# An Interval of No-Arbitrage Prices in Financial Markets with Volatility Uncertainty

Authors: Hanlei Hu; Zheng Yin; Weipeng Yuan
Journal: Mathematical Problems in Engineering
 (2017)

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## Abstract

In financial markets with volatility uncertainty, we assume that their risks are caused by uncertain volatilities and their assets are effectively allocated in the risk-free asset and a risky stock, whose price process is supposed to follow a geometricG-Brownian motion rather than a classical Brownian motion. The concept of arbitrage is used to deal with this complex situation and we consider stock price dynamics with no-arbitrage opportunities. For general European contingent claims, we deduce the interval of no-arbitrage price and the clear results are derived in the Markovian case.


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## Body

## 1. Introduction

Though many choice situations show uncertainty, owing to the Ellsberg Parasox, the impacts of ambiguity aversion on economic decisions are established and Beissner [1] considered general equilibrium economies with a primitive uncertainty model that features ambiguity about continuous-time volatility. Under uncertainty, multiple priors can be used to model decisions. Recently, these multiple priors models have attracted much attention. The decision theoretical setting of multiple priors was introduced by Gilboa and Schmeidler [2] and Artzner et al. [3] adapted it to monetary risk measures. Afterwards, Maccheroni et al. [4] generalized multiple priors to preferences. In diffusion models, Girsanov’s theorem was employed to consider stochastic processes by Chen and Epstein [5], but these multiple priors can only lead to uncertainty. When these multiple priors are used in finance areas, they result in drift uncertainty for stock prices. In the risk-neutral world, when we assess financial claims, the uncertainty of this drift will disappear.Under the assumption of no arbitrage and volatility uncertainty, Fernholz and Karatzas [6] considered to outperform the market. Compared with this, our paper is to model volatility uncertain financial markets which have no arbitrage. Epstein and Ji [7] or Vorbrink [8] used a specific example to illustrate an uncertain volatility model. On the basis of our predecessors, our paper solves a few basic problems of the volatility uncertainty in finance markets. Our aim is to analyze the volatility uncertain financial markets and we take advantage of the framework of sublinear expectation and G-Brownian motion which is introduced by Peng [9] to deal with the model in financial markets. The G-Brownian motion is no longer a classical Brownian motion. The construction of stochastic integration, Ito^’s lemma, and martingale theory is utilized to the framework of G-Brownian motion. In order to control the model risk, the G-Brownian motion is employed to concern the model and evaluate claims by means of G-expectation which is a sublinear expectation.In our financial markets with volatility uncertainty, the wealth is invested in risk-free asset and risky asset, in which the risky asset, that is, stockSt and its price process St, is given by the following geometric G-Brownian motion:(1)dSt=rStdt+νtStdBt,S0=x0>0,where constant interest rate r⩾0 is an expected instantaneous return of the stock and νt is the volatility of S which is associated with t. The canonical process B=Bt is a G-Brownian motion relating to a sublinear expectation EG, called G-expectation (see [9, 10] for a detailed construction). The stochastic calculus with respect to G-Brownian motion can also be established, especially Ito^ integral [9]. The ordinary martingales are replaced by G-martingales. Denis et al. [11] developed the G-framework of Peng [10] (see [12]) in the framework of quasi-sure analysis. An upper expectation of classical expectations is used to represent the sublinear expectation EG established by Denis et al. [11]; that is to say, there exists a set of probability measures P such that EGX=supP∈P⁡EPX.In this paper, we prove that the considered financial market does not admit any arbitrage opportunity, but it allows for uncertain volatility. In our analysis, the notion ofG-martingale which replaces the notion of martingale in classical probability theory plays a major role.One of our aims is to solve(2)supP∈P⁡EPDTVT,supP∈P⁡EP-DTVT,where VT denotes the payoff of contingent claims at maturity T and DT is a discounting. P presents a series of different probability measures.The stochastic environment can bring about a set of probability measures that are not equivalent but even mutually singular. To illustrate this, letB be a Brownian motion under a measure P and think about the processes Sσ_≔σ_Bt and Sσ¯≔σ¯Bt. Using Pσ_=P∘Sσ_-1 and Pσ¯=P∘Sσ¯-1, we describe the distributions over continuous trajectories which are induced by the two processes. These measures describe two possible hypotheses of real probability measure which drives the volatility uncertainty by (1). Therefore, we have(3)Pσ_BT=σ_2T=1=Pσ¯BT=σ¯2T,where both priors are mutually singular.The definition of trading strategy and portfolio process is applied to obtain the wealth equation. Defining the concept of no-arbitrage in financial markets and the hedging classes, we gain the interval of no-arbitrage price for general European contingent claims. Finally, the connection of the lower and upper arbitrage prices is presented.In such an ambiguous financial market, our subject is to analyze the European contingent claim concerning pricing and hedging. The asset pricing is extended to the financial markets with volatility uncertainty. The notion of no-arbitrage plays an important role in our analysis. Owing to the fact that the volatility uncertainty leads to additional source of risk, the classical definition of arbitrage will no longer be adequate. For this reason, a new arbitrage definition is presented to adjust our multiple priors model with mutually singular priors which are shown in (3). In this modified sense, we confirm that our volatility uncertain financial markets do not admit any arbitrage opportunity.Utilizing the notion of no-arbitrage, we have obtained several results, which provide us with a better economic understanding of financial markets under volatility uncertainty. For general contingent claims, we determine an interval of no-arbitrage prices. The bounds of this interval are the upper and lower arbitrage pricesvup and vlow, which are obtained as the expected value of the claim’s discounted payoff with respect to G-expectation (see (2)). They specify the lowest initial capital. We use the capital to hedge a short position in the claim or long position, respectively. Generally speaking, because EG is a sublinear expectation, we have vlow≠vup. This verifies the market’s incompleteness. In a few words, no arbitrage will be generated when price is in the interval vlow,vup for a European contingent claim. In Section 4, when the contingent claim’s payoff is only determined by the current stock price, we deduce a more clear structure about the upper and lower arbitrage prices by a partial differential equation (PDE for short). We derive an explicit representation for the corresponding supper-hedging strategies and consumption plans. Given the special situation when the payoff function shows convexity (concavity), the upper arbitrage price solves the classical Black-Scholes PDE with a volatility equal to σ_(σ¯), and vice versa concerning the lower arbitrage price.The novelties of this paper are that the volatility ofS in our model is a variable which is related to t. This is different from works of Vorbrink [8] in which the volatility of S is a constant. We employ the G-framework including G-expectation, G-Brownian motion, and the concept of arbitrage to study the financial markets with volatility uncertainty; we gain the interval of no arbitrage, which is different from that in Denis and Martini [12].This paper is organized as follows. Section2 introduces the financial markets. We focus on and extend the terminology from mathematical finance. Section 3 applies a series of definitions and lemmas to derive the interval of no arbitrage. Section 4 restricts us to the Markovian case and derives results which are analogy to those in Avellaneda et al. [13] or Vorbrink [8]. Conclusions are given in Section 5.


