for the case of exponential utility: U.x/ _D 1 e ˛x, giving the duality equality under (2.5) and the boundedness from above ofB. This assumption is weakened by [1] to the condition corresponding to our (A4). More recently, Owari [13] extends this framework to therobust exponential utility maximization.

General Semimartingales.Without doubt, the duality theory can be extended to the case with non-locally bounded S. In this case, however, the duality equality holds only in a generalized senseas (Biagini et al. [3]):

sup 2HW EŒU.ST CB/D inf >0Q2infMW E V dQ r dP CQ.B/_CkQs_k :

whereHW is the set of integrands of which S is bounded from below by asuitable random variableW,MW is a subset ofba,Q.B/is the “integral” ofBw.r.t. a finitely additive measureQ, andQr _{(resp. Q}s_{) denotes the regular (resp. singular) part ofQin}

the Hewitt-Yosida decomposition. Our integrability assumption (A4) appears in [3]. In this respect, Theorem2.1states that, in the case of locally boundedS, the singular term automatically disappears, wheneverBsatisfies (A4), although the case whereBsatisfies (2.5) and (2.6) for “₈" > 0” is covered by [3].

Other Case.Yet another approach is proposed by [14]. There the problem (1.1) is consid-ered under the assumption that there existsx0; x00₂Rand0; 00₂V such that

(2.11) x0_C0ST Bx00C00ST;

and0_{S is a martingale under everyQ 2 MV. This has no apparent relation to our}

assumption. In contrast to this formulation, our approach has an advantage that we need only the integrability conditions forB, which are easily checkeda priori, while (2.11) is hard to verify.

Remark 2.4. Since we focus only on the case of utility onR, articles on the case of utility onR_C are omitted. For this direction, see e.g., Cvitani´c et al. [4], Hugonnier and Kramkov [7], Hugonnier et al. [8] and references therein.

3. P

We first give the outline of the proof, which may help the understanding. Roughly speak-ing, our idea is based on Bellini and Frittelli [2], but exploits Rockafellar’s theorem [15] on convex integral functionals to arandomutility function.

As most of literature on this subject, we first reduce the problem to a maximization of a concave functional defined onL1, and then appeal to the.L1; ba/-duality. Define (3.1) C_{WD f}X ₂L1_{W 9 2}bbsuch thatX STg;

which is a convex cone containingL1andK_{WD f} ST W 2 bg(see e.g., [2]). As in

[2], we can show (Lemma3.6below): sup

bb

EŒU.ST CB/Dsup

Letı_C.X /_D0ifX ₂Cand_{D C1}otherwise (i.e.,ı_Cis the indicator function ofCin the sense of convex analysis), and define (formally) a concave functionaluB onL1by

(3.2) uB.X /WDEŒU.X CB/: Then we have sup X2CuB.X /D sup X2L1 .uB.X / ıC.X //;

Now ifuB is well-defined and regular enough, Fenchel’s duality theorem shows that

sup X2L1 .uB.X / ıC.X //D min 2ba.ı C./ uB.//D min 2ba.vB./Cı C.//;

wherevBis the conjugate ofuB defined onbaby

vB./WD sup X2L1

.uB.X / .X //; 2ba:

(3.3)

Thus, the key step is to verify the regularity ofuBand to derive the explicit form ofvB. We

will do this (Proposition3.8) by exploiting Rockafellar’s theorem touBwhich is a concave

integral functional defined by therandomconcave functionUB on˝ R: UB.!; x/WD

U.x_CB.!//. In this step, the assumption (A4) plays a crucial role, giving the estimates betweenU,UB andV (Lemma3.4).

3.2. PRELIMINARIES ANDIMPORTANTESTIMATES

We first introduce some additional notations and concepts used in the proof of Theorem2.1. The first one is the description of the spaceba.

