Dynamically Consistent Nonlinear Evaluations and Expectations

0.52 MB·English

by  Shi-Ge Peng

DYNAMICALLY CONSISTENT NONLINEAR EVALUATIONS
AND EXPECTATIONS
5
0
0
SHIGEPENG
2

n
Abstract. How an agent (or a firm, an investor, a financial market)
a
J evaluatesacontingentclaim,sayaEuropeantypeofderivativesX,with
maturityt? Inthispaperwestudyadynamicevaluationofthisproblem.
4
2 We denote by {Ft}t≥0, the information acquired by this agent. The
value X is known at thematurity t means that X is an Ft–measurable
] random variable. WedenotebyEs,t[X] theevaluated valueof X at the
R time s ≤ t. Es,t[X] is Fs–measurable since his evaluation is based on
P his information at the time s. Thus Es,t[·] is an operator that maps an
. Ft–measurable random variable to an Fs–measurable one. A system
h of operators {Es,t[·]}0≤s≤t<∞ is called Ft–consistent evaluations if it
at satisfies the following conditions: (A1) Es,t[X] ≥ Es,t[Y], if X ≥ Y;
m (A2) Et,t[X] = X; (A3) Er,sEs,t[X] = Er,t[X], for r ≤ s ≤ t; (A4)
1AEs,t[X1A]=1AEs,t[X], if A∈Fs.
1 [ posIentthheessoit-ucaaltlieodngw–heevraeluFattiiosngedneefirnaetdedbbyyEasg,Bt[Xro]w:n=iayns,mwohtieorne,yweisptrhoe-
solution of thebackward stochastic differential equationwith generator
v
5 g and with the terminal condition yt = X. This g–evaluation satis-
fies (A1)–(A4). We also provide examples to determine the function
1
g=g(y,z) by testing.
4
01 evaTluhaetimonaiinsEregµsu–ldtoomfitnhaitsedp,aip.ee.r,i(sAa5s)fEosl,lto[wXs]:−iEfsa,t[gXiv′e]n≤FEtg–µc[oXns−istXen′]t,
5 for a large enough µ > 0, where gµ = µ(|y|+|z|), then Es.t[·] is a g–
evaluation
0
/
h
t
a
m Contents
:
v 1. Introduction 2
i 2. Basic setting and Eg–evaluations by BSDE 5
X
2.1. Basic setting 5
ar 2.2. Eg–evaluations induced by BSDE 7
3. Main result: E [·] is determined by a function g 8
s,t
4. A more general formulation: Eg [·;K]–evaluation 11
s,t
5. E [·;K] and related properties 16
s,t
Date:Firstversion: 16August2003, Thisversion31March2004.
1991 Mathematics Subject Classification. primary60H10.
Key words and phrases. BSDE, nonlinear expectation, nonlinear expected utilities,
g–expectation, nonlinear evaluation, g-martingale, nonlinear martingale, Doob-Meyer
decomposition.
Thisresearchissupported inpartbyTheNational Natural ScienceFoundation ofChinaNo.
10131040. Thisreversionis made after the author’s visit,during November 2003, toInstitute of
Mathematics and System Science, Academica Sinica, where he gives a series of lectures on this
paper. He thanks Zhiming Ma and Jia-an Yan, for their fruitful suggestions, critics and warm
encouragements. Healsothanks toClaudeDellacherieforhissuggestions andcritics.
1
2 SHIGEPENG
5.1. E [·;K] and it’s main properties 16
s,t
5.2. Two corollaries from Theorem 3.1 19
6. E[·;K]–martingales 20
7. BSDE under E[·] 24
8. E–supermartingale decomposition theorem: intrinsic formulation 26
9. Proof of Theorem 3.1 30
10. Proof of Theorem 8.2 and optional stopping theorem for E [·] 34
σ,τ
10.1. Simple case: Eg [·] with σ, τ ∈S0 34
σ,τ T
10.2. E [·] with σ, τ ∈S0 36
σ,τ T
10.3. S case: Proof of Theorem 8.2 and optional stopping theorem 41
T
References 45
1. Introduction
We are interested in the following dynamically consistent evaluation of risky
assets: Let η = (η ) be a d–dimensional process, it may be the prices of stocks
t t≥0
ina financialmarket,the ratesofexchanges,the ratesoflocalandglobalinflations
etc. We assume that at each time t ≥ 0, the information for of an agent (a firm,
a group of people or even a financial market) is the history of η during the time
interval [0,t]. Namely, his actual filtration is
F =σ{η ;s≤t}.