## 2. The Market Model and the Mathematical Setting

### 2.1.G-Brownian Motion and the Multiple Priors Setting

In the whole paper, the one-dimensional case is considered and we fix an intervalσ_,σ¯ with σ_>0. This interval describes the volatility uncertainty. σ_ and σ¯ denote a lower and upper bound for volatility, respectively.Definition 1 (see [9]).
LetΩ≠∅ be a given set. Let H be a linear space of real valued functions defined on Ω with c∈H for all constants c, and X∈H if X∈H. (H is considered as the space of random variables.) A sublinear expectation E^ on H is a functional E^:H→R satisfying the following properties: for any X,Y∈H, it has(1)
Monotonicity: ifX⩾Y then E^X⩾E^Y.(2)
Constant preserving:E^c=c.(3)
Subadditivity:E^X+Y⩽E^X+E^Y.(4)
Positive homogeneity:E^μX=μE^X∀μ⩾0.The triple Ω,H,E^ is called a sublinear expectation space.Definition 2 (see [10] (G-normal distribution)).
In a sublinear expectation spaceΩ,H,E^, a random variable X is called (centralized) G-normal distributed if for any a,b⩾0(4)aX+bX¯~a2+b2X,where X¯ is an independent copy of X. Here the letter G denotes the function (5)Ga≔12E^aX2:R⟶R.Note thatX has no mean-uncertainty; that is, it has E^X=E^-X=0. Moreover, the following important identity holds: (6)Ga=12σ¯2a+-12σ_2a-with σ_2≔-E^-X2 and σ¯2≔E^X2. We write that X is N0×σ_2,σ¯2 distributed. Therefore, we say that G-normal distribution is characterized by the parameters 0<σ_⩽σ¯.Remark 3 (see [10]).
The random variableX which is defined in (4) is generated by the following parabolic PDE defined in 0,T×R.
For anyCl,LipR, define ut,x≔E^φx+tX; then u is the unique (viscosity) solution of(7)∂tu-G∂xxu=0,ut=0=φ.Equation (7) is called a G-equation.Definition 4 (see [10] (G-Brownian motion)).
A processBtt⩾0 in a sublinear expectation space Ω,H,E^ is called a G-Brownian motion if the following properties are satisfied: (i)
B0=0.(ii)
For eacht,s⩾0 the increment Bt+s-Bt is N0×σ_2s,σ¯2s distributed and independent from Bt1,Bt2,…,Btn for each n∈N, 0⩽t1⩽⋯tn⩽t.Condition (ii) can be replaced by the following three conditions giving a characterization of G-Brownian motion: (i)
For eacht,s⩾0:Bt+s-Bt~Bt and E^Bt3/t→0 as t→0.(ii)
The incrementBt+s-Bt is independent from Bt1,Bt2,…,Btn for each n∈N, 0⩽t1⩽⋯tn⩽t.(iii)
E^Bt=-E^-Bt=0,∀t⩾0.For eacht0>0, it has that Bt+t0-Bt0t⩾0 is a G-Brownian motion.Let us briefly depict the construction ofG-expectation and its corresponding G-Brownian motion. As in the previous sections, we fix a time horizon T>0 and set ΩT=C00,T,R-the space of all real valued continuous paths starting at zero. Considering the canonical process Btω≔ωt,t⩽T,ω∈Ω, we define (8)LipΩT≔φBt1,…,Btn∣n∈N,t1,…,tn∈0,T,φ∈Cl,LipRn.A G-Brownian motion is firstly constructed in LipΩT. For this purpose, let ξii∈N be a sequence of random variables in a sublinear expectation space Ω~,H~,E~ such that ξi is G-normal distributed and ξi+1 is independent of ξ1,…,ξi for each integer i⩾1. Then a sublinear expectation in LipΩT is constructed by the following procedure: for each X∈LipΩT with X=φBt1-Bt0,Bt2-Bt1,…,Btn-Btn-1 for some φ∈Cl,LipRn, 0⩽t0<t1<⋯<tn⩽T, set (9)EGφBt1-Bt0,Bt2-Bt1,…,Btn-Btn-1≔E~φt1-t0ξ1,t2-t1ξ2,…,tn-tn-1ξn.The related conditional expectation ofX∈LipΩT as above under Ωti,i∈N, is defined by (10)EGφBt1-Bt0,Bt2-Bt1,…,Btn-Btn-1∣Ωti≔ψBt1-Bt0,…,Bti-Bti-1,where ψx1,…,xi≔E~[φ(x1,…,xi,ti+1-tiξi+1, …,tn-tn-1ξn)]. One checks that EG consistently defines a sublinear expectation in LipΩT and the canonical process B represents a G-Brownian motion.LetΘ≔σ_,σ¯ and A0,TΘ be the collection of all Θ-valued Ft-adapted processes on 0,T. We write (11)Bt0,σ≔∫0tσsdBs,and Pσ as the law of B0,σ=∫0·σsdBs‍; that is, Pσ=P0∘B0,σ-1 is distribution over trajectories. Let the set of multiple priors P be the closure of {Pσ∣σ∈A0,TΘ} under the topology of weak convergence.Theorem 5 (see [8]).
For anyφ∈Cl,LipRn,n∈N, 0⩽t1⩽⋯⩽tn⩽T, it holds that (12)EGφBt1,…,Btn-Btn-1=supθ∈A0,TΘ⁡EPφBt10,θ,…,Btntn-1,θ=supθ∈A0,TΘ⁡EPθφBt1,…,Btn-Btn-1=supPθ∈P⁡EPθφBt1,…,Btn-Btn-1.Furthermore,(13)EGX=supPθ∈P⁡EPX,∀X∈LG1ΩT.We use the set of priorsP to define the G-expectation EG. It is given by (14)EGX=supP∈P⁡EPX,where X is any random variable. So the G-expectation can be defined. Relative to the G-expectation, the space of random variable is denoted by LG1ΩT.In this paper, we consider the tuple asΩT,F,Ft,P and the canonical process B=Bt is a G-expectation motion with respect to P as given in the previous. The G-framework enables the analysis of stochastic processes for all priors of P. The terminology of“quasi-surely” (q.s.) is proved to be very useful.Unless there are special instructions, all equations should also be understood as “quasi-sure.” This means a property almost surely for all conceivable scenarios.As mentioned in the preceding,G-expectation can be defined in the space LGpΩT,p⩾1. It is the completion of CbΩT, the set of bounded continuous functions on ΩT under the norm ξ≔EGξ21/2<∞. Because the stochastic integrals are required to define trading strategies in the next sections, we briefly introduce the basic concepts about stochastic calculus and the construction of Itô integral with respect to G-Brownian motion.Forp⩾1, let MGp,00,T be the collection of simple processes η of the following form: for a given partition 0,T, t0,t1,…,tN,N∈N, for any t∈0,T the process η is defined by(15)ηtω≔∑j=0N-1ξiω1tj,tj+1t,where ξiω∈LGpΩti,i=0,1,…,N-1. For each η∈MGp,00,T, let (16)ηMGp≔EG∫0Tηspds1/p.We denote by MGp,00,T the completion of MGp,00,T under the norm ·MGp.Definition 6 (see [14]).
Forη∈MG2,00,T with the presentation in (15), the integral mapping is defined by I:MG2,00,T→LG2ΩT and (17)Iη=∫0TηsdBs≔∑j=0N-1ξiBtj+1-Btj.We consider the quadratic variation processBt of G-Brownian motion. It has (18)Bt=Bt2-2∫0tBsdBs,∀t⩽T.It is a continuous, increasing process, absolutely continuous with respect to dt. It contains all the statistical uncertainty of the G-Brownian motion. For s,t⩾0 we have Bs+t-Bs~Bt and it is independent of Ωs.Definition 7 (see [9]).
Letx∈R, z∈MG20,T and η∈MG10,T. Then the process (19)Mt≔x+∫0tzsdBs+∫0tηsdBs-∫0t2Gηsds,t⩽T,is a G-martingale.Specially, the nonsymmetric part-Kt≔∫0tηsdBs‍-∫0t2Gηsds,t∈0,T, is a G-martingale, which is quite a surprising result because -Kt is continuous, nonincreasing with quadratic variation equal to zero.Remark 8 (see [15]).
M is a symmetric G-martingale if and only if K≡0.Theorem 9 (see [15] (martingale representation)).
Letα⩾0 and ξ∈LGαΩT. Then the G-martingale X with Xt≔EGξ∣Ft,t∈0,T, has the following unique representation: (20)Xt=X0+∫0tzsdBs-Kt,where K is a continuous, increasing process with K0=0, KT∈LGβΩT, z∈HGβ0,T,∀β∈1,α, and -K a G-martingale.Ifα=2 and ξ bounded from above, z∈MG20,T and KT∈LG2ΩT (see [14]).A construction of the stochastic integral for the domainHGp0,T,p⩾1 is established by Song [15]. Although the structure of these spaces is similar as before, the norm for completion is different and the random variables ξiω in (15) are elements of a subset of LGpΩti. We will also use the domain HG10,T which is necessary for the martingale representation in the G-framework (see Theorem 9). For p=2, both domains coincide (see Song [15]). As a consequence, we can define the stochastic integral since MG20,T is contained in HG10,T. In financial fields, more trading strategies will be feasible.


