Hyperfinite Construction of G -expectation
HHYPERFINITE CONSTRUCTION OF G -EXPECTATION TOLULOPE FADINA AND FREDERIK HERZBERG
Abstract.
The hyperfinite G -expectation is a nonstandard discrete analogueof G -expectation (in the sense of Robinsonian nonstandard analysis). A lift-ing of a continuous-time G -expectation operator is defined as a hyperfinite G -expectation which is infinitely close, in the sense of nonstandard topology,to the continuous-time G -expectation. We develop the basic theory for hyperfi-nite G -expectations and prove an existence theorem for liftings of (continuous-time) G -expectation. For the proof of the lifting theorem, we use a new dis-cretization theorem for the G -expectation (also established in this paper, basedon the work of Dolinsky, Nutz and Soner [Stoch. Proc. Appl. 122, (2012),664–675]). Keywords: G -expectation; Volatility uncertainty; Weak limit theorem; Lift-ing theorem; Nonstandard analysis; Hyperfinite discretization. Introduction
Dolinsky et al. [8] showed a Donsker-type result for G -Brownian motion by in-troducing a notion of volatility uncertainty in discrete time and defined a discreteversion of Peng’s G -expectation. In the continuous-time limit, the resulting sublin-ear expectation converges weakly to G -expectation. In their discretization, Dolinskyet al. [8] allow for martingale laws whose support is the whole set of reals in a d -dimensional setting. In other words, they only discretize the time line, but not thestate space of the canonical process. Now for certain applications, for example, ahyperfinite construction of G -expectation in the sense of Robinsonian nonstandardanalysis, a discretization of the state space would be necessary. Thus, we developa modification of the construction by Dolinsky et al. [8] which even ensures thatthe sublinear expectation operator for the discrete-time canonical process corre-sponding to this discretization of the state space (whence the martingale laws aresupported by a finite lattice only) converges to the G -expectation. Further, weprove a lifting theorem, in the sense of Robinsonian nonstandard analysis, for the G -expectation. Herein, we use the discretization result for the G -expectation.Nonstandard analysis makes consistent use of infinitesimals in mathematicalanalysis based on techniques from mathematical logic. This approach is verypromising because it also allows, for instance, to study continuous-time stochas-tic processes as formally finite objects. Many authors have applied nonstandardanalysis to problems in measure theory, probability theory and mathematical eco-nomics (see for example, Anderson and Raimondo [3] and the references thereinor the contribution in Berg [4]), especially after Loeb [20] converted nonstandard We are very grateful to Patrick Beissner, Yan Dolinsky, and Frank Riedel for helpful com-ments and suggestions. This work was supported by the International Graduate College (IGK)
Stochastics and Real World Models (Bielefeld–Beijing) and the Rectorate of Bielefeld University(Bielefeld Young Researchers’ Fund). a r X i v : . [ q -f i n . M F ] O c t TOLULOPE FADINA AND FREDERIK HERZBERG measures (i.e. the images of standard measures under the nonstandard embedding ∗ ) into real-valued, countably additive measures, by means of the standard part op-erator and Caratheodory ’s extension theorem. One of the main ideas behind theseapplications is the extension of the notion of a finite set known as hyperfinite set or more causally, a formally finite set. Very roughly speaking, hyperfinite sets aresets that can be formally enumerated with both standard and nonstandard naturalnumbers up to a (standard or nonstandard, i.e. unlimited) natural number.Anderson [2], Keisler [16], Lindstrøm [19], Hoover and Perkins [14], a few to men-tion, used Loeb’s [20] approach to develop basic nonstandard stochastic analysisand in particular, the nonstandard Itˆo calculus. Loeb [20] also presents the con-struction of a Poisson processes using nonstandard analysis. Anderson [2] showedthat Brownian motion can be constructed from a hyperfinite number of coin tosses,and provides a detailed proof using a special case of Donsker’s theorem. Anderson[2] also gave a nonstandard construction of stochastic integration with respect to hisconstruction of Brownian motion. Keisler [16] uses Anderson’s [2] result to obtainsome results on stochastic differential equations. Lindstrøm [19] gave the hyperfi-nite construction ( lifting ) of L standard martingales. Using nonstandard stochasticanalysis, Perkins [24] proved a global