and Quantitative Risk DOI 10.1186/s41546-017-0026-3
R E S E A R C H Open Access
Financial asset price bubbles under model
uncertainty
Francesca Biagini¡ Jacopo Mancin
Received: 10 January 2017 / Accepted: 3 December 2017 /
Š The Author(s). 2017Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Abstract We study the concept of financial bubbles in a market model endowed with a setPof probability measures, typically mutually singular to each other. In this setting, we investigate a dynamic version of robust superreplication, which we use to introduce the notions of bubble and robust fundamental value in a way consistent with the existing literature in the classical caseP = {P}. Finally, we provide concrete examples illustrating our results.
Keywords Financial bubbles¡Model uncertainty
AMS classification 60G48¡60G07
Introduction
Financial asset price bubbles have been intensively studied in the mathematical litera-ture in recent years. The description of a financial bubble is usually built on two main ingredients: the market price of an asset and its fundamental value. As the first is sim-ply observed by the agent, the modeling of the intrinsic value is where the differences between various approaches arise. In the classical setup, where models with one prior are considered, the main approach is given by the martingale theory of bubbles, see Biagini et al. (2014); Jarrow et al. (2010); Loewenstein and Willard (2000) and
J. Mancin ()¡F. Biagini
Workgroup Financial and Insurance Mathematics, Department of Mathematics, Ludwig-Maximilians Universit¨at, TheresienstraĂe 39, 80333 Munich, Germany
e-mail: mancin@math.lmu.de F. Biagini
Protter (2013). According to this theory, the fundamental value is defined as the expected sum of future discounted payoffs under a risk neutral measure. Other approaches within the martingale framework make use of further assumptions, such as portfolio constraints or defaultable claims, to explain how a bubble origi-nates in the market, see Biagini and Nedelcu (2015); Hugonnier (2012) and Jarrow et. al. (2012). Recently, however, another definition has been proposed in Herdegen and Schweizer (2016), where the fundamental value is assumed to be given by the superreplication price of the asset.
In this paper, we aim to contribute to the existing literature by introducing a frame-work for the formation of bubbles under model uncertainty. We consider a continuous (discounted) asset priceSin a market endowed with a setPof local martingale mea-sures forS, which are possibly mutually singular to each other and support a time consistent sublinear expectation. Each one of these priors is to be interpreted as a possible law for the dynamics ofS.
The market price will still be exogenous, but the fundamental valueSâwill have a fairly different interpretation from the classical literature and generate unexpected consequences. We describe in factSâby mean of theconditional sublinear expecta-tionof the terminal value ofS. By using the characterisation of sublinear operator of Theorem 1,Sârepresents in this way amaximalfundamental value, taking in account of each fundamental value under the different priors. Proposition 1 then shows that aggregation can be done on the set of probability measures having the same family of null sets up to timet, i.e. sharing the same view on the âimpossibleâ events up to timet. A financial interpretation and a strong link with the classical literature on bubble is then provided by Theorem 4, where we study the problem of dynamic superreplication, by extending Theorem 3.2 in Nutz (2015) to the dynamic case. This result provides an extension of the approach of Herdegen and Schweizer (2016) to the case of mutually singular priors.
The resulting bubble, defined asSâSâ, hasP-local submartingaledynamics under each P-market forP â P. This generalizes in a natural way the local-martingale dynamics which an asset price bubble displays in classical models under equiva-lent priors. Furthermore it allows to describe the birth of a bubble and its growth in size in astatic model, i.e. without changing the investorâs risk neutral measure over time, similarly to the approach of Herdegen and Schweizer (2016). The same submartingale behavior is in fact described for some cases also in Biagini et al. (2014), but it is the result of a smooth shift from an equivalent pricing measure to another. To the best of our knowledge this description of bubbles is new, as it dis-tinguishes itself also from the robust setting outlined in Cox et al. (2016), where bubbles arise as a consequence of constraints on possible trading strategies in a different setup.
Finally, we investigate the notion of no dominance, proposing its robust coun-terpart in the model uncertainty framework and studying its consequences on the concept of bubble.
The paper is organized as follows. In âThe settingâ section, we outline the notation and present an alternative characterization of the conditional sublinear expectation operator in Proposition 1. In âBubbles under uncertaintyâ section, after reviewing the existing literature, we discuss and study our concept of robust fundamental value, bubble under uncertainty, and robust no dominance, illustrating our results through concrete examples. In âInfinite time horizonâ section, we examine the situation in which the time horizon is not bounded. In the Appendix finally, we study the problem of dynamic superreplication.
The setting
We consider a familyP of probability measures, typically non-dominated, on= D0
R+,Rd, the space of c`adl`ag pathsĎ =(Ďs)sâĽ0inRd withĎ0 =0 endowed with the topology of weak convergence. We denote withFthe BorelĎ-field on. Given aF-measurable functionΞ, we are interested in sublinear expectations
Ξ âE0(Ξ):= sup
PâPEP[Ξ],
inducing time consistent conditional sublinear expectations. For this reason, some conditions have to be enforced both on the set of priors and on the random variables we take into account. Given a stopping timeĎof the filtrationF:= {Ft}tâĽ0generated by the canonical process, the main technical issue is to guarantee that the conditional sublinear expectation operator
EĎ(Ξ)= ess sup
PâP(Ď,P)EP[Ξ|FĎ] Pâa.s.for allPâP, (1)
where the essential supremum is taken with respect to the probability measurePand
P(Ď,P)denotes the set of probabilities{PâP : P=PonFĎ}first introduced in Soner et al. (2013), is well-defined. This problem is solved in the literature by means of different approaches, generally by shrinking the set of priorsP or by requiring strong regularity of the random variables, see Cohen (2012); Nutz and Soner (2012); Nutz and Van Handel (2013); Peng (2010), and Soner et al. (2011b). We choose to place ourselves in the context of Nutz (2015), as it generalizes the frameworks of G-expectation and randomG-expectation and provides some tractability of stopping times, which remains an open question in theG-setting.
For the sake of completeness, we then summarize the notation and the hypoth-esis we enforce on the setP, as stated in Nutz (2015), together with the notation introduced thereby. We start with the definition ofpolar set.
LetP() be the set of all probability measures on(,F) equipped with the topology of weak convergence. For any stopping timeĎ, the concatenation ofĎ,ĎË â
atĎ is the path
(ĎâĎĎ)Ë u:=Ďu1[0,Ď(Ď))(u)+ĎĎ(Ď)+ ËĎuâĎ(Ď)1[Ď(Ď),â)(u), uâĽ0.