t s
WedenotethesetofallrealvaluedF –measurablerandomvariablesbymF . Under
t t
this notation an η–underlying derivative X, with maturity T ∈ [0,∞), is an F –
T
measurable random variable, i.e., X ∈ mF . We denote this evaluated value at
T
the time t by E [X]. Itis reasonabletoassume thatE [X]is F –measurable. In
t,T t,T t
other words,
E [X]:mF −→mF .
t,T T t
In particular
E [X]:mF −→R.
0,T T
We will make the following Axiomatic Conditions for (E [·]) :
t,T 0≤t≤T<∞
(A1) Monotonicity: E [X]≥E [X′],if X ≥X′;
t,T t,T
(A2) E [X]=X, ∀X ∈mF . ParticularlyE [c]=c;
T,T T 0,0
(A3) Dynamical consistency: E [E [X]]=E [X], if s≤t≤T;
s,t t,T s,T
(A4) “Zero–one law”: for each t≤T, 1 E [X]=1 E [1 X], ∀A∈F .
A t,T A t,T A t
or, more specially,
(A4’) for each t≤T, 1 E [X]=E [1 X], ∀A∈F .
A t,T t,T A t
Remark 1.0.1. The meaning of (A1) and (A2) are obvious. Condition (A3) means
thatthe evaluated valueE [X]canbe also treatedas a derivative with the matu-
t,T
rity t. At a time s ≤ t, the “price” of this derivative evaluated by E [E [X]] is
s,t t,T
the same as the “price” of the originalderivative X with maturity T, i.e., E [X].
s,T
Remark 1.0.2. Themeaningofcondition(A4)is: attimet,theagentknowswhether
η is in A. If η is in A, then the value E [X] is the same as E [1 X] since
·∧t ·∧t t,T t,T A
the two outcomes X and 1 X are exactly the same.
A
DYNAMICALLY CONSISTENT NONLINEAR EVALUATIONS AND EXPECTATIONS 3
It is clear that, to investigate this abstract evaluation problem, we need to in-
troduce some regulation condition of E. In this paper the information F will be
t
limited to the σ–filtration of some d–dimensional Brownianmotion, and X will be
assumed to be square–integrable,i.e., X ∈L2(F ).
T
A condition stronger than (A2) is:
(A2’) E [X]=X, ∀0≤s≤t, ∀X ∈mF .
s,t s
The meaning is that the market has zero interest rate for a non–risky asset
X. In this case we can define E[X|F ] := E [X], for a sufficiently large T, and
t t,T
E[X]:=E[X|F ]. It is easy to check that
0
E[1 E[X|F ]]=E[1 X].
A t A
E[X|F ] is called the E–conditional expectation of X under F . It satisfies all
t t
properties of a classical expectation, with one exception that it can be a nonlinear
operator. {E[X|F ]} is called an F –consistent nonlinear expectation.
t 0≤t≤T t
A typical filtration-consistent nonlinear expectation, called g-expectation and
denoted by {E [X|F ]} , was introduced in [28, Peng1997]. A significant fea-
g t 0≤t≤T
tureofthis g–expectationis thatthe value ofE [X|F ]is uniquelydeterminedbya
g t
simple function g(t,y,z)with g(t,y,0)≡0. In fact (E [X|F ]) is the solution
g t 0≤t≤T
of the backwardstochastic differential equation (BSDE in short)with the function
g as its generator and with X as its terminal condition at the terminal time T. It
is then not surprising that the behavior of E [·|F ] is entirely characterizedby this
g t
concrete function g. For example, E [·|F ] is a linear (conventional) expectation if
g t
and only if g is independent of y and is a linear function of z, i.e., g has a form
g =b ·z; E [X|F ] is concave (resp. convex)in X if and only if g is concave (resp.