characterization of (standard) Brownian localtime. In this paper, we do not work on the Loeb space because the G -expectationand its corresponding G -Brownian motion are not based on a classical probabilitymeasure, but on a set of martingale laws.The aim of this paper is to give two approximation results on G -expectation.First, to refine the discretization of G -expectation by Dolinsky et al. [8], in order toobtain a discretization of the sublinear expectation where the martingale laws aredefined on a finite lattice rather than the whole set of reals. Second, to give an al-ternative, combinatorially inspired construction of the G -expectation based on thediscretization result. We hope that this result may eventually become useful for ap-plications in financial economics (especially existence of equilibrium on continuous-time financial markets with volatility uncertainty) and provides additional intuitionfor Peng ’s G -stochastic calculus. We begin the nonstandard treatment of the G -expectation by defining a notion of S -continuity, a standard part operator, andproving a corresponding lifting (and pushing down) theorem. Thereby, we showthat our hyperfinite construction is the appropriate nonstandard analogue of the G -expectation.The rest of this paper is divided into two parts: in the first part, Section 2,we define Peng’s G -expectation and introduce a discrete-time analogue of a G -expectation in the spirit of Dolinsky et al. [8]. Unlike in Dolinsky et al. [8], werequire the discretization of the martingale laws to be defined on a finite latticerather than the whole set of reals. In the continuous-time limit, the resulting sub-linear expectation converges weakly to the continuous-time G -expectation. In thesecond part, Section 3, we develop the basic theory for hyperfinite G -expectationsand prove an existence theorem for liftings of (continuous-time) G -expectation. Weextend the discrete time analogue of the G -expectation in Section 2 to a hyperfinitetime analogue. Then, we use the characterization of convergence in nonstandardanalysis to prove that the hyperfinite discrete-time analogue of the G -expectationis infinitely close in the sense of nonstandard topology to the continuous-time G -expectation. YPERFINITE CONSTRUCTION OF G -EXPECTATION 3 Weak approximation of G -expectation with discrete state space Peng [23] introduced a sublinear expectation on a well-defined space L G , thecompletion of Lip b.cyl (Ω) (bounded and Lipschitz cylinder function) under the norm (cid:107) · (cid:107) L G , under which the increments of the canonical process ( B t ) t> are zero-mean,independent and stationary and can be proved to be ( G )-normally distributed.This type of process is called G-Brownian motion and the corresponding sublinearexpectation is called
G-expectation .The G -expectation ξ (cid:55)→ E G ( ξ ) is a sublinear operator defined on a class ofrandom variables on Ω. The symbol G refers to a given function(1) G ( γ ) := 12 sup c ∈ D cγ : R → R where D = [ r D , R D ] is a nonempty, compact and convex set, and 0 ≤ r D ≤ R D < ∞ are fixed numbers. The construction of the G -expectation is as follows. Let ξ = f ( B T ), where B T is the G -Brownian motion and f a sufficiently regular function.Then E G ( ξ ) is defined to be the initial value u (0 ,
0) of the solution of the nonlinearbackward heat equation, − ∂ t u − G ( ∂ xx u ) = 0 , with terminal condition u ( · , T ) = f , Pardoux and Peng [22]. The mapping E G can be extended to random variables of the form ξ = f ( B t , · · · , B t n ) by a step-wise evaluation of the PDE and then to the completion L G of the space of all suchrandom variables (cf. Dolinsky et al. [8]). Denis et al. [7] showed that L G is thecompletion of C b (Ω) and Lip b.cyl (Ω) under the norm (cid:107) · (cid:107) L G , and that L G is thespace of the so-called quasi-continuous function and contains all bounded continu-ous functions on the canonical space Ω, but not all bounded measurable functionsare included. Ruan [27] introduced the invariance principle of G -Brownian motionusing the theory of sublinear expectation. There also exists an equivalent alterna-tive representation of the G -expectation known as the dual view on G -expectationvia volatility uncertainty , see Denis et al. [7]:(2) E G ( ξ ) = sup P ∈P G E P [ ξ ] , ξ = f ( B T ) , where P G is defined as the set of probability measures on Ω such that, for any P ∈ P G , B is a martingale with the volatility d (cid:104) B (cid:105) t /dt ∈ D P ⊗ dt a.e.2.1. Continuous-time construction of sublinear expectation.