Given a functionΞ onandĎâ, we define the functionΞĎ,Ďonby
ΞĎ,Ď(Ď)Ë :=Ξ(ĎâĎĎ),Ë ĎË â.
For any probability measurePâ P(), there exists a regular conditional proba-bility distribution{PĎĎ}ĎâgivenFĎ, i.e.,PĎĎ âP()for eachĎ, whileĎâPĎĎ(A) isFĎ-measurable for anyAâFand
EPĎĎ[Ξ] =EP[Ξ|FĎ](Ď) for Pâa.e. Ďâ,
whenever Ξ is F-measurable and bounded. Moreover, PĎĎ can be chosen to be concentrated on the set of paths that coincide withĎup to timeĎ(Ď),
PĎ
Ď{Ďâ: Ď=Ďon[0, Ď(Ď)]} =1 for allĎâ.
We define the probability measurePĎ,ĎâP()by
PĎ,Ď(A):=PĎĎ(ĎâĎ A), AâF, whereĎâĎ A:= {ĎâĎ ĎË : ËĎâ A}. We then have the the identities
EPĎ,Ď[ΞĎ,Ď] =EPĎĎ[Ξ] =EP[Ξ|FĎ](Ď) for Pâa.e. Ďâ. (2) We next recall, as in Nutz and Van Handel (2013), some basic notions from the theory of analytic sets. A subset of a Polish space is calledanalyticif it is the image of a Borel subset of another Polish space under a Borel-measurable mapping. In particular, any Borel set is analytic. The collection of analytic sets is stable under countable intersections and unions, but, in general, not under complementation and therefore does not constitute aĎ-algebra.
For each(s, Ď)âR+Ăwe fix a setP(s, Ď)âP(). Assume that
P(s, Ď)=P(s,Ď)Ë if Ď|[0,s]= ËĎ|[0,s],
andP(0, Ď)=Pfor allĎâ. We next define theuniversal completionof aĎ-field.
Definition 2Given aĎ-fieldG, the universal completion ofGis theĎ-fieldGâ=
âŠPGP, wherePranges over all probability measures onGandGPis the completion ofGunderP.
We then state Condition (A) from Nutz (2015).
Assumption 1Let(s,Ď)ÂŻ âR+Ă, letĎbe a stopping time such thatĎ âĽs and
PâP(s,Ď)ÂŻ . Setθ:=Ďs,ĎÂŻ âs.
(ii) Invariance: We havePθ,Ď âP(Ď,ĎÂŻ âsĎ)forP-a.eĎâ.
(iii) Stability under pasting: Ifν : â P() is aFθ-measurable kernel and
ν(Ď)âP(Ď,ĎÂŻ âsĎ)forP-a.eĎâ, then the measure defined by
ÂŻ
P(A)= (1A)θ,Ď(Ď)ν(dĎ;Ď)P(dĎ), AâF (3)
is an element ofP(s,Ď)ÂŻ .
Exploiting the previous conditions, by Proposition 3.1 in Nutz (2015) we have the following
Theorem 1LetĎ â¤Ďbe stopping times andΞ :â ÂŻRbe an upper semianalytic function1. Then, under Assumption1the function
EĎ(Ξ)(Ď):= sup
PâP(Ď,Ď)EP[Ξ
Ď,Ď], Ďâ (4)
isFĎâ-measurable and upper semianalytic. Moreover,
EĎ(Ξ)(Ď)=EĎ(EĎ(Ξ))(Ď) for all Ďâ. (5)
Furthermore,
EĎ(Ξ)= ess sup
PâP(Ď,P)EP[Ξ|FĎ] Pâa.s. for all PâP, (6)
whereP(Ď,P)= {PâP: P=PonFĎ}, and in particular,
EĎ(Ξ)= ess sup
PâP(Ď,P)EP[EĎ(Ξ)|FĎ] Pâa.s. for all PâP. (7)
We finally callP-martingale orrobust martingale, an adapted stochastic process M=(Ms)sâĽ0such thatE0(Mt)is finite for everytand
Mt =Et(MT) Pâq.s.
for anyT âĽt. The particularP-martingales for which also(âM)is aP-martingale are calledP-symmetric martingales.
We now provide another characterization of the conditional sublinear expectation given in (6). To this end, given an arbitraryPâPand a stopping timeĎ, we introduce the set
Peq(Ď,P):=
PâP : PâźPonFĎâP(Ď,P) (8) of the measures inPwhich areequivalenttoPrestricted toFĎ. In the case in which
P is composed by local martingale measures relative to some price processSand every priorPdescribes a complete market model, it holdsPeq(Ď,P)=P(Ď,P), but in general the inclusion in (8) is strict.
Proposition 1LetĎ be a stopping time andΞ : â ÂŻRan upper semianalytic function. Then under Assumption1for everyPâPit holds
EĎ(Ξ)= ess sup
PâPeq(Ď,P)EP[Ξ|FĎ] Pâa.s.
ProofIt is sufficient to prove ess sup
PâP(Ď,P)EP[Ξ|FĎ] âĽPess supâPeq(Ď,P)EP[Ξ|FĎ] Pâa.s. (9)
as the reverse inequality is evident. Let forPâP
AP:= {Ďâ|EĎ(Ξ)(Ď)= ess sup
PâP(Ď,P)EP[Ξ|FĎ](Ď)}.
It follows from Theorem 1 that
EĎ(Ξ)â ess sup
PâP(Ď,P)EP[Ξ|FĎ]
isFĎâ-measurable, so thatAPâFĎââFĎP. This implies that there existF âFĎ and NâNĎP(see Ash (1972) page 18), where
NĎP:= {Nâ| âC âFĎ such thatN âCandP(C)=0}, such thatAP=FâŞN and for everyQâPeq(Ď,P)
1=PAP=P(F)=Q(F)=QAP. This implies that for allQ,PâPeq(Ď,P)
EĎ(Ξ)⼠EP[Ξ|FĎ] Qâa.s. since by (4) and (2) it holds
EĎ(Ξ)⼠EP[Ξ|FĎ] Pâa.s. Hence we have
EĎ(Ξ)⼠ess sup
PâPeq(Ď,P)EP[Ξ|FĎ] Pâa.s. (10)
becauseEĎ(Ξ)is definedĎperĎin (3), i.e. it is independent of the choice ofQ,Pâ
Peq(Ď,P). Inequality (9) holds for every Q â Peq(Ď,P), i.e. in particular for the initially chosen probabilityPâP.