t g t
convex)in(y,z),etc. Foraninterestingapplicationofg–expectationstotheutility
in stochastic continuous–time setting with ambiguity (or “model uncertainty” re-
ferredbyHansenandSargentandAnderson,Hansenand[1, Sargent],see[7,Chen
and Epstein, 2002].
g–expectations also have very interesting mathematical properties. A nonlinear
Doob–Meyer’sdecompositiontheoremforg–supermartingaleswasobtainedby[29,
Peng, 1999], for the case of Brownian filtration, and then by Chen and Peng 1998
[11, Chen & Peng, 1998]for a general filtration. In the case where the assumption
g(t,y,0) ≡ 0 does not hold, we have to denote the solution of the related BSDE
by Eg [X] instead of E [X|F ]. {Eg [·]} satisfies (A1)–(A4). Eg [·] can be
s,t g s s,t 0≤s≤t≤T s,t
applied to a wider situation in economics and finance.
TheapplicationofBSDEtothepricingofcontingentclaimsinafinancialmarket
was studied in [20, El Karoui et al., 1997]. Most of the results in [20] can be
interpreted in the language of Eg [·]. Other recent results in g–expectations are in
s,t
[5], [6], [8], [11], [12], [13], [28], [29], [30], [31], [32] where some cases are studied
in depth. For nonlinear evaluations, see [30, Peng, 2002], [31, Peng, 2003] and [32,
Peng, 2003]).
An interesting problem is: are the notions of g–expectations and g-evaluations
generalenough to representall “enoughregular”filtration-consistentnonlinear ex-
pectations and evaluations? In this case we can then concentrate ourselves to find
the corresponding function g which determine entirely the evaluation.
For the case of filtration–consistent expectations, we have partially solved the
problem in [8]: If the assumptions (A1)–(A4) plus (A2’) hold and if for a large
enough µ>0, the nonlinear expectation E[·] is dominated by the ‘gµ–expectation’
4 SHIGEPENG
Egµ[·] with g = µ|z|, and furthermore, if E[X +η|F ] = E[X|F ]+η for all F -
t t t
measurableη,then,thereexistsauniqueg,independentofy,suchthatE[·]=E [·].
g
The main objective of this paper is to prove this problem for the general case
of filtration–consistentevaluation: (see Theorem 3.1) if a filtration consistent eval-
uation {Es,t[·]}0≤s≤t≤T satisfies (A1)–(A4) plus the corresponding Egµ–dominated
conditions (see (A5) in Section 3), then the mechanism of this seemingly very ab-
stract evaluation E [·] can be entirely determined by a simple function g(t,y,z).
s,t
Thismeansthat,thereexistsaunique functiong suchthat,foreach0≤s≤t≤T
and for each X ∈ L2(F ), the value E [X] is the solution of the BSDE with
t s,t
generatorg andwith terminalconditionX. The resultofthis paper havenontriv-
ially generalized our previous result of [8]: condition (A2’) and “E [X +θ] = θ,
t,T
∀X ∈mF and η ∈mF ” are not at all required.
T t
Itisworthtopointoutthatthewell–knownBlack–Scholesoptionpricingformula
is a case where g is a linear function. But in our axiomatic condition (A1)–(A4)
aswell asthe regularitycondition(A5), neither the arbitragefree condition, which
is a principle argumentin Black–Scholestheory, nor utility maximizationhas been
involved. Another point is that the model of the price η of the underlying stocks
is not specified. This gives us a large freedom to determine the function g in each
specificsituation. Wealsoexplainhowthe functiong canbe determinedbysimply
testing the agent’s evaluation. This testing method is very useful to determine an
agent’s behavior under risk.