### 2.2. The Financial Market Model

We consider the following financial marketM which includes a risk-free asset and a single risky asset and two assets are traded continuously over 0,T. Assume that the risk-free asset is a bond and its interest rate is r. So the discount process Dt can be defined to satisfy the following formula:(21)dDt=-rDtdt,D0=1,where constant r⩾0 is the interest rate of the riskless bond as in the classical theory.Assume that the risky asset is a stock with priceSt at time t, whose price process St is given by the following equation:(22)dSt=rStdt+νtStdBt,S0=x0>0,where B=Bt denotes the canonical process which is a G-Brownian motion under EG or P, respectively, with parameters σ¯>σ_>0.SinceB=Bt is a G-Brownian motion, the volatility of S is related to t which is different from that of Vorbrink [8], where the volatility of stock price is a constant 1. Consequently, the stock price evolution involves not only risk modeled by the noise part but also ambiguity about the risk due to the unknown deviation of the process B from its mean. According to financial fields, this ambiguity is called volatility uncertainty.Compared with the classical stock price process, (22) does not contain any volatility parameter σ. This is due to the characteristics of the G-Brownian motion B. Apparently, if we choose σ=σ_=σ¯, then we will be in the classical Black-Scholes model.Remark 10.
Take notice of the discounted stock price processDtSt which is a symmetric G-martingale relative to the corresponding G-expectation EG. As everyone knows, both the pricing and hedging of contingent claims are treated under a risk-neutral measure. This leads to a favorable situation in which the discounted stock price process is a (local) martingale [16]. In our ambiguous setting, this is also allowed. In order to model DtSt as a symmetric G-martingale (see Definition 7), we do not need to change the sublinear expectation. A symmetric G-martingale is required to make sure that the stock is the same for all participants, whether they sell or buy.Definition 11 (see [17]).
In the marketM, a trading strategy is an Ft-adapted vector process α,β=αt,βt, β a member of HG10,T such that βtSt∈HG10,T, and αt∈R for all t⩽T.
A cumulative consumption processC=Ct,0⩽t⩽T is a nonnegative Ft-adapted process with values in LG1ΩT, and with increasing, right-continuous, and C0=0, CT<∞ q.s.A basic assumption in the marketM is that the stock price process St defined by (22) is an element of MG20,T=HG20,T. We impose the so-called self-financing condition. In other words, consumption and trading in M satisfy(23)Ht≔Dt-1αt+βtSt=α0D0-1+β0S0+∫0tαudDu-1+∫0tβudSu-Ct,∀t⩽Tq.s.,where Ht denotes the value of the trading strategy at time t. The meaning of (23) is that, starting with an initial amount D0-1α0+β0S0 of wealth, all changes in wealth are due to capital gains (appreciation of stocks and interest from the bond), minus the amount consumed. The q.s. means quasi-surely, which is the same as before.For economic and mathematical considerations, it is more appropriate to introduce wealth and a portfolio process which presents the proportions of wealth invested in the risky stock.Remark 12 (see [17]).
A portfolio processπ represents proportions of a wealth X which is invested in the stock. If we define (24)βt≔XtπtSt,αt≔XtDt1-πt,∀t⩽T,then we have Xt=Ht. As long as π constitutes a portfolio process with corresponding wealth process X, the α,β is a trading strategy in the sense of (23).Definition 13 (see [8]).
A portfolio process is anFt-adapted real valued process if π=πt with values in LG1ΩT.Definition 14.
For a given initial capitalm, a portfolio process π, and a cumulative consumption process C, consider the wealth equation (25)dXt=Xt1-πtDtdDt-1+XtπtdStSt-dCt=Xtrdt+XtπtνtdBt-dCtwith initial wealth X0=m. Or equivalently,(26)DtXt=m-∫0tDudCu+∫0tDuXuπuνudBu,∀t⩽T.If this equation has a unique solution X=Xt≔Xm,π,C, then it is called the wealth process corresponding to the triple m,π,C.In the setup of Definition14, notice that the ∫0TXt2πt2νt2dt‍<∞ must hold quasi-surely. Thus, we need to impose requirements πtXtνt∈HGp0,Tp⩾1, or πtXtνt∈MGp0,T,p⩾2.Definition 15.
A portfolio/consumption process pairπ,C is called admissible for an initial capital m∈R if (i)
the pair obeys the conditions of Definitions11, 13, and 14,(ii)
πtXtm,π,Cνt∈HG10,T,(iii)
the solutionXtm,π,C satisfies(27)Xtm,π,C⩾-J,∀t⩽T,q.s.,whereJ is a nonnegative random variable in LG2ΩT.We then haveπ,C∈Am.In the above Definitions11 and 13–15, it is necessary to guarantee that the financial fields and related stochastic analysis can be well defined. In particular, condition (ii) of Definition 15 makes sure that the mathematical framework does not collapse by allowing for many portfolio processes.