Example 1We illustrate the result of Proposition1in the case where the filtration is generated by a finite partition of. In particular, we consider a toy model with time horizon T = 2. Given a Polish space , we sett := t to be the t-fold
Cartesian product offor t â {0,1,2}with the convention that0is a singleton.
Let thenFT =B(T)andF1=Ď(A1, . . . ,An), where(Ai)i=1,...,nis a partition of
and nâN.
Similarly toBouchard and Nutz (2015)we assume that for each t â {0,1}and
Ďât we have a nonempty setPt(Ď):=P(t, Ď)âP()of probability measures
Pt(Ď)for allĎât. Therefore, we can consider a set of priorsPfor this two-period
market such that for everyPâPand AâFT it holds that
P(A)=
1A(Ď1, Ď2)P1(Ď1;dĎ2)P0(dĎ1),
whereĎ =(Ď1, Ď2)indicates a generic element inT andPt(¡)â Pt(¡), for t â
{0,1}. Alternatively stated, everyPâPis of the type
P=P0âP1, (11)
for someP0(¡)âP0(¡)andP1(¡)âP1(¡).
Given aFT-measurable functionΞand a priorPâP, it holds that
EP[Ξ|F1]=
n
i=1
EΞ1Ai
P(Ai)
1Ai. (12)
Every term on the right side of (12)can be further rewritten as E[Ξ1Ai]
P(Ai) =
1Ai(Ď1, Ď2)Ξ(Ď1, Ď2)P1(Ď1;dĎ2)P0(dĎ1)
1Ai(Ď1, Ď2)P1(Ď1;dĎ2)P0(dĎ1)
=
1Ai(Ď1)Ξ(Ď1, Ď2)P1(Ď1;dĎ2)P0(dĎ1)
1Ai(Ď1)P1(Ď1;dĎ2)P0(dĎ1)
(13)
=
Ai
Ξ(Ď1, Ď2)P1(Ď1;dĎ2)P0(dĎ1)
AiP0(dĎ1)
=
AiΞP1(Ď1)P0(dĎ1)
AiP0(dĎ1)
,
(14)
whereΞP1(Ď1):=
Ξ(Ď1, Ď2)P1(Ď1;dĎ2). AsΞP1(Ď1)is constant on Ai, we denote
for simplicityΞP1(Ď1)=ΞPAi
1 forĎ1â Ai. It follows from(14)that
E[Ξ1Ai]
P(Ai) =
ΞAi
P1
AiP0(dĎ1)
AiP0(dĎ1)
=ΞAi
P1,
from which EP[Ξ|F1] =ni=1Ξ
Ai
P11Ai. Because of (11), a probabilityP
âPcan be
written asP =P0âP1. This implies that, in order to haveP âP(1,P), we must enforce a constraint for the componentP0, whileP1is arbitrary. The same argument holds forPÂŻ âPeq(1,P). Hence it follows that
ess sup
PâP(1,P)
EP[Ξ|F1] = ess sup ¯
PâPeq(1,P)
EPÂŻ[Ξ|F1] Pâa.s.
for allPâP.
Finally we define the filtrationsG=(Gt)0â¤tâ¤T, where
Gt :=Ftââ¨NP
Bubbles under uncertainty
Here, we begin our study of financial bubbles under model uncertainty.In order to be consistent with the classical literature, we consider a discounted risky asset given by a non-negative,Rd-valued,Fâ-adapted, andP-q.s. continuous processS =(St)tâĽ0. LetĎ > 0 q.s. be a stopping time describing the maturity of the risky asset andXĎ
be the final payoff or liquidation value at timeĎ. The bank accountS0is assumed to be constant and equal to 1. The wealth processW =(Wt)tâĽ0generated from owning the asset is given by
Wt :=St1{Ď>t}+XĎ1{Ďâ¤t},
see also Jarrow et al. (2007).
In the standard literature on bubbles it is usually assumed that the No Free Lunch With Vanishing Risk condition (NFLVR) holds. When working in the context of multiple priors models the situation becomes more involved. In fact, there does not yet exist a robust counterpart to NFLVR. There is actually just one well stud-ied concept of arbitrage (arbitrage of the first kind (NA1)) in the continuous time setting under uncertainty introduced in Biagini et al. (2017). However in Biagini et al. (2017), the existence of absolutely continuous martingale measures requires introducing a stopping time Îś that causes a jump to a cemetery state, which is invisible under all P â P but may be finite under some Q â Q, where Q is an appropriate set of local martingale measures. Therefore, despite the possibility to start with a familyP of physical measures, the results of Biagini et al. (2017) require to reserve some particular care to the tractability of Îś. This is one of the reasons why, while working in the setting outlined, we assume the following for simplicity.
Assumption 2We consider a family P of probability measures, possibly non dominated, satisfying Assumption1and such that the wealth process is aQ-local martingale for everyQâ P. Thus, the setP is made of local martingale measures, enforcing NFLVR under allQ-market.
By doing this we guarantee at the same time thatW is justified by no arbitrage arguments under all probability scenarios.
In order to have a better understanding of what should be the right notion of asset fundamental value under model uncertainty, we first present a short survey on how this concept is modeled in the classical literature of financial bubbles.
For simplicity, we will start by considering a finite time horizon. Let thenT âR+ be such thatĎ â¤ T. We note that in this case, Wt = St for everyt â [0,T], if
XĎ =SĎ, which will be assumed throughout this section. Moreover, we assumeST to
be measurable with respect to the BorelĎ-field onT, in order to be able to compute
its conditional sublinear expectation.
H = (Ht)tâ[0,T] such that the stochastic integral process
t
0 Hsd Ss
tâ[0,T] is
well-defined2for allPâP.
Definition 3(see Nutz (2015), Section 3.2)We say that aG-predictable process H â L(S,P)is admissible if0t Hsd Ss
tâ[0,T] is aP-supermartingale for every
PâP. We denote byHthe set of all admissible processes.
Classical setting for bubble modeling
In the classical setting the elements of P are assumed to be all equivalent to a probability measureP, i.e., P = Mloc(S), whereMloc(S)denotes the set of the
equivalent local martingale measures forS. Under this assumption there are two main approaches for defining the fundamental value of a financial asset. In the setting of Biagini et al. (2014); Jarrow et al. (2007), and Jarrow et al. (2010), the fundamental value is defined as the assetâs discounted future payoffs under a risk neutral measure. This means that ifQâMloc(S), the fundamental valueSâ,Q=
Stâ,Q
tâ[0,T]under
Qis given by
Stâ,Q=EQ[ST|Ft]
for everytâ [0,T], whereTis a fixed finite time horizon. For anyQâMloc(S)the
non-negative process
βt :=StâStâ,Q
is called theQ-bubble. The concept of bubble depends on the following distinction
Mloc(S)=MU I(S)âŞMN U I(S),
whereMU I(S)is the class of measuresQâPsuch thatSis a uniformly integrable
martingale underQandMN U I(S)=Mloc(S)\MU I(S).