The paper is organizedas follows: in section 2, we give a rigoroussetting of the
notion of F –consistent nonlinear evaluation and its special case: F –expectations
t t
in subsection 2.1. We then give a concrete F –evaluation: Eg–evaluations in sub-
t
section 2.2. The main result, Theorem 3.1, will be presented in section 3. We also
provide some examples and explain how to find the function g through by test-
ing the input–output data. This main theorem will be proved in Section 9, with
several propositions served as lemmas for the proof given in Sections 4–10. Al-
though the whole paper is focused to proveTheorem 3.1, many preparativeresults
of this paper have their own interests, e.g., the existence and uniqueness of BSDE
under E (Theorem 7.1); the new nonlinear supermartingale decomposition theo-
rem of Doob–Meyer’s type (Theorem 8.1), using a new and intrinsic formulation
(see Remark 8.1.2 after Theorem 8.1). This decomposition theorem has also the
extension of E [·] to E [·] with stopping times σ and τ and the related optional
s,t σ,τ
stopping theorem (Theorem 8.2 and Theorem 10.19). Mathematically, some of
themaremorefundamentalthanTheorem3.1. Inparticular,thenonlineardecom-
position theorems of Doob–Meyer’s type, i.e., Proposition 4.10 and Theorem 10.2
play crucialroles in the proof of Theorem3.1. Theorem10.2 has alsoan intersting
interpretation in finance (see Remark 8.1.1).
Another application of the dynamical expectations and evaluations is to risk
measures. Axiomatic conditions for a (one step) coherent risk measure was intro-
ducedbyArtzner,Delbaen,EberandHeath1999[2]and,foraconvexriskmeasure,
by F¨ollmer and Schied (2002)citeFo-Sc. RosazzaGianin (2003)studied dynamical
risk measures using the notion of g–expectations in [Roazza2003] (see also [32]) in
which (B1)–(B4) are satisfied. In fact conditions (A1)-(A4), as well as their spe-
cial situation (B1)–(B4) (see Proposition 2.4) provides an ideal characterizationof
the dynamical behaviors of a the a risk measure. But in this paper we emphasis
DYNAMICALLY CONSISTENT NONLINEAR EVALUATIONS AND EXPECTATIONS 5
the study of the mechanism of the evaluation to a further payoff, for which is, in
general, the translation property in risk measure is not satisfied.
2. Basic setting and Eg–evaluations by BSDE
2.1. Basic setting. Let (Ω,F,P) be a probability space and let (B ) be a d–
t t≥0
dimensional Brownian motion defined in this space. We denote by (F ) the
t t≥0
natural filtration generated by B, i.e.,
F :=σ{σ{B , s≤t}∩N}.
t s
Here N is the collection of all P–null subsets. For each t∈[0,∞), we denote by
• L2(F ):={thespaceofallrealvaluedF –measurablerandomvariablessuch
t t
that E[|ξ|p]<∞}.
Definition 2.1. A system of operators:
E [X]:X ∈L2(F )→L2(F ), T ≤s≤t≤T
s,t t s 0 1
is called an F –consistent nonlinear evaluation defined on [T ,T ] if it satisfies the
t 0 1
following properties: for each T ≤r ≤s≤t and for each X, X′ ∈L2(F ),
0 t
(A1) E [X]≥E [X′], a.s., if X ≥X′, a.s.;
s,t s,t
(A2) E [X]=X, a.s.;
t,t
(A3) E [E [X]]=E [X], a.s.;
r,s s,t r,t
(A4) 1 E [X]=1 E [1 X], a.s. ∀A∈F .
A s,t A s,t A s
We will often consider (A1)–(A4) plus an additional condition:
(A4 ) E [0]=0, a.s. ∀0≤s≤t≤T.
0 s,t
Proposition 2.2. (A4) plus (A4 ) is equivalent to
0
(A4’) 1 E [X]=E [1 X], a.s. ∀A∈F .
A s,t s,t A s
Proof. It is clear that (A4’) implies (A4). E [0] ≡ 0 can be derived by putting
s,t
A=∅ in (A4’). On the other hand, (A4) plus the additional condition implies
1 E [1 X]=1 E [1 1 X]=0.
AC s,t A AC s,t Ac A
We thus have
E [1 X] = 1 1 E [X]+1 1 E [X]
s,t A AC A s,t A A s,t
= 1 E [X].
A s,t
(cid:3)
Proposition 2.3. (A4) is equivalent to, for each 0≤s≤t and X,X′ ∈L2(F ),
t
(2.1) E [1 X +1 X′]=1 E [X]+1 E [X′], a.s.∀A∈F .
s,t A AC A s,t AC s,t s
Proof. (A4) ⇒ (2.1): We let Y =1 X+1 X′. Then, by (A4)
A AC
1 E [Y]=1 E [1 Y]=1 E [1 X]=1 E [X].
A s,t A s,t A A s,t A A s,t
Similarly
1 E [Y]=1 E [1 Y]=1 E [1 X′]=1 E [X′].
AC s,t AC s,t AC AC s,t AC AC s,t
Thus (2.1) from 1 E [Y]+1 E [Y]=1 E [X]+1 E [X′].