## 2.1.G-Brownian Motion and the Multiple Priors Setting

In the whole paper, the one-dimensional case is considered and we fix an intervalσ_,σ¯ with σ_>0. This interval describes the volatility uncertainty. σ_ and σ¯ denote a lower and upper bound for volatility, respectively.Definition 1 (see [9]).
LetΩ≠∅ be a given set. Let H be a linear space of real valued functions defined on Ω with c∈H for all constants c, and X∈H if X∈H. (H is considered as the space of random variables.) A sublinear expectation E^ on H is a functional E^:H→R satisfying the following properties: for any X,Y∈H, it has(1)
Monotonicity: ifX⩾Y then E^X⩾E^Y.(2)
Constant preserving:E^c=c.(3)
Subadditivity:E^X+Y⩽E^X+E^Y.(4)
Positive homogeneity:E^μX=μE^X∀μ⩾0.The triple Ω,H,E^ is called a sublinear expectation space.Definition 2 (see [10] (G-normal distribution)).
In a sublinear expectation spaceΩ,H,E^, a random variable X is called (centralized) G-normal distributed if for any a,b⩾0(4)aX+bX¯~a2+b2X,where X¯ is an independent copy of X. Here the letter G denotes the function (5)Ga≔12E^aX2:R⟶R.Note thatX has no mean-uncertainty; that is, it has E^X=E^-X=0. Moreover, the following important identity holds: (6)Ga=12σ¯2a+-12σ_2a-with σ_2≔-E^-X2 and σ¯2≔E^X2. We write that X is N0×σ_2,σ¯2 distributed. Therefore, we say that G-normal distribution is characterized by the parameters 0<σ_⩽σ¯.Remark 3 (see [10]).
The random variableX which is defined in (4) is generated by the following parabolic PDE defined in 0,T×R.
For anyCl,LipR, define ut,x≔E^φx+tX; then u is the unique (viscosity) solution of(7)∂tu-G∂xxu=0,ut=0=φ.Equation (7) is called a G-equation.Definition 4 (see [10] (G-Brownian motion)).
A processBtt⩾0 in a sublinear expectation space Ω,H,E^ is called a G-Brownian motion if the following properties are satisfied: (i)
B0=0.(ii)
For eacht,s⩾0 the increment Bt+s-Bt is N0×σ_2s,σ¯2s distributed and independent from Bt1,Bt2,…,Btn for each n∈N, 0⩽t1⩽⋯tn⩽t.Condition (ii) can be replaced by the following three conditions giving a characterization of G-Brownian motion: (i)
For eacht,s⩾0:Bt+s-Bt~Bt and E^Bt3/t→0 as t→0.(ii)
The incrementBt+s-Bt is independent from Bt1,Bt2,…,Btn for each n∈N, 0⩽t1⩽⋯tn⩽t.(iii)
E^Bt=-E^-Bt=0,∀t⩾0.For eacht0>0, it has that Bt+t0-Bt0t⩾0 is a G-Brownian motion.Let us briefly depict the construction ofG-expectation and its corresponding G-Brownian motion. As in the previous sections, we fix a time horizon T>0 and set ΩT=C00,T,R-the space of all real valued continuous paths starting at zero. Considering the canonical process Btω≔ωt,t⩽T,ω∈Ω, we define (8)LipΩT≔φBt1,…,Btn∣n∈N,t1,…,tn∈0,T,φ∈Cl,LipRn.A G-Brownian motion is firstly constructed in LipΩT. For this purpose, let ξii∈N be a sequence of random variables in a sublinear expectation space Ω~,H~,E~ such that ξi is G-normal distributed and ξi+1 is independent of ξ1,…,ξi for each integer i⩾1. Then a sublinear expectation in LipΩT is constructed by the following procedure: for each X∈LipΩT with X=φBt1-Bt0,Bt2-Bt1,…,Btn-Btn-1 for some φ∈Cl,LipRn, 0⩽t0<t1<⋯<tn⩽T, set (9)EGφBt1-Bt0,Bt2-Bt1,…,Btn-Btn-1≔E~φt1-t0ξ1,t2-t1ξ2,…,tn-tn-1ξn.The related conditional expectation ofX∈LipΩT as above under Ωti,i∈N, is defined by (10)EGφBt1-Bt0,Bt2-Bt1,…,Btn-Btn-1∣Ωti≔ψBt1-Bt0,…,Bti-Bti-1,where ψx1,…,xi≔E~[φ(x1,…,xi,ti+1-tiξi+1, …,tn-tn-1ξn)]. One checks that EG consistently defines a sublinear expectation in LipΩT and the canonical process B represents a G-Brownian motion.LetΘ≔σ_,σ¯ and A0,TΘ be the collection of all Θ-valued Ft-adapted processes on 0,T. We write (11)Bt0,σ≔∫0tσsdBs,and Pσ as the law of B0,σ=∫0·σsdBs‍; that is, Pσ=P0∘B0,σ-1 is distribution over trajectories. Let the set of multiple priors P be the closure of {Pσ∣σ∈A0,TΘ} under the topology of weak convergence.Theorem 5 (see [8]).
For anyφ∈Cl,LipRn,n∈N, 0⩽t1⩽⋯⩽tn⩽T, it holds that (12)EGφBt1,…,Btn-Btn-1=supθ∈A0,TΘ⁡EPφBt10,θ,…,Btntn-1,θ=supθ∈A0,TΘ⁡EPθφBt1,…,Btn-Btn-1=supPθ∈P⁡EPθφBt1,…,Btn-Btn-1.Furthermore,(13)EGX=supPθ∈P⁡EPX,∀X∈LG1ΩT.We use the set of priorsP to define the G-expectation EG. It is given by (14)EGX=supP∈P⁡EPX,where X is any random variable. So the G-expectation can be defined. Relative to the G-expectation, the space of random variable is denoted by LG1ΩT.In this paper, we consider the tuple asΩT,F,Ft,P and the canonical process B=Bt is a G-expectation motion with respect to P as given in the previous. The G-framework enables the analysis of stochastic processes for all priors of P. The terminology of“quasi-surely” (q.s.) is proved to be very useful.Unless there are special instructions, all equations should also be understood as “quasi-sure.” This means a property almost surely for all conceivable scenarios.As mentioned in the preceding,G-expectation can be defined in the space LGpΩT,p⩾1. It is the completion of CbΩT, the set of bounded continuous functions on ΩT under the norm ξ≔EGξ21/2<∞. Because the stochastic integrals are required to define trading strategies in the next sections, we briefly introduce the basic concepts about stochastic calculus and the construction of Itô integral with respect to G-Brownian motion.Forp⩾1, let MGp,00,T be the collection of simple processes η of the following form: for a given partition 0,T, t0,t1,…,tN,N∈N, for any t∈0,T the process η is defined by(15)ηtω≔∑j=0N-1ξiω1tj,tj+1t,where ξiω∈LGpΩti,i=0,1,…,N-1. For each η∈MGp,00,T, let (16)ηMGp≔EG∫0Tηspds1/p.We denote by MGp,00,T the completion of MGp,00,T under the norm ·MGp.Definition 6 (see [14]).
Forη∈MG2,00,T with the presentation in (15), the integral mapping is defined by I:MG2,00,T→LG2ΩT and (17)Iη=∫0TηsdBs≔∑j=0N-1ξiBtj+1-Btj.We consider the quadratic variation processBt of G-Brownian motion. It has (18)Bt=Bt2-2∫0tBsdBs,∀t⩽T.It is a continuous, increasing process, absolutely continuous with respect to dt. It contains all the statistical uncertainty of the G-Brownian motion. For s,t⩾0 we have Bs+t-Bs~Bt and it is independent of Ωs.Definition 7 (see [9]).
Letx∈R, z∈MG20,T and η∈MG10,T. Then the process (19)Mt≔x+∫0tzsdBs+∫0tηsdBs-∫0t2Gηsds,t⩽T,is a G-martingale.Specially, the nonsymmetric part-Kt≔∫0tηsdBs‍-∫0t2Gηsds,t∈0,T, is a G-martingale, which is quite a surprising result because -Kt is continuous, nonincreasing with quadratic variation equal to zero.Remark 8 (see [15]).
M is a symmetric G-martingale if and only if K≡0.Theorem 9 (see [15] (martingale representation)).
Letα⩾0 and ξ∈LGαΩT. Then the G-martingale X with Xt≔EGξ∣Ft,t∈0,T, has the following unique representation: (20)Xt=X0+∫0tzsdBs-Kt,where K is a continuous, increasing process with K0=0, KT∈LGβΩT, z∈HGβ0,T,∀β∈1,α, and -K a G-martingale.Ifα=2 and ξ bounded from above, z∈MG20,T and KT∈LG2ΩT (see [14]).A construction of the stochastic integral for the domainHGp0,T,p⩾1 is established by Song [15]. Although the structure of these spaces is similar as before, the norm for completion is different and the random variables ξiω in (15) are elements of a subset of LGpΩti. We will also use the domain HG10,T which is necessary for the martingale representation in the G-framework (see Theorem 9). For p=2, both domains coincide (see Song [15]). As a consequence, we can define the stochastic integral since MG20,T is contained in HG10,T. In financial fields, more trading strategies will be feasible.