The market bubbliness thus is built upon the investorâs views: if she acts accord-ingly to a Q â MU I(W), then she would see no bubble; on the contrary, if
Q â MN U I(W)is perceived to be the right market view, then there would be an
asset bubble. In this sense, the concept of bubble is dynamic: bubbles are born or burst depending on how the investor changes her perspectives on the market.
If the market is complete the situation simplifies. AsMloc(S)is made of a unique
element (and it must exist if the usual NFLVR condition holds), either there is a bub-ble from the beginning or there is no bubbub-ble at all. This result agrees with the second main approach to financial bubbles (see Herdegen and Schweizer (2016) and the ref-erences therein), in which the fundamental value coincides with the superreplication
2t
0Hsd Ss denotes the usual stochastic (ItËo) integral under the probabilityP. Note that this definition
price. Denoting byL0+(Ft)the set ofFt-measurable random variables takingP-a.s.
values in[0,â), the superreplication price is given by
Ďt(S):=ess inf
v âL0
+(Ft) : âθ âwithv+
T
t θ
ud Su âĽST Pâa.s.
(15) fort â [0,T]anddenoting the class ofF-predictable andS-integrable processes. Given the duality (see Kramkov 1996)
Ďt(S)= ess sup
QâMloc(S)
EQ[ST|Ft], (16)
where the essential supremum of the family of random variables EQ[ST|Ft],Q â
Mloc(S), is taken under P, if there exists a bubble in a complete market, then the
superreplication price must be lower than the actual price of the asset observed in the market.
Differences between the two approaches emerge in the context of incomplete mar-kets. Given the superreplication duality (16), as soon asMU I(S)= â , then there is
no bubble. The concepts of bubble itself and bubble birth change. It might be that the superreplication price and the market price are equal att =0, but they may differ at a later timet >0 (see Example 3.7 in Herdegen and Schweizer (2016)): at timetis the bubble is born. Here the âbubble mispriceâ on an asset is given by the fact that we can get the same wealth at the end, but by exploiting an investment strategy with a lower initial price. Still, we cannot profit from this opportunity because its gener-ating strategy is not admissible: we need to go short on the asset and long on the superreplicating strategy to generate a sure profit at terminal time but by doing this we also face the risk of unbounded losses in(0,T). In this setting, if a bubble exists then it is perceived by any investor, independently from the particular choice of the pricing measureQâMloc(S).
Finally, we present an interesting result that links the two settings described above. We prove that if there is noQâ Mloc(S)that excludes the presence of a bubble in
the sense of Biagini et al. (2014) or Jarrow et al. (2010), then there is also a bubble in the case when fundamental values are given by superreplication prices.
Proposition 2Let S=(St)tâ[0,T]be a continuous, non-negative, adapted process
in a filtered probability space (,F,F,P), where the filtration F = (Ft)tâ[0,T]
satisfies the usual conditions. IfMloc(S) = MN U I(S), then there is a t â [0,T)
such that with positive probability
St > ess sup
QâMloc(S)
EQ[ST|Ft].
ProofWe argue by contradiction. If we suppose that
St = ess sup
QâMloc(S)
for everyt â [0,T], the processĎ(S)defined in (16) is aQ-local martingale for each
Qâ Mloc(S). This implies, according to Theorem 3.1 in Kramkov (1996), that the
minimal superreplicating portfolioV =(Vt)tâ[0,T], where
Vt = sup
QâMloc(S)
EQ[ST] +
t
0
Hsd Ss
andHis a predictable,S-integrable process, is non-negative, andself-financingand thus that there existsQÂŻ â Mloc(S)âŠMU I(S)(see Kramkov 1996, Section 3.2),
contradicting the hypothesis.
Remark 1It is also possible to obtain the same result when, instead of consider-ingMloc(S), we look at
P= {QâP()|QP, S is aQ-local martingale}. (17) In this context, ifPis m-stable3, it is possible to define the asset fundamental value as
Stâ=ess sup
QâP EQ[ST|Ft],
for any t â [0,T](for this result we refer to Delbaen (2006)). As the measures inP which are equivalent toPare dense inP (see again Delbaen (2006)), it holds
Stâ=ess sup
QâP EQ[ST|Ft] =Qess supâMloc(S)
EQ[ST|Ft],
so that Proposition2also applies in this case.
Robust fundamental value
We start this section with a discussion on suitable requirements for defining a bubble under uncertainty. We note that whenP =Mloc(S), the model should collapse into
one of the two approaches mentioned in the previous section. This already tells us that the fundamental value under uncertainty should be defined in terms of some conditional expectation.
A first attempt would be to define the existence of an asset bubble when there exists aQ-bubble for at least oneQâ P in the classical sense of âClassical setting for bubble modelingâ section. However, this would imply that any classical bubble will turn into a bubble under uncertainty. To overcome this problem, it is crucial to introduce a meaningful concept for the fundamental valueSâ = (Stâ)tâ[0,T]under
uncertainty. We could set
Stâ=EQ[ST|Ft] Qâa.s. (18)
3Pdefined in (17) is m-stable if for elementsQ0âP,QâPsuch thatQâźP, with associate martingales Z0
t =E
dQ0 dP|Ft
andZt=E
dQ
dP|Ft
, and for each stopping timeĎ, the elementLdefined asLt=Z0t fortâ¤ĎandLt=ZT0Zt/ZTfortâĽĎis a martingale that defines an element inP. We also assume that everyF0-measurable non-negative functionZ0such thatEP[Z0] =1, defines an elementdQ= Z0dP
for eachQ â P. In this way we would be consistent with the existing literature and recover the traditional setup of Jarrow et al. (2010) whenP consists of only one probability measure. This class ofQ-fundamental values cannot eventually be aggregated (this is already the case with theG-setting, see Soner et al. 2011a). In this setting, we could define a bubble as the situation in which
Q(Stâ<St) >0
for eachQâPand somet>0. Alternatively stated, we would say that the assetSis aP-bubble if it is aQ-bubble for everyQâP. However, this seems to be too strong of a requirement.