Definition 3.1 (ba.˝;FT; P /). ba WDba.˝;FT; P /is the set of all bounded finitely

additive measures absolutely continuous w.r.t. P, i.e., ₂ba.˝;FT; P /if and only if

is a real valued function onFT such that (1) supA2FTj.A/j<1, (2) for everyA2FT,

P .A/ _D 0 implies.A/ _D 0, (3) if A; B ₂ FT andA\B D ;, then.A[B/ D

.A/_C.B/. Also,ba_C(resp.ba) denotes the set of positive (resp.-additive) elements ofba, and setba

C WDbaC\ba,ba;1C WD f 2baCW.˝/D1g, and

QV WD f2ba_CWEŒV .d=dP / <1g:

Only facts which will be used here are: (1) bais a Banach space equipped with the total variation norm, andba _' .L1_{/, (2) every 2 ba has a unique decomposition}

_Dr _Cs, wherer ₂ ba ands is purely finitely additive. ba_C;1is nothing but the set of probabilitiesQon.˝;FT/withQ P. Also, as a direct consequence of (2.4),

QV is a convex cone having the following representation:

Lemma 3.2.

1. IfV .0/_DU.₁/ <₁,

(3.4) QV D fQW 0; Q2ba;1_C ; EŒV .dQ=dP / <1g:

2. IfV .0/_{D C1},

(3.5) QV D fQW > 0; Q2ba;1_C ; EŒV .dQ=dP / <1g:

Recall that the setC(defined by (3.1)) is a convex cone containingL1. The following relation betweenCandMlocis well-known (e.g., [2, Lemma 1.1]): for everyQ2ba_C;1,

Letı_Cbe the conjugate of the indicator functionı_C, i.e.,

ı_C./_D sup

X2L1

..X / ı_C.X //_D sup

X2C.X /; 82ba:

The above observations immediately yield the next lemma.

Lemma 3.3. ı

C./D C1if62baC, and for all 2baC,

(3.7) ı_C./_D (

0 if₂cone.Mloc/

C1 otherwise. Here

cone.Mloc/D fQW 0; Q2Mlocg:

Proof. If₆₂ba, there existsXN ₂L1_C with.X / < 0N . SinceL1C, we have XN ₂C

for all > 0, hence

ı_C./sup

>0

.X /N _{D C1}:

The fact thatC is a cone implies thatı

C isf0;C1g-valued, andıC./ D 0if and only

if.X / 0for allX ₂ C. If ₂ ba_C, the latter condition is equivalent to saying that

₂cone.Mloc/by (3.6). ¤

The following estimates are elementary, but play a key role in the proof of theorem.

Lemma 3.4. Let" > 0.

(a) For every random variableY 0,

" 1_C".V .Y / V .1//CU. .1C"/B /V .Y /CYB 1C" " V .Y / 1 "U. "B C_/: (3.8)

(b) For everyY 0and every random variableX,

" 1_C"U 1_C" " X C 1 1_C"U. .1C"/B /U.XCB/ 1C" " V .Y /CX Y 1 "U. "B C_/: (3.9)

Remark 3.5. We make some remarks on the consequences of (A4).

1. (3.8) implies that V .Y / ₂ L1 _{if and only if V .Y /C YB 2 L}1_{, and in this case,}

YB ₂L1andEŒV .Y /_CYB_DEŒV .Y /_CEŒYB. In particular, for anyQ₂ba;1_C ,

EŒV .dQ=dP / <₁impliesB₂L1.Q/.

2. The map.; Q/_7!EŒV .dQ=dP /_C.dQ=dP /BonR_Cba_C;1to. ₁;_C1is well-defined (note thatV is bounded from below), and is finite if and only ifQ₂QV.

Let; Qbe such a pair. Then by Jensen’s inequality, (3.10) E V dQ dP CdQ dPB ₁ " C".V ./ V .1//CEŒU. .1C"/B /:

In particular, inf0;Q2MV EŒV .dQ=dP /C.dQ=dP /B > 1, since againV is

bounded from below.