## 2.2. The Financial Market Model

We consider the following financial marketM which includes a risk-free asset and a single risky asset and two assets are traded continuously over 0,T. Assume that the risk-free asset is a bond and its interest rate is r. So the discount process Dt can be defined to satisfy the following formula:(21)dDt=-rDtdt,D0=1,where constant r⩾0 is the interest rate of the riskless bond as in the classical theory.Assume that the risky asset is a stock with priceSt at time t, whose price process St is given by the following equation:(22)dSt=rStdt+νtStdBt,S0=x0>0,where B=Bt denotes the canonical process which is a G-Brownian motion under EG or P, respectively, with parameters σ¯>σ_>0.SinceB=Bt is a G-Brownian motion, the volatility of S is related to t which is different from that of Vorbrink [8], where the volatility of stock price is a constant 1. Consequently, the stock price evolution involves not only risk modeled by the noise part but also ambiguity about the risk due to the unknown deviation of the process B from its mean. According to financial fields, this ambiguity is called volatility uncertainty.Compared with the classical stock price process, (22) does not contain any volatility parameter σ. This is due to the characteristics of the G-Brownian motion B. Apparently, if we choose σ=σ_=σ¯, then we will be in the classical Black-Scholes model.Remark 10.
Take notice of the discounted stock price processDtSt which is a symmetric G-martingale relative to the corresponding G-expectation EG. As everyone knows, both the pricing and hedging of contingent claims are treated under a risk-neutral measure. This leads to a favorable situation in which the discounted stock price process is a (local) martingale [16]. In our ambiguous setting, this is also allowed. In order to model DtSt as a symmetric G-martingale (see Definition 7), we do not need to change the sublinear expectation. A symmetric G-martingale is required to make sure that the stock is the same for all participants, whether they sell or buy.Definition 11 (see [17]).
In the marketM, a trading strategy is an Ft-adapted vector process α,β=αt,βt, β a member of HG10,T such that βtSt∈HG10,T, and αt∈R for all t⩽T.
A cumulative consumption processC=Ct,0⩽t⩽T is a nonnegative Ft-adapted process with values in LG1ΩT, and with increasing, right-continuous, and C0=0, CT<∞ q.s.A basic assumption in the marketM is that the stock price process St defined by (22) is an element of MG20,T=HG20,T. We impose the so-called self-financing condition. In other words, consumption and trading in M satisfy(23)Ht≔Dt-1αt+βtSt=α0D0-1+β0S0+∫0tαudDu-1+∫0tβudSu-Ct,∀t⩽Tq.s.,where Ht denotes the value of the trading strategy at time t. The meaning of (23) is that, starting with an initial amount D0-1α0+β0S0 of wealth, all changes in wealth are due to capital gains (appreciation of stocks and interest from the bond), minus the amount consumed. The q.s. means quasi-surely, which is the same as before.For economic and mathematical considerations, it is more appropriate to introduce wealth and a portfolio process which presents the proportions of wealth invested in the risky stock.Remark 12 (see [17]).
A portfolio processπ represents proportions of a wealth X which is invested in the stock. If we define (24)βt≔XtπtSt,αt≔XtDt1-πt,∀t⩽T,then we have Xt=Ht. As long as π constitutes a portfolio process with corresponding wealth process X, the α,β is a trading strategy in the sense of (23).Definition 13 (see [8]).
A portfolio process is anFt-adapted real valued process if π=πt with values in LG1ΩT.Definition 14.
For a given initial capitalm, a portfolio process π, and a cumulative consumption process C, consider the wealth equation (25)dXt=Xt1-πtDtdDt-1+XtπtdStSt-dCt=Xtrdt+XtπtνtdBt-dCtwith initial wealth X0=m. Or equivalently,(26)DtXt=m-∫0tDudCu+∫0tDuXuπuνudBu,∀t⩽T.If this equation has a unique solution X=Xt≔Xm,π,C, then it is called the wealth process corresponding to the triple m,π,C.In the setup of Definition14, notice that the ∫0TXt2πt2νt2dt‍<∞ must hold quasi-surely. Thus, we need to impose requirements πtXtνt∈HGp0,Tp⩾1, or πtXtνt∈MGp0,T,p⩾2.Definition 15.
A portfolio/consumption process pairπ,C is called admissible for an initial capital m∈R if (i)
the pair obeys the conditions of Definitions11, 13, and 14,(ii)
πtXtm,π,Cνt∈HG10,T,(iii)
the solutionXtm,π,C satisfies(27)Xtm,π,C⩾-J,∀t⩽T,q.s.,whereJ is a nonnegative random variable in LG2ΩT.We then haveπ,C∈Am.In the above Definitions11 and 13–15, it is necessary to guarantee that the financial fields and related stochastic analysis can be well defined. In particular, condition (ii) of Definition 15 makes sure that the mathematical framework does not collapse by allowing for many portfolio processes.