There is also a deeper issue, which is peculiar to the nature of our framework. In a market model that is intrinsically incomplete, it is not immediate to transpose the concept of ârisk neutral expectation of future discounted payoffsâ from the classical setting to the modeling under uncertainty, because of the absence of a linear pricing system. We refer to Beissner (2012) for a discussion on linear pricing systems and the fundamental theorem in the context of uncertain volatility. For a thorough analy-sis on the foundations of financial economics under Knightian Uncertainty, see also Burzoni et al. (2017). The first naive definition of fundamental value under uncer-tainty that we propose above is actually linked to this notion, as it coincides with the approach of Jarrow et al. (2010) whenPreduces to a singleton.
Definition 4We call robust fundamental value the process Sâ = (Stâ)tâ[0,T],
where
Sât = ess sup
QâP(t,Q)
EQ[ST|Ft], Qâa.s. (19)
for everyQâP, withP(t,Q)= {QâP : Q=QonFt}.
Theorem 1, Proposition 1, and Theorem 4 play a fundamental role in our specifi-cation of the robust fundamental value. Definition 4 is motivated by the intuition that in presence of uncertainty we should consider amaximalfundamental value, taking in account of each fundamental value under the different priors. This can be only achieved if aggregation of the conditional expectations is possible, which is guaran-teed by the results of Theorem 1. Furthermore, Theorem 1 ensures that the resulting robust fundamental value is sub-linear, which extends the classical concept of linear pricing operator. This sublinearity property plays a fundamental role in allowing an arbitrage-free setting, see again Beissner (2012) and Burzoni et al. (2017). We then obtain by Proposition 1 that the aggregation can be done on the set of priors having the same family of null sets up to timet, i.e. sharing the same view on the âimpossi-bleâ events up to timet. This result extends Theorem 1, where aggregation is done on the set of probability measures coinciding up to timet, i.e. exactly sharing the same view onallmeasurable sets up to timet. A financial interpretation and a link to the classical literature on bubble is then provided by Theorem 4, which shows whenSâ coincides with the superreplication value.
Definition 5(see Nutz 2015, Section 2) A setPof probability measures on(,F) is called saturated if the following condition is satisfied: for allP â P, if Qis a sigma-martingale measure for S equivalent toP, thenQâP.
If the familyP is saturated, we have that by Theorem 3.2 in Nutz (2015)
S0â=inf
xâR : â HâHwithx+
T
0
Hsd Ss âĽST Qâa.s.for allQâP
,
see also Appendix for further details. In this way our approach extends in a natural way the setting of Herdegen and Schweizer (2016) to the case of mutually singular priors.
Furthermore we also have a strong link to the literature in the classical case, as we show in the following proposition.
Proposition 3Let P = Mloc(S), then the robust fundamental value (19)
coincides with the classical superreplication price, i.e., ess sup
QâMloc(S)
EQ[ST|Ft] = ess sup
QâMloc(S,t,Q)
EQ[ST|Ft] q.s., (20)
whereMloc(S,t,Q):= {QâMloc(S)|Q=QonFt}.
ProofWe prove that ess sup
QâMloc(S,t,Q1)
EQ[ST|Ft] = ess sup
QâMloc(S,t,Q2)
EQ[ST|Ft] q.s. (21)
for everyQ1,Q2 â Mloc(S)by a measure pasting argument similar to Proposition
9.1 in Delbaen (2006). This suffices to conclude as
ess sup
QâMloc(S)
EQ[ST|Ft] ⼠ess sup
QâMloc(S,t,Q)
EQ[ST|Ft],
but (21) also guarantees
ess sup
QâMloc(S)
EQ[ST|Ft] ⤠ess sup
QâMloc(S)
ess sup
QâP(t,Q)
EQ[ST|Ft]
= ess sup
QâMloc(S,t,Q)
EQ[ST|Ft].
AssumeQ2âMloc(S)\Mloc(t,Q1), otherwise the claim is trivial. We note that
dQ dP
Ft =
dQi dP
Ft
:=Zit,
for everyQâ Mloc
S,t,Qi,i =1,2. This is clear as, for everyA â Ft, it must
hold
EP[Zit1A] =EQi[1A] =Qi(A)=Q(A)=EQ[1A] =EP
dQ dP
F
t
1A
Now, define
Zs :=Z1s1{sâ¤t}+Zt1
Zs2 Zt2
1{t<s},
which is the Radon-Nykodim derivative of an ELMM, as proven in Proposition 9.1 in Delbaen (2006). The measureQ associated to (Zs)sâ[0,T] thus belongs to
Mloc(S,t,Q1)and satisfies
EQ[ST|Ft] =
EP[STZT|Ft]
Zt =
EP
STZt1 Z2
T
Z2
t
Ft
Zt1
= EP
STZT2Ft
Z2t
=EQ2[ST|Ft].
This shows that for everyQâMlocS,t,Q2there exists aQâMlocS,t,Q1
such thatEQ[ST|Ft] =EQ[ST|Ft]. This is enough to establish (21).
Remark 2Note that in Proposition 3we can consider quasi-sure equalities as
P=Mloc(S).
We now introduce the notion of bubble in this setting.
Definition 6The asset price bubbleβ =(βt)tâ[0,T]for S is given by
βt :=St âStâ, (22)
where Sâis defined in(19).
It then follows that the asset price bubble is different from zero if there exists a stopping timeĎ such that
Q(SĎ >SĎâ) >0
for aQâ P. It is not necessary to have a bubble under all scenarios to have a bub-ble under uncertainty. The parallel with the notion of robust arbitrage now becomes evident. AsSis a non-negativeQ-local martingale, hence aQ-supermartingale, we have
St ⼠Stâ, Qâa.s.
for everyt â [0,T]andQâ P. There is now a bubble in the market at a stopping timeĎ if there exists a scenario (a probability measureQÂŻ âP) such that the robust fundamental value is smaller than the market value with positive probability and all probabilities that coincide withQÂŻ onFĎ agree on this view. In the case of no duality gap, our definition of bubble extends the approach where fundamental prices are given by superreplication prices as in Herdegen and Schweizer (2016) to the framework under uncertainty.
other cases the occurrence of a bubble may be caused either by a difference between the market price and the superreplication price or by a duality gap in
sup
QâPEQ[ST] â¤inf{xâR : âHâHwithx+
T
0
Hsd SsâĽST Qâa.s.for allQâP}.
This second situation is precisely the one considered in Cox et al. (2016) in order to detect a bubble. This means thatSâ can always be viewed at least as the worst model price, among the models considered by the investor.
Properties and examples
Lemma 1The bubbleβis a non-negativeQ-local submartingale for everyQâP, such thatβT =0q.s. Moreover, if there exists a bubble, S is not aP-martingale.