A s,t AC s,t A s,t AC s,t
6 SHIGEPENG
(2.1) ⇒ (A4): It is simply because of
1 E [1 X] = 1 E [1 X +1 (1 X)]
A s,t A A s,t A AC A
= 1 (1 E [X]+1 E [1 X])
A A s,t AC s,t AC
= 1 E [X].
A s,t
(cid:3)
Remark 2.3.1. At time t, the agent knows the value of 1 . (A4) means that, if
A
1 =1 then the evaluated value E [1 X] should be the same as E [X] since the
A s,t A s,t
two outcomes X(ω) and (1 X)(ω) are exactly the same. (A4) is applied to the
A
evaluation of a final outcome X plus some “dividend” (D ) .
s s≥0
If, instead of (A2), we set
(A2’) E [X]=X, a.s., for each T ≤s≤t≤T, and X ∈L2(F ).
s,t 0 s
Then we define
E[X|F ]:=E [X], X ∈L2(F ).
t t,T T
WeobservethatthisnotiondescribesallE [X]since,whenX ∈L2(F ),E [X]=
s,t t s,t
E[X|F ].
t
Proposition 2.4. With (A2’), the system of operators
E[·|F ]:L2(F )→L2(F )
t T t
is a F –consistent nonlinear expectation, i.e., it satisfies, for each T ≤s≤t≤T,
t 0
and X ∈L2(F )
T
(B1) E[X|F ]≥E[X|F ], a.s., if X ≥X′, a.s.
t t
(B2) E[X|F ]=X, a.s., if X ∈L2(F );
t t
(B3) E[E[X|F ]|F ]=E[X|F ], a.s.;
t s s
(B4) E[1 X|F ]=1 E[X|F ], a.s. ∀A∈F .
A t A t t
Proof. (B1)–(B3) are easy. Since (A2’) implies E [0] = 0, thus, by Proposition
s,t
2.2, (A4’) and then (B4) holds. (cid:3)
We have the following immediate result
Proposition2.5. LetT <T <T <···<T begivenand,fori=0,1,2,··· ,N−
0 1 2 N
1, let
Ei [X]:X ∈L2(F )→L2(F ), T ≤s≤t≤T
s,t t s i i+1
be an F –consistent evaluation defined on [T ,T ] in the sense of Definition 2.1.
t i i+1
Then there exists a unique F –consistent evaluation E[·] defined on [T ,T ]
t 0 N
E [X]:X ∈L2(F )→L2(F ), T ≤s≤t≤T
s,t t s 0 N
such that, for each i=0,1,··· ,N −1, and for each T ≤s≤t≤T ,
i i+1
(2.2) E [X]=Ei [X], ∀X ∈L2(F ).
s,t s,t t
Proof. It suffices to prove the case N = 2. Because after we then can apply this
result to prove the cases [T ,T ] = [T ,T ]∪[T ,T ], ···, [T ,T ] = [T ,T ]∪
0 3 0 2 2 3 0 N 0 N−1
[T ,T ]. We define
N−1 N
(i) E1 [X], T ≤s≤t≤T ;
s,t 0 1
(2.3) Es,t[X]= (ii) Es2,t[X], T1 ≤s≤t≤T2;
 (iii) E1 [E2 [X]] T ≤s<T <t≤T .
s,T1 T1,t 1 1 2

DYNAMICALLY CONSISTENT NONLINEAR EVALUATIONS AND EXPECTATIONS 7
It is clear that, on [T ,T ], E [·] satisfies (A1) and (A2). To prove (A3) we only
0 2 s,t
need to check the relation
E [E [X]]=E [X], T ≤r ≤s≤t≤T
r,s s,t r,t 0 1
for two cases: T ≤ r ≤ s ≤ T ≤ t ≤ T and T ≤ r ≤ T ≤ s ≤ t ≤ T . For the
0 1 2 0 1 2
first case
E [E [X]] = E1 [E1 [E2 [X]]]
r,s s,t r,s s,T1 T1,t
= E1 [E2 [X]]
r,T1 T1,t
= E [X].
r,t
For the second case
E [E [X]] = E1 [E2 [E2 [X]]]
r,s s,t r,T1 T1,s s,t
= E1 [E2 [X]]
r,T1 T1,t
= E [X].
r,t
We now prove (A4). Again it suffices to check the case T ≤ s ≤ T ≤ t ≤ T .