## 3. Arbitrage and Contingent Claims

Definition 16 (see [8] (arbitrage in M)).
We say that there is an arbitrage opportunity inM if there exist an initial wealth m⩽0 and an admissible pair π,C∈Am with C≡0 such that, at some time T>0, (28)XTm,π,0⩾0q.s.,PXTm,π,0>0>0for  at  least  oneP∈P.Lemma 17 (no arbitrage).
In the financial marketM, there does not exist any arbitrage opportunity.Proof.
Assume that there exists an arbitrage opportunity; that is to say, there existm⩽0 and a pair π,C∈Am with C≡0 such that XTm,π,0⩾0 quasi-surely for some T>0. Then we have EGXTm,π,0⩾0. By definition of the wealth process, it has (29)0⩽EGDTXTm,π,0⩽m+EG∫0TDtXtm,π,0πtνtdBt=m.Since the G-expectation of an integral with respect to G-Brownian motion is zero, we have EGDTXTm,π,0=0. This implies DTXTm,π,0=0q.s. Therefore, m,π,0 cannot constitute an arbitrage.In the financial marketM, we consider a European contingent claim V and assume that its payoff at maturity time T is VT. Here, VT represents a nonnegative, Ft-adapted random variable. Regardless of any time, we impose the assumption VT∈LG2ΩT. The price of the claim at time 0 is denoted by V0. For the sake of finding reasonable prices for V, we need to utilize the concept of arbitrage. Considering that the financial market M,V contains the original market M and the contingent claim V. Similar to the above, an arbitrage opportunity needs to be defined in the financial market M,V.Definition 18 (see [17] (arbitrage in (M,V))).
We say that there is an arbitrage opportunity in (M,V) if there exist an initial wealth m⩾0 (m⩽0, resp.), an admissible pair π,C∈Am, and a constant a=-1 (a=1, resp.), such that (30)m+a·V0⩽0at time 0, and (31)XTm,π,C+a·VT⩾0q.s.,PXTm,π,C+a·VT>0>0for  at  least  oneP∈Pat time T.The valuesa=±1 in Definition 18 indicate short or long positions in the claims V, respectively. This definition of arbitrage is standard in the literature [17]. For the same reasons as before, we again require quasi-sure dominance for the wealth at time T and again with positive probability for only one possible scenario.In the following, we show that there exist no-arbitrage prices for a claimV. Under these prices, there is no-arbitrage opportunity. Because the uncertainty caused by the quadratic variation cannot be dispelled, generally speaking, there is no self-financing portfolio strategy which replicates the European claim or a risk-free hedge for the claim in our ambiguous market M.Roughly stated, since there is only one kind of situation where stocks will be traded, the measures induced by theG-framework result in market’s incompleteness.Definition 19 (see [17]).
Given a European contingent claimV, the upper hedging class is defined by (32)U≔m⩾0∣∃π,C∈Am:XTm,π,C⩾VTq.s.and the lower hedging class is defined by (33)L≔m⩾0∣∃π,C∈A-m:XT-m,π,C⩾-VTq.s..In addition, the upper arbitrage price is defined by (34)vup≔inf⁡m∣m∈Uand the lower arbitrage price is defined by (35)vlow≔sup⁡m∣m∈L.Lemma 20 (see [17]).
m
1
∈
L and 0⩽n1⩽m1 implies n1∈L. Analogously, m2∈U and n2⩾m2 implies n2∈U.The proof uses the idea that one “just consumes immediately the difference between the two initial wealth” (see [17] for the complete proof process).For anyσ∈σ_,σ¯, we define the Black-Scholes price of a European contingent claim V as follows: (36)u0σ=EPσDTVT.Similar to the constrained circumstances [17], we prove the next three lemmas which are related to the European contingent claim V.Lemma 21.
For anyσ∈σ_,σ¯, it holds that u0σ belongs to the interval vlow,vup.Proof.
Letm∈U. From the definition of U, we know that there exists a pair π,C∈Am such that XTm,π,C⩾VT q.s. Employing the properties of G-expectation as stated in Definition 1, we obtain for any σ∈σ_,σ¯ that (37)m=EGm+∫0TDtXtm,π,CπtνtdBt⩾EGm+∫0TDtXtm,π,CπtνtdBt-∫0TDtdCt=EGDTXTm,π,C⩾EGDTVT=supP∈P⁡EPDTVT⩾u0σ.Therefore, u0σ⩽m. We know vup≔inf⁡m∣m∈U; hence, u0σ⩽vup.
Analogously, letm∈L. By definition of L, there exists a pair π,C∈A-m such that XT-m,π,C⩾-VT q.s. For the same reason, we obtain for any σ∈σ_,σ¯ that (38)-m=EG-m+∫0TDtXt-m,π,CπtνtdBt⩾EG-m+∫0TDtXt-m,π,CπtνtdBt-∫0TDtdCt=EGDTXT-m,π,C⩾EG-DTVT⩾-EPσ-DTVT=-u0σ.Therefore, m⩽u0σ. We know vlow≔sup⁡m∣m∈L; hence, vlow⩽u0σ.Lemma 22.
For any priceV0>vup, there exists an arbitrage opportunity. Also for any price V0<vlow, there exists an arbitrage opportunity.Proof.
The idea of proving this lemma comes from [8]. We only consider the case V0<vlow since the argument is similar. Assume V0<vlow, m⩽0 and let -m∈V0,vlow. By definition of vlow and Lemma 20, we deduce that -m∈L. Hence, there exists a pair π,C∈A-m with (39)XT-m,π,C⩾-VTq.s.,-m-V0>0.This implies the existence of arbitrage in the sense of Definition 18. If ∃0<a<1 with -am=V0, then π,aC∈A-am and XT-am,π,aC=aXT-m,π,C. Let P∈P; without loss of generality, we may assume PVT>0>0. Due to (40)1=PXT-m,π,C⩾-VT⩽PXT-m,π,C=-VT+PaXT-m,π,C>-VT⩽1,we deduce (41)PXT-m,π,C=-VT+PaXT-m,π,C>-VT=1.Assume PXT-m,π,C=-VT=1 and we deduce XT-m,π,C=-VT, q.s. This contradicts XT-m,π,C⩾-VT q.s. Hence, PXT-am,π,aC>-VT>0. Hence, -am,π,aC constitutes an arbitrage.Lemma 23.