Let Ω = { ω ∈C ([0 , T ]; R ) : ω = 0 } be the canonical space endowed with the uniform norm (cid:107) ω (cid:107) ∞ = sup ≤ t ≤ T | ω t | , where | · | denotes the absolute value on R . Let B be thecanonical process B t ( ω ) = ω t , and F t = σ ( B s , ≤ s ≤ t ) the filtration generated by B . A probability measure P on Ω is a martingale law provided B is a P -martingaleand B = 0 P a.s. Then, P D is the set of martingale laws on Ω and the volatilitytakes values in D , P ⊗ dt a.e; P D = { P martingale law on Ω: d (cid:104) B (cid:105) t /dt ∈ D , P ⊗ dt a.e. } . Discrete-time construction of sublinear expectation.
We denote L n = (cid:26) jn √ n , − n (cid:112) R D ≤ j ≤ n (cid:112) R D , for j ∈ Z (cid:27) , TOLULOPE FADINA AND FREDERIK HERZBERG and L n +1 n = L n × · · · × L n ( n + 1 times), for n ∈ N . Let X n = ( X nk ) nk =0 be thecanonical process X nk ( x ) = x k defined on L n +1 n and ( F nk ) nk =0 = σ ( X nl , l = 0 , . . . , k )be the filtration generated by X n . We note that R D = sup α ∈ D | α | . D (cid:48) n = D ∩ (cid:18) n N (cid:19) is a nonempty bounded set of volatilities. A probability measure P on L n +1 n is amartingale law provided X n is a P -martingale and X n = 0 P a.s. The increment∆ X nk = X nk − X nk − . Let P n D be the set of martingale laws of X n on R n +1 , i.e., P n D = (cid:8) P martingale law on R n +1 : r D ≤ | ∆ X nk | ≤ R D , P a.s. (cid:9) , such that for all n , L n +1 n ⊆ R n +1 .. In order to establish a relation between the continuous-time and discrete-timesettings, we obtained a continuous-time process (cid:98) x t ∈ Ω from any discrete path x ∈ L n +1 n by linear interpolation. i.e., (cid:98) x t := ( (cid:98) nt/T (cid:99) + 1 − nt/T ) x (cid:98) nt/T (cid:99) + ( nt/T − (cid:98) nt/T (cid:99) ) x (cid:98) nt/T (cid:99) +1 where (cid:98) : L n +1 n → Ω is the linear interpolation operator, x = ( x , . . . , x n ) (cid:55)→ (cid:98) x = { ( (cid:98) x ) ≤ t ≤ T } , and (cid:98) y (cid:99) denotes the greatest integer less than or equal to y. If X n isthe canonical process on L n +1 n and ξ is a random variable on Ω , then ξ ( (cid:98) X n ) definesa random variable on L n +1 n . Strong formulation of volatility uncertainty.
We consider martingalelaws generated by stochastic integrals with respect to a fixed Brownian motion asin Dolinsky et al. [8], Nutz [21] and a fixed random walk as in Dolinsky et al. [8].Continuous-time construction; let Q D be the set of martingale laws: Q D = (cid:26) P ◦ ( M ) − ; M = (cid:90) f ( t, B ) dB t , and f ∈ C ([0 , T ] × Ω; √ D ) is adapted (cid:27) .B is the canonical process under the Wiener measure P .Discrete-time construction; we fix n ∈ N , Ω n = { ω = ( ω , . . . , ω n ) : ω i ∈ {± } , i =1 , . . . , n } equipped with the power set and let P n = δ − + δ +1 ⊗ · · · ⊗ δ − + δ +1 (cid:124) (cid:123)(cid:122) (cid:125) n times be the product probability associated with the uniform distribution where δ x ( A )is a Dirac measure for any A ⊆ R and a given x ∈ A . Let ξ , . . . , ξ n be an i.i.dsequence of {± } -valued random variables. The components of ξ k are orthonormalin L ( P n ) and the associated scaled random walk is X = 1 √ n k (cid:88) l =1 ξ l . We denote by Q n D (cid:48) n the set of martingale laws of the form: Q n D (cid:48) n = (cid:110) P n ◦ ( M f, X ) − ; f : { , . . . , n } × L n +1 n → (cid:112) D (cid:48) n is F n -adapted. (cid:111) (3)where M f, X = (cid:16)(cid:80) kl =1 f ( l − , X )∆ X l (cid:17) nk =0 . YPERFINITE CONSTRUCTION OF G -EXPECTATION 5 Results and proofs.