0 1 2
In this case, for each A∈F ⊂F , (A4) is derived from
s T1
1 E [X] = 1 E1 [E2 [X]]
A s,t A s,T1 T1,t
= 1 E1 [1 E2 [X]]
A s,T1 A T1,t
= 1 E1 [E2 [1 X]]
A s,T1 T1,t A
= 1 E [1 X].
A s,t A
It remains to prove the uniqueness of E[·]. Let Ea[·] be an F –consistentevalua-
t
tion such that,
Ea [X]=Ei [X], ∀X ∈L2(F ), i=1,2.
s,t s,t t
Wethenhave,whenT ≤s≤t≤T andT ≤s≤t≤T ,Ea [X]≡E [X], ∀X ∈
0 1 1 2 s,t s,t
L2(F ). For the remaining case, i.e., T ≤s<T <t≤T , since Ea satisfies (A3),
t 0 1 1
Ea [X] = Ea [Ea [X]]
s,t s,T1 T1,t
= E1 [E2 [X]]
s,T1 T1,t
= E [X], ∀X ∈L2(F ).
s,t t
Thus Ea [·]=E [·]. This completes the proof. (cid:3)
s,t s,t
Remark 2.5.1. (i) In the remaining of this paper, we mainly consider the situation
t∈[0,T]for afixed T. The conclusionscanbe extended to [0,∞),using the above
Proposition. (ii) The argumentof the above Proposition2.5 can be also applied to
afiltrationdifferentfrom{F } ,e.g.,{F } ,whereτ isanF –stoppingtime.
t t≥0 t∧τ t≥0 t
2.2. Eg–evaluations induced by BSDE. In the remaining of this paper, we
limitedourselveswithinthetimeinterval[0,T]forsomefixedT >0. Theresultsof
this paper can be extended to the whole interval [0,∞), using Proposition2.5. We
need the following notations. Let p≥1 and τ ≤T be a given F –stopping time.
t
• Lp(F ;Rm) :={the space of all Rm–valued F –measurable random vari-
τ τ
ables such that E[|ξ|p]<∞};
• Lp(0,τ;Rm) :={Rm–valued and F –predictable stochastic processes such
F t
that E τ |φ |pdt<∞};
0 t
R
8 SHIGEPENG
• Dp(0,τ;Rm):={allRCLLprocessesinLp(0,τ;Rm)suchthatE[sup |φ |p]<
F F 0≤t≤τ t
∞};
• Sp(0,τ;Rm):={all continuous processes in Dp(0,τ;Rm) };
F F
• S :={the collection of all F –stopping times bounded by T};
T t
• S0 :={τ ∈ S and ∪n {τ = t } = Ω, with some deterministic 0 ≤ t <
T T i=1 i 1
···<t }.
N
In the case m=1, we denote them by Lp(F ), Lp(0,τ), Dp(0,τ) and Sp(0,τ).
τ F F F
We recall that all elements in D2(0,T) are F –predictable.
F t
For each given t∈[0,T] and X ∈L2(F ), we solve the following BSDE
t
t t
(2.4) Y =X + g(r,Y ,Z )dr− Z dB , s∈[0,t],
s r r r r
Z Z
s s
where the unknown is the pair of the adapted processes (Y,Z). Here the function
g :(ω,t,y,z)∈Ω×[0,T]×R×Rd →R
satisfies the following basic assumptions for each ∀y,y′ ∈R, z,z′ ∈Rd
(i) g(·,y,z)∈L2(0,T);
(2.5) F
(cid:26) (ii) |g(t,y,z)−g(t,y′,z′)|≤µ(|y−y′|+|z−z′|).
In some cases it is interesting to consider the following situation:
(a) g(·,0,0) ≡0,
(2.6)
(cid:26) (b) g(·,y,0) ≡0, ∀y ∈R.