For anyV0⊈L∪U, there is no arbitrage in the financial market M,V.Proof.
The idea of proving this lemma also comes from [8]. We prove it by contradiction. Assume V0⊈U, V0⊈L and that there exists an arbitrage opportunity in M,V. We suppose that it satisfies Definition 18 for a=1. The case a=-1 works similarly.
By definition of arbitrage, there existsm⩽0, π,CA∈-m with (42)m=X0-m,π,C⩽-V0,XT-m,π,C⩾-VTq.s.Therefore, m∈L. By Lemma 21, it has V0∈L. This contradicts our assumption.Theorem 24.
For the financial marketM,V, the following identities hold:(43)vup=EGDTVT,vlow=-EG-DTVT.Proof.
Firstly, let us begin with the identityvup=EGDTVT. As seen in the proof of Lemma 22, for any m∈U we have m⩾EGDTVT. Therefore, vup=inf⁡m∣m∈U⩾EGDTVT.
To show the opposite inequality we need to define theG-martingale M by (44)Mt≔EGDTVT∣Ft,∀t⩽T.By the martingale representation theorem [15] (see Theorem 9), we know there exists z∈HG10,T and continuous, increasing processes K=Kt with KT∈LG1ΩT such that for any t⩽T(45)Mt=EGDTVT+∫0tzsdBs-Ktq.s.
For anyt⩽T, we set m=EGDTVT⩾0, Xtπtνt=ztDt-1∈HG10,T, and Ct=∫0tDs-1dKs‍∈LG1ΩT. Then the wealth process Xm,π,C satisfies (46)DtXtm,π,C=m+∫0tDsXsm,π,CπsνsdBs-∫0tDsdCs=Mt.The properties of K and C obey the conditions of a cumulative consumption process in the sense of Definition 11. Due to DtXtm,π,C=Mt⩾0, for ∀t⩽T, the wealth process is bounded from below, where π,C is admissible for m.
AsXTm,π,C=DT-1MT=VT quasi-surely, we have m=EGDTVT∈U. Due to the definition of U, we conclude that vup⩽EGDTVT.
The proof for the second identity is analogous. Again, using the proof of Lemma22, we obtain m⩽-EG-DTVT for any m∈L. Hence, vlow⩽-EG-DTVT.
In order to obtainvlow⩾-EG-DTVT, we define a G-martingale M by (47)Mt=EG-DTVT∣Ft,∀t⩽T.The remaining part is almost a copy of the above. By the martingale representation theorem [15], there exist z∈HG10,T, and a continuous, increasing process K=Kt with KT∈LG1ΩT such that, for any t⩽T, (48)Mt=EG-DTVT+∫0tzsdBs-Ktq.s.As the above, for any t⩽T, let (49)-m=EG-DTVT⩾0,Xtπtνt=ztDt-1∈HG10,T,and Ct=∫0tDs-1dKs‍∈LG1ΩT. Then the wealth process X-m,π,C satisfies (50)DtXt-m,π,C=-m+∫0tDsXs-m,π,CπsνsdBs-∫0tDsdCs=Mt,where C obeys the condition of a cumulative consumption process due to the properties of K. Moreover, for any t⩽T, it has(51)DtXt-m,π,C=EG-DTVT∣Ft⩾EG-VT∣Ft,which is bounded from below in the sense of item (iii) in Definition 15 because -VT∈LG2ΩT. Therefore, the wealth process is bounded from below. Consequently, π,C is admissible for -m.
SinceXT-m,π,C=DT-1MT=-VTq.s., it has m=-EG-DTVT∈L.
Due to the definition ofL, we conclude vlow⩾-EG-DTVT. So far, we have completed the proof of Theorem 24.The proof of Theorem24 is different from that of Vorbrink [8], because the volatility of stock price is related to t rather than a constant.Remark 25.
Because of sublinear expectationEG, by Theorem 24 we have vlow≠vup. This means that the market is not complete implying that not all claims can be hedged perfectly. Therefore, there are many no-arbitrage prices for V. As long as EGDTVT∣Ft is not a symmetric G-martingale, it has vlow≠vup. Under other circumstances, the process K is identically equal to zero (see Remark 8), meaning that EGDTVT∣Ft is symmetric and VT can be hedged perfectly owing to Remark 8 and Theorem 9.Theorem 26.
For any priceV0∈vlow,vup≠∅ of a European contingent claim at time zero, there does not exist any arbitrage opportunity in M,V. For any price V0⊈vlow,vup≠∅ there exists an arbitrage in the market.Proof.
The first part directly follows from Lemma23. From Lemma 22, we know that V0⊈vlow,vup implies the existence of an arbitrage opportunity. Thus, we only need to show that V0=vup and V0=vlow admit an arbitrage opportunity.
We only treat the caseV0=vup=inf⁡m∣m∈U. Then m⩾V0; that is, -m+V0⩽0. The second case is similar. Comparing the proof of Theorem 24 and letting m>0, for -m=EG-DTVT, there exists a pair π,C∈A-m such that (52)DtXt-m,π,C=-m+∫0tDsXs-m,π,CπsνsdBs-∫0tDsdCs=Mt=-DTVTq.s.Then KT=∫0tDsdCs‍, where K is an increasing, continuous process with EG-KT=0. So we can select P∈P such that EP-KT<0 (see Remark 25). Then the pair π,C∈A-m satisfies(53)EPDTXT-m,π,0>EPDTXT-m,π,C=EP-DTVT.Thus, PXTm,π,0>-VT>0 and we conclude that π,C∈A-m constitutes an arbitrage.On account of Theorem26, we call vlow,vup≠∅ the arbitrage free interval. Particularly, in the Markovian case where VT=ΦST for some Lipschitz function Φ:R→R, we can give more structural details about the bounds vup and vlow. We investigate this issue in Section 4.