3. (A3) and (A4) implies thatU.X _CB/ ₂ L1for everyX ₂L1. Indeed, the LHS of (3.9) is integrable for anyX ₂L1_{sinceU is monotone, while the RHS is integrable}

Proof of Lemma. (a) For anyY 0,

"YB Y ."BC/V .Y / U. "BC/;

(3.11)

by Young’s inequality, thus,

V .Y /_CYB 1C" " V .Y /

1 "U. "B

C_/;

and we get the second inequality in (3.8). On the other hand,

YB _D Y 1_C".1C"/B " 1_C"C 1 1_C"Y .1_C"/B V " 1_C"C 1 1_C"Y U. .1_C"/B / 1 1_C"V .Y /C " 1_C"V .1/ U. .1C"/B /: Using this, V .Y /_CYB V .Y / YB " 1_C"V .Y / " 1_C"V .1/CU. .1C"/B /:

These prove the assertion (a).

(b) For any random variableXand positive random variableY,

U.X_CB/V .Y /_CY .X_CB/

1C_" "V .Y /_CX Y 1 "U. "B

C_/;

by (3.11). Also, sinceU is concave and monotone increasing,

U.X_CB/_DU " 1_C" 1_C" " X C 1 1_C".1C"/B ₁ " C"U 1_C" " X C ₁ 1 C"U..1C"/B/ ₁ " C"U 1_C" " X C ₁ 1 C"U. .1C"/B /:

This completes the proof. ¤

We now reduce the problem to a minimization inC.

Lemma 3.6. We have (3.12) sup

2bb

EŒU.ST CB/D sup

X2CEŒU.XCB/:

Proof. The inequality “” is immediate from the definition ofCand the monotonicity of

U. Let ₂bb. Then for anyk 2N,Xk WD.ST/^kis inC. Since 2bb, there

existsx > 0withS xuniformly, a.s., henceXk x, a.s. We have

U.XkCB/%U.ST CB/; a.s.

(3.13)

Now Lemma3.4(b) implies thatU.Xk CB/ 1C""U. .1C"/

" x/C 1

1C"U. .1C"/B /,

for each k, which is in L1 _{by (A4). On the other hand, taking Q 2 M}

V (by (A3)),

U. ST CB/ 1C_""V .dQ=dP /CSTdQ=dP 1_"U. "BC/2L1, sinceS is a

Q-supermartingale, andU. "BC/₂L1by (A4). Therefore, the convergence (3.13) takes place in L1 by the dominated convergence theorem, hence limk!1EŒU.Xk CB/ D

The final lemma in this subsection states that the infimum in the dual problem must not attained neither by_D0nor byQ₆P.

Lemma 3.7. If ₂QV nQVe, there existsQ2QeV such that

E V d_Q dP C dQ dPB < E V d dP C d dPB :

Proof. This is trivial ifV .0/ _{D C1}since thenQV DQeV, thus we assumeV .0/ <1.

Let₂QV andN 2Qe_V(¤ ;by (A3)). Set˛WD˛NC.1 ˛/2QV (˛2Œ0; 1). Note

that˛2Qe_V for˛2.0; 1/. Set also,

'˛WDV d˛ dP C d˛ dP B2L 1_:

Since˛_7!'˛.!/is convex for a.e.!,˛7!.'˛ '0/=˛is increasing in˛, hence

'˛ '0

˛ &Z; a.s.,

for some random variable Z. Since .'1 '0/=˛ 2 L1, we can apply the monotone

convergence theorem to get

(3.14) lim ˛&0E h_{'˛ '0} ˛ i DEŒZ:

On the other hand,

Z _D V0 d dP CB dN dP d dP D 1 on d dP D0 ;

sinceV0.0/_{D 1}(by (A1)) and_{N 2} Qe_V. Therefore, (3.14) shows that if ₆P, there exists˛₂.0; 1/such that

E V d˛ dP C d˛ dP B E V d dP C d dPB < ˛:

Since˛2QeV, we have the desired result. ¤

3.3. DESCRIPTION OFTHECONJUGATEFUNCTIONAL

We now come to the key step, namely, the regularity of uB defined by (3.2), and the

description of its conjugatevB defined by (3.3).

Proposition 3.8. Assume (A1) – (A4). Then

(a) uBis well-defined and continuous onL1w.r.t. the norm topology.