Obviously (b) implies (a). This kind of BSDE was intrduced by Bismut [3], [4]
for the case where g is a linear function of (y,z). Pardoux and Peng [25] obtained
the following result (see Theorem 4.1 for a more general situation): for each X ∈
L2(F ), there exists a unique solution (Y,Z)∈S2(0,t)×L2(0,t;Rd) of the BSDE
t F F
(2.4).
Definition 2.6. We denote by Eg [X]:=Y , 0≤s≤t.
s,t s
We thus define a system of operators
Eg [·]:L2(F )→L2(F ), 0≤s≤t≤T.
s,t t s
Wewillprovethat(Eg [·]) formsanF –consistentevaluationon[0,T]. This
s,t 0≤s≤t≤T t
evaluation is entirely determined by the simple function g.
3. Main result: E [·] is determined by a function g
s,t
From now on the system E [·] : L2(F ) → L2(F ), 0 ≤ s ≤ t ≤ T, is always a
s,t t s
fixedF –consistentnonlinearevaluation,i.e.,satisfying(A1)–(A4),withadditional
t
assumptions (A40) and the following Egµ–domination assumption:
(A5) there exists a sufficiently large number µ>0 such that, for each0≤s≤t≤
T,
(3.1) E [X]−E [X′]≤Egµ[X−X′], ∀X,X′ ∈L2(F ),
s,t s,t s,t t
where the function g is
µ
(3.2) g (y,z):=µ|y|+µ|z|, (y,z)∈R×Rd.
µ
The main theorem of this paper is:
DYNAMICALLY CONSISTENT NONLINEAR EVALUATIONS AND EXPECTATIONS 9
Theorem 3.1. Let E [·] : L2(F ) → L2(F ), 0 ≤ s ≤ t ≤ T, satisfy (A1)–(A4),
s,t t s
(A4 ) and (A5). Then there exists a function g(ω,t,y,z) satisfying (2.5) with
0
g(s,0,0)≡0, such that, for each 0≤s≤t≤T,
(3.3) E [X]=Eg [X], ∀X ∈L2(F ).
s,t s,t t
Remark 3.1.1. The case where E [·] satisfy (A1)–(A5), without (A4 ), can be
s,t 0
obtained as corollaries of the this main theorem. This will be given in Corollaries
5.11 and 5.12. In this more general situation the condition g(s,0,0) ≡ 0 is not
imposed.
We consider some special situations of our theorem.
Example 3.2. If moreover, g(s,y,0) ≡ 0. Then, by [28], (A2’) holds. Thus,
accordingtoProposition 2.4, Eg [·]becomesanF –consistentnonlinearexpectation:
s,t t
E[X|F ]=E [X|F ]:=Eg [X]=Eg [X].
t g t s,t s,T
This is the so called g–expectation introduced in [28].
This extends non trivially the result obtained in [8], (see also [32] for a more
systematical presentation and explanations in finance), where we needed a more
strict domination condition plus the following assumption
E[X +η|F ]=E[X|F ]+η, ∀η ∈F .
t t t
Under these assumptions we have proved in [8] that there exists a unique function
g =g(s,z), with g(s,0)≡0, such that E [X]≡E[X]=E[X|F ].
g 0
Example 3.3. Consider a financial market consisting of d+1 assets: one bond
and d stocks. We denote by P (t) the price of the bond and by P (t) the price of the
0 i
i-th stock at time t. We assume that P is the solution of the ordinary differential
0
equation: dP (t)=r(t)P (t)dt, and {P }d is the solution of the following SDE
0 0 i i=1
d
dP (t) = P (t)[b (t)dt+ σ (t)dBj],
i i i ij t
j=1
X
P (0) = p , i=1,··· ,d.
i i
Here r is the interest rate of the bond; {b }d is the rate of the expected return,
i i=1
{σ }d the volatility of the stocks. We assume that r, b, σ and σ−1 are all F –
ij i,j=1 t
adapted and uniformly bounded processes on [0,∞). Black and Scholes have solved
theproblem of the market evaluation of an European type of derivative X ∈L2(F )
T
with maturity T. In the point of view of BSDE, the problem can be treated as
follows: consider an investor who has, at a time t ≤ T, n (t) bonds and n (t) i-
0 i
stocks, i=1,··· ,d, i.e., he invests n (t)P (t) in bond and π (t)=n (t)P (t) in the
0 0 i i i
i-th stock. π(t) = (π (t),··· ,π (t)), 0 ≤ t ≤ T is an Rd valued, square-integrable
1 d
and adapted process. We define by y(t) the investor’s wealth invested in the market
at time t:
d
y(t)=n (t)P (t)+ π (t).