ProofThis immediately follows from Definition 4, asβis the difference between aQ-local martingale and aQ-supermartingale.
The local submartingale characterization is not at all a contradiction as it may seem at first sight. In fact, as opposed to non-negative local supermartingales, non-negative local submartingales do not have to be true submartingales. There are a variety of examples of local submartingales with nonstandard behavior, such as decreasing mean, as shown in Elworthy et al. (1999) and Pal and Protter (2010). To mention a clear example, it suffices to consider the class of non-negative local martingales: such processes are non-negative local submartingale and also supermartingales.
We present an example of a bubble by adapting one result of Cox and Hobson (2005) to the context ofG-expectation.
Remark 3We highlight that in Example 2,3, and 4the asset price S is a Q -local martingale for everyQ â P under the completed filtrationFQ =
FtQ
tâĽ0.
However, being a non-negative process adapted toFâ â FQ, S is also a Q-local martingale with respect to the filtrationFâ, thanks to a result from (Stricker 1977) that we report in the formulation of Theorem10from F¨ollmer and Protter(2011).
Theorem 2Let X be a non-negative local martingale forGand assume that X is adapted to the subfiltrationFâG. Then, X is also a local martingale forF.
Example 2LetP =PDas in Proposition7, whereD= [Ď2, Ď2] âR+\ {0}. Let
S0=s>0and
St =s+
t
0
Su
â
T âud Bu, t â [0,T). (23) The process in(23)is well defined for any tâ [0,Tâ], for >0, as a consequence of the results of Luo and Wang(2014). We show that S is a price process with a bubble by showing that S is a non-negativeQ-local martingale for everyQâP, with terminal value equal to zero. To this purpose, let us fix a priorQ. We have that
St =se
t
0Ďsd Bsâ12 t
The stochastic integral0¡1/âT âsd Bs is aQ-local martingale on[0,T), such
that the quadratic covariation is
¡
0
1/âT âsd Bs,
¡
0
1/âT âsd Bs
u
⼠âĎ2ln1â u
T
,
and continuous on[0,T). Using the same argument as in Lemma5from Jarrow et al. (2007), which exploits the Dubins-Schwarz theorem together with the law of the iterated logarithm, we can argue that
lim
uâT Su=0 Qâa.s. (24)
Hence, we set ST = 0so that S is q.s. continuous on [0,T]. This follows as
the set{Ď â : limuâT Su = 0}is polar: the existence ofQ â P such that
Q(limuâT Su = 0) > 0is in fact in contradiction with(24). Hence, S is not aQ
-martingale for anyQâ P as EQ[ST] = 0< EQ[S0], and in particular it is not a
robust martingale.
Another example comes from adapting the concept of the Bessel process to the context of model uncertainty.
Example 3We consider P = PD, whereD â R3Ă3 consists of the matrices (ai,j)i,j=1,2,3such that ai,j =0for all i = j , a1,1=a2,2=a3,3â [1,2]in order to
fix some values.
LetB = (Bt)tâ[0,T] =
Bt1,Bt2,Bt3tâ[0,T]. We consider the process given by f(B) = (f(Bt))tâ[0,T], where B0 = (1,0,0)and f(x) = xâ1. As f is
Borel-measurable, we can compute the sublinear expectationE0(f(Bt))for any t â [0,T],
according to Theorem1. It is a well known result that f(B)is a strictQ-local mar-tingale for allQâ PD(see for example Revuz and Yor1999, Exercise XI.1.16). To
prove that the price process has a bubble, it suffices to show that f(B)is not aPD
-martingale. This can be done using an argument from F¨ollmer and Protter(2011), where the inverse three-dimensional Bessel process is projected on the filtration gen-erated by the first component of the Brownian motion. Theorem14from F¨ollmer and Protter(2011)in particular proves that
EP
1
Bt
=2
1
â
t
â1, (25)
whereis the cumulative distribution function of a standard normal random vari-able. Using the invariance by rotation of the Brownian motion it is possible to show that, forWĎ =ĎBwhereĎ â RandWnow issued at a generic pointxâR3, the equality(25)generalizes to
EP
1
WĎt
= 1
x
2
x Ďât
â1
. (26)
Let us now consider a priorQâPDand a sequenceBnof processes such that
EQBT âBnT
2 2
andBnti âBntiâ1 âź N0, (ti âtiâ1)Ď2tiâ1
, where ti = T in andĎtiâ1is aD-valued, Ftiâ1-measurable function. We can now compute
EQ
1
BnT
=EQ
EQ
1
BnT Ftnâ1
=EQ
EQ
1
BnT âBntnâ1+B
n tnâ1
Ftnâ1
=EQ
1
Bntnâ1
2
Bn tnâ1 [Ďtnâ1]11
â
tnâtnâ1
â1
â¤EQ
1
Bntnâ1
2
Bn tnâ1 â
tnâtnâ1
â1
=EQ
EQ
1
ËBnT Ftnâ1
=EQ
1
ËBnT
,
where[Ďtnâ1]11stands for the first entry of the diagonal matrixĎtnâ1whose elements
different from zero are all equal andBËnis the process whereĎtnâ1is replaced by the
identity matrix inR3Ă3. We can iterate the previous computation by noting that
EQ
1
BnT
⤠EQ
1
ËBnT
=EQ
EQ
1
ËBnT Ftnâ2
=EQ
EQ
1
BnTâ1+BT âBtnâ1
âBnTâ2+BnTâ2 Ftnâ2
=EQ
1
Bntnâ2
2
Bn
tnâ2 [Ďtnâ2]11
â
tnâ1âtnâ2+âtnâtnâ1
â1
⤠EQ
1
Bntnâ2
2
Bn
tnâ2 â
tnâtnâ1+âtnâ1âtnâ2
â1
.
After n iterations we finally obtain EQ
1
BnT
â¤2
1
n i=1
â
ti âtiâ1
â1=2
1
â
nT
â1ââ0for nâ â. (28) We can use this result to show that
sup
QâPDEQ
1
BT
< sup
QâPDEQ
1
B0
=1 (29)
thus proving that f(B) is not a robust martingale. If we denote with (fm)mâN
a sequence of bounded and continuous functions converging increasingly to f, the convergence(27)ensures that
EQ
1 fm(Bn
T)
ââEQ
1 fm(B
T)
On the other hand, as f dominates fm and because of (28), EQfm(1B T)
<c, with 0< c < 1, for m big enough. It is then possible to use monotone convergence to prove
EQ
1 fm(B
T)
ââEQ
1
BT
<c for mâ â.