## 4. The Markovian Case

For the European contingent claimsV, we have the form VT=ΦST for some Lipschitz function Φ:R→R. We use a nonlinear Feynman-Kac formula which is established in Peng [9]. Let us rewrite the dynamics of S in (22) as (54)dSut,x=rSut,xdu+νuSut,xdBu,u∈t,T,Stt,x=x>0.Analogy to the lower and upper arbitrage prices at time 0, at time t∈0,T, the lower and upper arbitrage prices are noted by vlowtx and vuptx, respectively. At a considered time t, the stock price St is replaced by the variable x. That is, St=x.Theorem 27.
Given a European contingent claimV=ΦST, its upper arbitrage price vuptx is given by ut,x, where u:0,T×R+→R is the unique solution of the following PDE:(55)∂tu+rx∂xu+Gν2x2∂xxu=ru,uT,x=Φx.A precise representation for the corresponding trading strategy in the stock and the cumulative consumption process is given by(56)βt=∂xut,St,∀t∈0,T,Ct=-12∫0tνs2Ss2∂xxus,SsdBs+∫0tνs2Ss2G∂xxus,Ssds,∀t∈0,T.Analogously,-u_t,x is the lower arbitrage price vlowtx, where u_:0,T×R+→R. vlowtx solves (55) but with terminal condition u_t,x=-Φx, ∀x∈R+.Proof.
Firstly, we consider the Backward Stochastic Differential Equation (in short BSDE):(57)Yst,x=EGΦSTt,x+∫sTfSrt,x,Yrt,xdr∣Fs,s∈t,T,where f:R×R→R is a given Lipschitz function. Peng [9] showed that the BSDE has a unique solution. So we can define a function u:0,T×R+→R by ut,x≔Ytt,x,t,x∈0,T×R+. In the light of the knowledge of the nonlinear Feynman-Kac formula [9], the function u is a viscosity solution of the following PDE:(58)∂tu+rx∂xu+Gν2x2∂xxu+fx,u=0,uT,x=Φx.
We define the function(59)u^t,x≔EGΦSTt,xDT.According to the above definition, for f≡0, u^ solves (58). Since the function G is nondegenerate, u^ turns into a classical C1,2-solution (see page 19 in [9]). Consequently, together with Itô’s formula (Theorem5.4 in [18]), it has(60)u^t,St0,x-u^0,x=∫0t∂tu^s,Ss0,x+rSs0,x∂xu^s,Ss0,xds+∫0tνsSs0,x∂xu^s,Ss0,xdBs+∫0t12νsSs0,x2∂xxu^s,Ss0,xdBs=∫0tνsSs0,x∂xu^s,Ss0,xdBs+12∫0tνsSs0,x2∂xxu^s,Ss0,xdBs-∫0tνsSs0,x2G∂xxu^s,Ss0,xds︸-Kt=-∫0tDsdCs.Now, consider the function (61)u~t,x≔Dt-1u^t,x,∀t,x∈0,T×R+.For t=0, based on Theorem 24, it has (62)u~t,x=u^t,x=EGΦSTt,xDT=EGDTVT=vuptx,∀t,x∈0,T×R+.Moreover, u~ can be used as a solution of (55). In addition, the function u defined by (63)ut,x≔Ytt,x=EGΦSTt,x-∫tTrYst,xds∣Ft,∀t,x∈0,T×R+,solves (55) owing to the nonlinear Feynman-Kac formula since fx,y=-ry. By uniqueness of the solution in (55) (see [19]; f is obviously bounded in x), we have u~=u. Thus, (64)ut,x=EGΦSTt,xDT-t=vuptx,∀t,x∈0,T×R+,and it uniquely solves (55).
In combination with the proof of Theorem24, by using its notions and Remark 12, we obtain the precise expressions for the trading strategy β and the cumulative consumption process C. That is, it has zt=νtSt0,x∂xu^t,St0,x=St0,xβtνtDt. Therefore, βt=∂xut,St,∀t∈0,T.
Analogously, we derive(65)Ct=∫0tDt-1dKs=-12∫0tνs2Ss2∂xxus,SsdBs+∫0tνs2Ss2G∂xxus,Ssds,∀t∈0,T.Due to Theorem27, the functions ut,x=vuptx and u_t,x=-vlowtx can be characterized as the unique solutions of (55). Under the circumstances of Φ being a convex or concave function, respectively, (55) simplifies greatly.Lemma 28.
(1) IfΦ is concave, then ut,· is convex for any t⩽T.
(2) IfΦ is concave, then ut,· is concave for any t⩽T. Similarly, if Φ is convex, then u_t,· is concave for any t⩽T. If Φ is concave, then u_t,· is convex for any t⩽T.Proof.
We only need to take into account the upper arbitrage price which is determined by the function(66)ut,x=EGΦSTt,xDT-t=vuptx,∀t,x∈0,T×R+.
First of all, letΦ be convex, t∈0,T, and x,y∈R+. For any θ∈0,1, it has (67)ut,θx+1-θy=EGΦSTt,θx+1-θye-rT-t=EGΦθx+1-θye-rT-t-1/2∫tTνu2dBu+∫tTνudBue-rT-t⩽EGθΦxe-rT-t-1/2∫tTνu2dBu+∫tTνudBu+1-θΦye-rT-t-1/2∫tTνu2dBu+∫tTνudBue-rT-t⩽EGθΦxe-rT-t-1/2∫tTνu2dBu+∫tTνudBue-rT-t+EG1-θΦye-rT-t-1/2∫tTνu2dBu+∫tTνudBu=θEGΦSTt,xe-rT-t+1-θEGΦSTt,ye-rT-t=θut,x+1-θut,y,where we used the convexity of Φ, the monotonicity of EG, and, in the second inequality, the sublinearity of EG. Therefore, ut,· is convex for all t∈0,T.
Secondly, letΦ be concave. For any t,x∈0,T×R+, we define (68)gt,x≔EPΦS~Tt,xe-rT-t,where(69)dS~st,x=rS~st,xds+σ_S~st,xdBs,s∈t,T,S~tt,x=x. Since B=Bt is a classical Brownian motion under P0, g solves the Black-Scholes PDE (7) with σ¯ replaced by σ_.
BecauseEP is linear, this is straightforward to mean that gt,· is concave for any t∈0,T. Consequently, g also solves (55). By uniqueness, we conclude that g=u. Therefore, ut,· is concave for any t∈0,T.


## 5. Conclusion

In order to analyse the financial markets with volatility uncertainty, we consider a stock price modeled by a geometricG-Brownian motion which features volatility uncertainty. This is all based on the structure of a G-Brownian motion. The “G-framework” is summarized by Peng [9] which gives us a useful mathematical setting. A little new arbitrage free concept is utilized to obtain the detailed results which give us an economically better understanding of financial markets under volatility uncertainty. We establish the connection of the lower and supper arbitrage prices by means of partial differential equations. The outcomes in this paper are only applied to European contingent claims. For other cases, we would extend these results to American contingent claims in our forthcoming paper.



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*Source: 5769205-2017-06-19.xml*