(b) vB has the expression:

(3.15) vB./D

˚

EhV _dPd_{C dP}dBi if ₂QV

C1 otherwise.

We shall prove this by exploiting Rockafellar’s theorem on convex integral functionals. We begin with some preparation.

Definition 3.9. A mapf _W˝R_!R_{[ fC1g}is called anormal convex integrandif: (a) f is jointly measurable (i.e.,FB.R/-measurable),

(b) x_7!f .!; x/is a lower semicontinuous proper convex function for a.e.!. Also, the conjugate random convex function off is defined by

(3.16) f.!; y/_WDsup

x2R

We cite here Rockafellar’s theorem in a form suited to our purpose.

Theorem 3.10(Rockafellar [15], Theorem 1, Corollary 2A).

1. Letf _W˝R_!Rbe a random convex function such that (a) there exists someX ₂L1_{such that,f .; X.//}C_2L1_,

(b) there exists someY ₂L1_{such that,f.Y .//}C _2L1_.

Then the map

(3.17) If.X /WDEŒf .X /D

Z

˝

f .!; X.!//P .d!/; X ₂L1

is well-defined as a convex functional onL1_{, and the conjugateI}

f Wba7!R[ fC1g is expressed as: (3.18) I_f./_DIf.r/Cı_{dom .If/}.s/; 2ba; where, If.r/DEŒf.dr=dP /D Z ˝ f !;d r dP .!/ P .d!/; ı_{dom .If/}.s/_D sup X2dom.If/ s.X /:

2. If in additionf .; X.//₂L1for everyX ₂L1, thenIf is continuous onL1and

(3.19) I_f./_D (

EŒf.d=dP / if₂ba

C1 otherwise.

Remark 3.11. In [15], the notion of normal convex integrands is introduced in a slightly different way, which is equivalent to our Definition3.9if the underlying probability space is complete as we assumed. See Rockafellar and Wets [16], Ch.14 for detail. Also, the original version of Theorem3.10in [15] is stated and proved on a-finite measure space, rather than a probability space.

Proof of Proposition3.8. We apply Rockafellar’s theorem to the random convex function

f .!; x/_D U. x_CB.!//;

which is clearly jointly measurable, convex and continuous inx, hence normal. The con-jugatefis given by

f.!; y/_DV .y/_CyB.!/;

andIf.X /DEŒ U. XCB/D uB. X /, thusI_fDvB.

For everyX ₂L1_{,f .X /D U. XCB/is integrable by Lemma3.4and Remark3.5.}

On the other hand, we can takeQx₂MV by (A3), so thatf.dQ=dP /x DV .dQ=dP /x C

.dQ=dP /Bx ₂L1, by Lemma3.4. Hence we can apply Theorem3.10to get the assertion (a), and that

vB./DIf./D

˚

EhV_dPd_{C dP}dBi if ₂ba

C1 otherwise.

It remains to show thatvB./D C1if2banQV. Suppose2banba_C. Since

f.d=dP /_DV .d=dP /_C.d=dP /B_{D C1}on the set_fd=dP < 0_gwhich has a pos-itive probability, the estimate (3.8) of Lemma3.4shows thatvB./DEŒf.d=dP /D

C1. Finally, for anyY 0 f_{.Y /2L}1_{if and only ifV .Y /2L}1_{by Remark3.5, hence}

3.4. PROOF OFMAINRESULTS

Proof of Theorem2.1. We apply Fenchel’s theorem for.L1; ba/touB andıC. By Propo-sition3.8, dom.uB/ D L1anduB is continuous, hence epi.uB/has non-empty interior

w.r.t. the product topology ofL1R. Indeed,.0; uB.0/ 1/is an interior point of epi.uB/.