As this holds for everyQâPD, we obtain(29).
In both Example 2 and 3, the risky asset is a strictQ-local martingale for every
QâP. This means that the bubble in those cases is perceived under all priors which are possible in the model.
By slightly modifying the framework of Example 3, we are able to obtain a bubble that is aQÂŻ-martingale for a particularQÂŻ âP. Thus, despite the existence of a bubble, an investor endowed only with the priorQÂŻ would not detect it. This is one of the main novelties of our model: when a bubble arises it will be identified by an agent whose significative setsare those with positive probability under anyQâ P; alternatively stated, this agent considers negligible only thepolar sets, i.e., thoseAâFsuch that
Q(A) = 0 for all Q â P. However, an investor affected by less uncertainty, who neglects only theQÂŻ-null sets, will not spot the bubble.
Example 4We considerP =PDas in Example3, but now we chooseDin a way
to allow for a degenerate case, where there existsQÂŻ â P such that the canonical process is constantly equal to its initial value. We do this by considering the same setup as in Example3, but choosing a1,1 = a2,2 = a3,3 â [0,2]. Exactly as in
Example 3, f(B)is a Q-local martingale for every Q â P. However, under the âdegenerate priorâQÂŻ, associated to a volatility constantly equal to 0, every process turns deterministic. This implies in particular that f(Bt) = f(B0) =1QÂŻ-a.s. for
every tâ [0,T], which in turn ensures that f(B)is a trueQÂŻ-martingale, while being a strictQ-local martingale for allQâP\ { ÂŻQ}.
The examples regarding financial bubbles are usually obtained by showing spe-cific asset dynamics with strict local martingale behavior. We give here an example of a bubble in uncertainty setting by focusing our attention on the choice of probability measures considered in the uncertainty framework.
Example 5We adopt here the financial model introduced in Nutz and
Soner (2012). The major difference with respect to the setting presented in âThe settingâ section is that we consider the setPSof laws
QÎą:=Q
0âŚ(XÎą)â1, where XtÎą:=
t
0 Îą 1/2
s d Bs, t â [0,T]. (30)
In (30), Q0 denotes the Wiener measure, while Îą ranges over all the F
-progressively measurable processes with values inS+d satisfying0T |Îąs|ds < â
Q0-a.s. HereS+d âRdĂdrepresents the set of all strictly positive definite matrices
and the stochastic integral in(30)is the ItËo integral underQ0. The setP is asked to
Definition 7The setPis stable underF-pasting if for allQâP,Ď stopping time taking finitely many values,âFĎ andQ1,Q2âP(Ď,Q), the measureQÂŻ defined
by
ÂŻ
Q(A):=EQ[Q1(A|FĎ)1+Q2(A|FĎ)1c], AâFT (31) is again an element ofP.
Otherwise we leave all the other definitions stated in the preceding sections unchanged. Consider a risky asset S such that there exists aQÎąË â PSfor which the
asset is a strictQÎąË-local martingale. For example, we can take S to be the process described in Example2. We then study how a bubble is generated in this model. To this purpose we consider a subsetP âPSgiven by thoseQÎąâPSfor which
Îąs = ËÎąs for s â(t,T] Q0âa.s.
for some tâ(0,T). With such requirement, the setPis stable under pasting, accord-ing to Definition7, thanks to the same proof of Lemma3.3in Nutz and Soner(2012). In other words, we are considering a subset ofPSwhere there is no uncertainty after
time t, and where the volatility on(t,T]implies a strict local martingale behavior under at least one prior. It follows that, for every s>t,
ess sup
QâP(s,QÎąË)
EQ[ST|Fs] =EQÎąË[ST|Fs]<Ss,
thus implying the presence of a bubble.
We now investigate another interesting relation between robust and classical bubbles. The arguments in âRobust fundamental valueâ section clarified that the existence ofQâPandt â [0,T)such that
St > ess sup
QâP(t,Q)
EQ[ST|Ft]
implies the presence of a bubble in the classical sense for all theQ-markets with
QâP(t,Q). This is evident for at least two situations: when everyQ-market admits
a unique ELMM or when fundamental prices are described as the expected value of future discounted payoffs. However, we now show that the reverse is not true, i.e., the presence of a bubble with respect to someQâPdoes not determine the existence of a bubble under uncertainty. We do that by showing that set of priorsQâPfor which Sis a strict local martingale cannot be a singleton. To show this result we consider the setting outlined in Example 5. This choice allows at the same time to ease the computations as well as to infer some conclusions about the framework outlined in âThe settingâ section, as both models can describe theG-setting.
Proposition 4Consider the financial model introduced in Example5. IfQÂŻ is the pasting ofQ,Q1, andQ2at the stopping timeĎ andâFĎ, as in(31), it holds that
EQÂŻ[Y|FĎ] =EQEQ1[Y1|FĎ]|FĎ
+EQEQ2[Y1c|FĎ]|FĎ
(32)
for any positiveFT-measurable random variable Y and stopping timeĎ such that
ProofWe follow a procedure similar to Lemma 6.40 in F¨ollmer and Schied (2011) to prove (32). Let Ď be a stopping time andY a positive FT-measurable random
variable. By (31) we have that
EQÂŻ[Y] =EQEQ1[Y|FĎ]1+EQ2[Y|FĎ]1c
,
so that, for every positiveFĎ-measurable random variableĎ, we can study the value of
EQÂŻ[YĎ1{Ďâ¤Ď}]. (33)
The expectation in (33) can then be written as
EQÂŻ[YĎ1{Ďâ¤Ď}] =EQEQ1[YĎ1{Ďâ¤Ď}|FĎ]1+EQ2[YĎ1{Ďâ¤Ď}|FĎ]1c
=EQEQ1[YĎ1{Ďâ¤Ď}1|FĎ] +EQ2[YĎ1{Ďâ¤Ď}1c|FĎ]
=EQ
EQEQ1[Y1|FĎ]|FĎ
Ď1{Ďâ¤Ď}+EQEQ2[Y1c|FĎ]|FĎ
Ď1{Ďâ¤Ď}
=EQÂŻ
EQEQ1[Y1|FĎ]|FĎĎ1{Ďâ¤Ď}+EQEQ2[Y1c|FĎ]|FĎĎ1{Ďâ¤Ď}
=EQÂŻEQEQ1[Y1|FĎ]|FĎ+EQEQ2[Y1c|FĎ]|FĎĎ1{Ďâ¤Ď}.