0 0 i
i=1
X
We make the so called self–financing assumption: in the period [0,T], the investor
does not withdraw his money from, or put his money in his account y . Under this
t
condition, his wealth y(t) evolves according to
d
dy(t)=n (t)dP (t)+ n (t)dP (t).
0 0 i i
i=1
X
10 SHIGEPENG
or
d d
dy(t)=[r(t)y(t)+ (b (t)−r(t))π (t)]dt+ σ (t)π (t)dBj.
i i ij i t
i=1 i,j=1
X X
We denote g(t,y,z):=−r(t)y− d (b (t)−r(t))σ−1(t)z . Then, by the variable
i,j=1 i ij j
change z (t)= d σ (t)π (t),Pthe above equation is
j i=1 ij i
P
−dy(t)=g(t,y(t),z(t))dt−z(t)dB .
t
We observe that the function g satisfies (2.5). It follows from the existence and
uniqueness theorem of BSDE (Theorem 4.1) that for each derivative X ∈L2(F ),
T
there exists a unique solution (y(·),z(·))∈L2(0,T;R1+d) with the terminal condi-
F
tiony =X. Thismeaningissignificant: inordertoreplicatethederivative X,the
T
investorneedsandonlyneedstoinvesty(t)atthepresenttimetandthen,duringthe
time interval [t,T] and then to perform the portfolio strategy π (s) = σ−1(s)z (s).
i ij j
Furthermore,byComparison Theorem of BSDE,if hewantstoreplicateaX′ which
isbiggerthanX,(i.e., X′ ≥X,a.s., P(X′ ≥X)>0),thenhemustpaymore, i.e.,
there this no arbitrage opportunity. This y(t) is called the Black–Scholes price, or
Black–Scholes evaluation, of X at the time t. We define, as in (4.8), Eg [X]=y .
t,T t
We observe that the function g satisfies (b) of condition (2.6). It follows from
Proposition 4.6 that Eg [·] satisfies properties (A1)–(A4) for F –consistent evalua-
t,T t
tion.
Example 3.4. An very important problem is: if we know that the evaluation of
an investigated agent is a g–evaluation Eg, how to find this function g. We now
consider a casewhere g depends only on z,i.e., g =g(z):Rd →R. In this case we
can find such g by the following testing method. Let z¯∈ Rd be given. We denote
Y :=Eg [z¯(B −B )], s∈[t,T], where t is the present time. It is the solution of
s s,T T t
the following BSDE
T T
Y =z¯(B −B )+ g(Z )du− Z dB , s∈[t,T].
s T t u u u
Z Z
s s
T
It is seen that the solution is Y =z¯(B −B )+ g(z¯)ds, Z ≡z¯. Thus
s s t s s
Eg [z¯(B −B )]=Y =gR(z¯)(T −t),
t,T T t t
or
(3.4) g(z¯)=(T −t)−1Eg [z¯(B −B )].
t,T T t
Thus the function g can be tested as follows: at the present time t, we ask the
investigated agent to evaluate z¯(B −B ). We thus get Eg [z¯(B −B )]. Then
T t t,T T t
g(z¯) is obtained by (3.4).
Remark 3.4.1. The above test works also for the case g :[0,∞)×Rd →R, or for
a more general situation g =γy+g (t,z).
0
Aninterestingproblemis,ingeneral,howtofindthefunctiongthroughatesting
of the input–output behaviour of Eg[·]? Let b : Rn 7−→ Rn, σ¯ : Rn 7−→ Rn×d be
two Lipschitz functions.
s s
Xt,x =x+ b(Xt,x)dr+ σ(Xt,x)dB , s≥t.
s Z r Z r r
t t
The following result was obtained in Proposition 2.3 of [5].