Also, dom.uB/\dom.ıC/D Chas an interior point, sinceL1_{C, andX 1is an}

interior point ofL1. Using (3.6), Lemma3.4, and (A4), sup X2L1 .uB.X / ıC.X //D sup X2CEŒU.XCB/ 1C_" "E " V dQx dP !# 1 "EŒU. "B C_/Csup X2CE x Q ŒX  1C" " E " V dQx dP !# 1 "EŒU. "B C_{/ <1:}

whereQxis an element ofMV(¤ ;by (A3)). Therefore, we can apply Fenchel’s theorem

to get sup X2C uB.X /D sup X2L1 .uB.X / ıC.X //D min 2ba.vB./Cı C.// D min 2QV E V d dP C d dPB Cı_C./ D min >0;Q2Me V E V dQ dP CdQ dP :

Here, the third equality follows from Proposition3.8, and the fourth from Lemma3.3and Lemma3.7. Now Theorem2.1follows from Lemma3.6. ¤ Proof of Corollary2.2. This is a direct consequence of the following minor modification of Kabanov and Stricker [11], Lemma 5.1. ¤

Lemma 3.12. Suppose thatU is bounded from above. Then for any ₂bb, there exists

a sequence.n_/

bsuch that.. n/S /T !0in probability and

(3.20) EŒU.ST CB/D lim n!1EŒU.

n

ST CB/:

Proof. SinceS is locally bounded, we can take a increasing sequence.n/nof stopping

times with S_n n, and n % T, stationarily, a.s. Then ..1J0;nK /S /T ! 0

in probability, for any ₂ L.S /. Thus, if S x, we have U. Sn

T CB/ !

U. ST CB/in probability, and this sequence is uniformly bounded from below (resp.

above) by _1C"_"U. ."_"C1/x/_C1C1_"U. .1_C"/B /₂L1(resp.U.₁/ <₁) by Lemma3.4 (b) and (A4). Hence the dominated convergence theorem shows that

lim

n!1EŒU.S

n

T CB/DEŒU. ST CB/:

This reduces the assertion to the case whereSis uniformly bounded by some constantc. Suppose thatSis uniformly bounded from below bya > 0. Set

Q

n_WD1_fjjng; nWDinfft WStng; nWDinfft W..Qn /S /t 1g ^T:

Note thatQn_Sn _{a 1. Indeed,}Qn_Sn _Sn _{1by the definition of} n, and

QnSn

hence Q nSn D QnSn CQnSn Sn ₁ C1_fjjngSn D1_fjjngSnC1fjj>ngSn 1a 1: Now letn_{WD Q}n₁ J0;n^nK. Then n_{S D Q}n_Sn^n _{a 1, and} nS_DnS _CnS _QnSn^n C1_fjjngSn^n Sn^n C1_C2cnn_C1_C2cn:

Hencen₂ b. On the other hand, we have..Qn /S /T D..1fjj>ng/S /T !0

in probability (note that ₂L.S /if and only if..1_fjjng/S /n2N is a Cauchy sequence

w.r.t. the semimartingale topology). This implies also thatP .n< T /!0( i.e.,n%T,

stationarily, a.s.), thus..n Qn/S /_{T !}0in probability. Hence..n /S /_{T !}0

in probability. Finally, sincenS is uniformly bounded from below bya 1, andU is bounded from above, we can use as above the dominated convergence theorem to conclude limn!1EŒU.nST CB/DEŒU.ST CB/. ¤

Proof of Corollary2.3. Let bb V. For any 2 , we have by Young’s

inequality, U.ST CB/V dQ dP CdQ dP.ST CB/; 8 > 0;8Q2MV; hence EŒU.STCB/E V dQ dP CdQ dPB CEQŒST E V dQ dP CdQ dPB ; _{8 2};₈ > 0;₈Q₂MV;

sinceSis a supermartingale under eachQ₂MV. Then Theorem2.1implies that

sup 2 EŒU. ST CB/ inf >0Q2infMV E V dQ dP CdQ dPB D sup 2bb EŒU.ST CB/;

The converse inequality follows from the inclusionbb . Finally, ifU.1/ <1, we

can replace allbb above byb, and the proof is complete. ¤

A

The author warmly thanks to Prof. Koichiro Takaoka for carefully reading the manuscript, and giving a number of valuable comments. Also, the financial support from the Global Center of Excellence program “the Research Unit for Statistical and Empirical Analysis in Social Sciences” of Hitotsubashi University is gratefully acknowledged.

R

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