(34) Hence, we can conclude that ifĎ â¤Ď the equality (32) holds.
No dominance
In this section, we investigate the implications of no dominance in our market model. This as a concept first appeared in Merton (1973). It is natural to transpose this concept to our setting with uncertainty, as we do in the following definition.
Definition 8Consider a market model under a set of priorsP. The i-th security Si is undominated on[0,T]if there is no admissible strategy H âHsuch that
S0i +
T
0
Hsd Ss âĽSTi Pâq.s.
and there exists aQâP such that
Q
Si0+
T
0
Hsd Ss >STi
>0.
A market satisfies robust no dominance (RND) on[0,T]if each Si, i â {1, . . . ,d}, is undominated on[0,T].
Definition 8 coincides with Definition 2 of Jarrow and Larsson (2012) for the classical situation in which a unique priorQexists.
Remark 4It is important to note that, as in the classical case, if Siis undominated on[0,T], it also undominated on[0,T], for T<T . Let Hibe given by
with1in position i. The trading strategy Hi is admissible, being Si aQ-local mar-tingale for everyQ â P. If there would be a dominating strategy H on[0,T], by applying the strategy K=H1{tâ¤T}+Hi1{t>T}, we would obtain
Si0+
T
0
Ksd Ss =STi +S0i + T
0
Hsd SsâSiT âĽSTi q.s.,
together with the existence of aQâP such that
Q
Si0+
T
0
Ksd Ss >STi
>0.
The ND assumption plays a key role in the classical literature on bubbles. We just mention two results by recalling that, if enforced, this concept rules out bubbles in the complete market models described by Jarrow et al. (2007); moreover, ND is precisely the ingredient needed to exclude bubbles in the setting of Herdegen and Schweizer (2016), where fundamental values are modeled with superreplication prices. Similar results can also be obtained in the present framework.
Lemma 2Suppose that for eachQâPtheQ-market model is complete. If robust no dominance holds, then there exists no bubble.
ProofObserve that, if each Q-market is complete, the results of Nutz (2015) guarantee the duality
sup
QâPEQ[ST] =inf{xâR : âH âHwithx+
T
0
Hsd Ss âĽST Qâa.s.for allQâP}.
(35) In the presence of a bubble, the superreplicating strategy would then dominateS, in contradiction with RND.
Hence, in the general case, under RND any bubble would be the result of a duality gap in (35), which is the case considered in Cox et al. (2016).
We remark how in general RND does not imply NFLVR for everyQ-market,Qâ
P. It is in fact well known that ND is stronger than NFLVR in the single prior setting, but it is a priori not necessary that RND implies ND for everyQ-market.
Infinite time horizon
ofSin the case{Ď = â}, we generalize the definition of robust fundamental value established in (19) by setting
Stâ=
ess sup
QâP(t,Q)
EQ[SĎ1{Ď<â}|Ft]
1{t<Ď}, Qâa.s. (36)
for everyt ⼠0 and Q â P. The fundamental value (36) includes the finite time horizon case (19) and is well defined, as shown in the following proposition.
Proposition 5The fundamental value (36) is well defined. In addition, Stâ§Ď
converges to SĎ q.s. for t â â.
ProofFixedQ â P, we know thatWt := Stâ§Ď,t ⼠0, is aQ-supermartingale,
which then convergesQ-a.s. toSĎ fort â â, because of the classical supermartin-gale convergence theorem (see Dellacherie et al. 1982, V.28 and VI.6). Therefore, Stâ§Ď âSĎq.s., thanks to the same argument used in Example 2, andSĎis Borel
mea-surable. Hence,SĎ1{Ď<â}is a Borel measurable random variable and we can compute its sublinear conditional expectation. Moreover, asWis a robust supermartingale, by Fatouâs Lemma we obtain
E0(SĎ)=E0
lim inf
tââ Stâ§Ď
= sup
QâPEQ
lim inf
tââ Stâ§Ď
⤠sup
QâPlim inftââ EQ(Stâ§Ď)
= sup
QâPlim inftââ EQ(Wt)â¤QsupâPEQ(W0) <â,
which guaranteesE0(SĎ1{Ď<â}) <â.
We also introduce the notion of robust fundamental wealth, by defining the process Wâ=(Wtâ)tâĽ0, where
Wtâ: =Stâ+SĎ1{Ďâ¤t}=
ess sup
QâP(t,Q)
EQ[SĎ1{Ď<â}|Ft]
1{t<Ď}+SĎ1{Ďâ¤t}
= ess sup
QâP(t,Q)
EQ[SĎ1{Ď<â}|Ft], Qâa.s.
(37)
for allQâP. A bubble is defined as in the finite time horizon case, i.e.,
βt =St âStâ=Wt âWtâ,
Example 6Let St =1for all tâR+be fiat money. Since money never matures,
we haveĎ = â, SĎ =1and Stâ=0q.s. for all tâĽ0. As
βt =StâStâ=1 q.s.
this means that the entire value of the asset comes from the bubble. We summarize these results in the following proposition.
Proposition 6It holds:
(i) In the case there exists aQÂŻ â P and t âĽ0such thatQÂŻ(Ď = â)= 1for all
ÂŻ
QâP(t,QÂŻ), there exists a bubble.
(ii) The bubbleβis aQ-local submartingale for everyQâP.
(iii) The wealth process W can be aP-symmetric martingale also in the presence of a bubble.
ProofThe proof of (i) follows from (37), noting that Wtâ=0 QÂŻ âa.s.,
as by hypothesis
SĎ1{Ď<â}=0 QÂŻâa.s.
for allQÂŻâP(t,QÂŻ). The local submartingale property follows from the definition of bubble and from Assumption 2. The wealth process can be aP-symmetric martingale as it can be seen in Example 6.
Appendix
Superhedging duality
In the framework of âThe settingâ section, we now study the problem ofrobust super-replicationof a contingent claim in a dynamical setting. The following results are of independent interest, but they also play an important role for studying financial bub-bles under model uncertainty, as explained in âRobust fundamental valueâ section. We consider, as done in Nutz (2015), an asset price processSwhich isRd-valued,
G+-adapted with c`adl`ag paths. We first present a brief recap regarding the duality result proved in Nutz (2015).
By Theorem 3.2 in Nutz (2015) we have that, if the setP is saturated
sup
PâPEP[f]=min{x âR: â HâHwithx+ T
0 Hsd Ss ⼠f Pâa.s.for allPâP}, (38) wheref is an upper semianalytic,GT-measurable function such thatE0(|f|) <â.
We begin by introducing some notations.