Theorem 1 states that a sublinear expectation withdiscrete-time volatility uncertainty on our finite lattice converges to the G -expectation. Lemma 2.1. Q n D = (cid:110) P n ◦ (cid:0) M f, X (cid:1) − ; f : { , . . . , n } × R n +1 → √ D is adapted (cid:111) . Then Q n D ⊆ P n D . Proposition 2.2.
Let ξ : Ω → R be a continuous function satisfying | ξ ( ω ) | ≤ a (1+ (cid:107) ω (cid:107) ∞ ) b for some constants a, b > . Then, ( i )(4) lim n →∞ sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] = sup P ∈Q D E P [ ξ ] . ( ii )(5) sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] = max Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] . To prove (4), we prove two separate inequalities together with a density argu-ment. The left-hand side of (5) can be written assup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] = sup f ∈A E P n ◦ ( M f, X ) − [ ξ ( (cid:98) X n )] , where A = (cid:110) f : { , . . . , n } × L n +1 n → (cid:112) D (cid:48) n /n is F n -adapted. (cid:111) . We prove that A isa compact subset of a finite-dimensional vector space, and that f (cid:55)→ E P n ◦ ( M f, X ) − [ ξ ( (cid:98) X n )]is continuous. Before then, we introduce a smaller space L ∗ that is defined as thecompletion of C b (Ω; R ) under the norm (cf. Dolinsky et al. [8]) (cid:107) ξ (cid:107) ∗ := sup Q ∈Q E Q | ξ | , Q := P D ∪ { P ◦ ( (cid:98) X n ) − ; P ∈ P n D /n , n ∈ N . } . This is because Proposition 2.2 will not hold if ξ just belong to L G , which is thecompletion of C b (Ω; R ) under the norm(6) (cid:107) ξ (cid:107) L G := sup P ∈P D E P [ | ξ | ] . Proof of Proposition 2.2. First inequality (for ≤ in (4)) : (7) lim sup n →∞ sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] ≤ sup P ∈Q D E P [ ξ ] . For all n , (cid:112) D (cid:48) n /n ⊆ (cid:112) D /n and Q n D (cid:48) n ⊆ Q n D . It is shown in Dolinsky et al. [8] thatlim sup n →∞ sup Q ∈P n D /n E Q [ ξ ( (cid:98) X n )] ≤ sup P ∈P D E P [ ξ ] . Since Q D ⊆ P D (see Dolinsky et al. [8, Remark 3 . Q n D ⊆ P n D (see Lemma2.1), (7) follows. Second inequality (for ≥ in (4)) : It remains to show thatlim inf n →∞ sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] ≥ sup P ∈Q D E P [ ξ ] . For arbitrary P ∈ Q D , we construct a sequence ( P n ) n such that for all n ,(8) P n ∈ Q n D (cid:48) n /n , TOLULOPE FADINA AND FREDERIK HERZBERG and(9) E P [ ξ ] ≤ lim inf n →∞ E P n [ ξ ( (cid:98) X n )] . For fixed n , we want to construct martingales M n whose laws are in Q n D (cid:48) n /n andthe laws of their interpolations tend to P. Thus, we introduce a scaled random walkwith the piecewise constant c`adl`ag property,(10) W nt := 1 √ n (cid:98) nt/T (cid:99) (cid:88) l =1 ξ l = 1 √ n Z n (cid:98) nt/T (cid:99) , ≤ t ≤ T, and we denote the continuous version of (10) obtained by linear interpolation by(11) (cid:99) W nt := 1 √ n (cid:98) Z n (cid:98) nt/T (cid:99) , ≤ t ≤ T. By the central limit theorem; ( W n , (cid:99) W n ) ⇒ ( W, W ) as n → ∞ on D ([0 , T ]; R )( ⇒ implies convergence in distribution). i.e., the law ( P n ) converges to the law P on the Skorohod space D ([0 , T ]; R ) Billingsley [5, Theorem 27 . g ∈C ([0 , T ] × Ω , √ D ) such that P = P ◦ (cid:90) g ( t, W ) dW t (cid:124) (cid:123)(cid:122) (cid:125) M − . Since g is continuous and (cid:99) W nt is the interpolated version of (10), (cid:18) W n , (cid:16) g (cid:16) (cid:98) nt/T (cid:99) T /n, (cid:99) W nt (cid:17)(cid:17) t ∈ [0 ,T ] (cid:19) ⇒ (cid:0) W, ( g ( t, W t )) t ∈ [0 ,T ] (cid:1) as n → ∞ on D ([0 , T ]; R ) . We introduce martingales with discrete-time integrals,(12) M nk := k (cid:88) l =1 g (cid:16) ( l − T /n, (cid:99) W n (cid:17) (cid:99) W nlT/n − (cid:99) W n ( l − T/n . In order to construct M n which is “close” to M and also is such that P n ◦ ( M n ) − ∈ Q n D (cid:48) n /n .We choose (cid:101) h n : { , · · · , n } × Ω → (cid:112) D (cid:48) n /n such that d J (cid:18)(cid:16)(cid:101) h n ( (cid:98) nt/T (cid:99) T /n, (cid:99) W nt ) (cid:17) t ∈ [0 ,T ] , (cid:16) g ( (cid:98) nt/T (cid:99) T /n, (cid:99) W nt ) (cid:17) t ∈ [0 ,T ] (cid:19) is minimal (this is possible because there are only finitely many choices for (cid:16)(cid:101) h n ( (cid:98) nt/T (cid:99) T /n, (cid:99) W nt ) (cid:17) t ∈ [0 ,T ] )and d J is the Kolmogorov metric for the Skorohod J topology. From Billingsley[6, Theorem 4 . . (cid:18) W n , (cid:16)(cid:101) h n (cid:16) (cid:98) nt/T (cid:99) T /n, (cid:99) W nt (cid:17)(cid:17) t ∈ [0 ,T ] (cid:19) ⇒ (cid:0) W, g ( t, W t ) t ∈ [0 ,T ] (cid:1) on D ([0 , T ]; R ) . We then define g n : { , . . . , n } × L n +1 n → (cid:112) D (cid:48) n /n by g n : ( (cid:96), (cid:126) X ) (cid:55)→ (cid:101) h n ( (cid:96), (cid:98) (cid:126)X ) . Let M n be defined by M nk = k (cid:88) l =1 g n (cid:18) l − , √ n Z n (cid:19) √ n ∆ Z nl , ∀ k ∈ { , · · · , n } . YPERFINITE CONSTRUCTION OF G -EXPECTATION 7 By stability of stochastic integral (see Duffie and Protter [9, Theorem 4 . . (cid:16) M n (cid:98) nt/T (cid:99) (cid:17) t ∈ [0 ,T ] ⇒ M as n → ∞ on D ([0 , T ]; R )because M n (cid:98) nt/T (cid:99) = (cid:98) nt/T (cid:99) (cid:88) l =1 (cid:101) h n (cid:16) ( l − T /n, (cid:16)(cid:99) W kT/n (cid:17) nk =0 (cid:17) ∆ (cid:99) W lT/n . In addition, as n goes to ∞ , the increments of M n uniformly tend to 0. Thus, (cid:99) M n ⇒ M on Ω . Since ξ is bounded and continuous,(13) lim n →∞ E P n ◦ ( M n ) − [ ξ ( (cid:98) X n )] = E P ◦ M − [ ξ ] . Therefore, (8) is satisfied for P n = P n ◦ ( M n ) − ∈ Q n D (cid:48) n /n . Taking the lim inf as n tends to ∞ and the supremum over P ∈ Q D , (13) becomes(14) sup P ∈Q D E P [ ξ ] ≤ lim inf n →∞ sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] . Combining (7) and (14),sup P ∈Q D E P [ ξ ] ≥ lim sup n →∞ sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] ≥ lim inf n →∞ sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] ≥ sup P ∈Q D E P [ ξ ] . Therefore,(15) sup P ∈Q D E P [ ξ ] = lim n →∞ sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] . Density argument : (4) is established for all ξ ∈ C b (Ω , R ). Since Q D ⊆ P D (seeDolinsky et al. [8, Remark 3 . Q n D ⊆ P n D (see Lemma 2.1), Q n D (cid:48) n ⊆ Q and Q D ⊆ Q . Thus, (4) holds for all ξ ∈ L ∗ , and hence, holds for all ξ that satisfycondition of Proposition 2.2. First part of 5 : A is closed and obviously bounded with respect to the norm (cid:107) · (cid:107) ∞ as D (cid:48) n is bounded. By Heine-Borel theorem, A is a compact subset of a N ( n, n )-dimensional vector space equipped with the norm (cid:107) · (cid:107) ∞ . Second part of 5 : Here, we show that F : f (cid:55)→ E P n ◦ ( M f, X ) − [ ξ ( (cid:98) X n )] is continu-ous. From Proposition 2.2 we know that ξ is continuous, (cid:98) X n is the interpolatedcanonical process, i.e., (cid:98) X : L n +1 n → Ω , thus (cid:98) X n is continuous and P n takes itvalues from the set of real numbers. For F : f (cid:55)→ E P n ◦ ( M f, X ) − [ ξ ( (cid:98) X n )] to be con-tinuous, ψ : f (cid:55)→ M f, X has to be continuous. Since A is a compact subset of a N ( n, n )-dimensional vector space for fixed n ∈ N and M f, X : Ω n → L n +1 n , for all f, g ∈ A , | M f, X − M g, X | = |(cid:107) f (cid:107) ∞ − (cid:107) g (cid:107) ∞ | ≤ (cid:107) f − g (cid:107) ∞ . Thus, ψ is continuous with respect to the norm (cid:107) · (cid:107) ∞ . Hence F is continuous withrespect to any norm on R N ( n,n ) . (cid:3) The cardinality of L n , L n = 2 n + 1, L n +1 n = (2 n + 1) n +1 , and { , . . . , n } × L n +1 n ) =( n + 1)(2 n + 1) n +1 = N ( n, n ) . TOLULOPE FADINA AND FREDERIK HERZBERG
Theorem 1.
Let ξ : Ω → R be a continuous function satisfying | ξ ( ω ) | ≤ a (1 + (cid:107) ω (cid:107) ∞ ) b for some constants a, b > . Then,(16) sup P ∈Q D E P [ ξ ] = lim n →∞ max Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] . Proof.
The proof follows directly from Proposition 2.2. (cid:3) Nonstandard construction of G -expectation Hyperfinite-time setting.
Here we present the nonstandard version of thediscrete-time setting of the sublinear expectation and the strong formulation ofvolatility uncertainty on the hyperfinite timeline.
Definition 3.1. ∗ Ω is the ∗ -image of Ω endowed with the ∗ -extension of the max-imum norm ∗ (cid:107) · (cid:107) ∞ . ∗ D = ∗ [ r D , R D ] is the ∗ -image of D , and as such it is internal .It is important to note that st : ∗ Ω → Ω is the standard part map, and st ( ω ) willbe referred to as the standard part of ω , for every ω ∈ ∗ Ω. ◦ z denotes the standardpart of a hyperreal z . Definition 3.2.
For every ω ∈ Ω, if there exists (cid:101) ω ∈ ∗ Ω such that (cid:107) (cid:101) ω − ∗ ω (cid:107) ∞ (cid:39) (cid:101) ω is a nearstandard point in ∗ Ω. This will be denoted as ns ( (cid:101) ω ) ∈ ∗ Ω.For all hypernatural N, let(17) L N = (cid:26) KN √ N , − N (cid:112) R D ≤ K ≤ N (cid:112) R D , K ∈ ∗ Z (cid:27) , and the hyperfinite timelime(18) T = (cid:26) , TN , · · · , − TN + T, T (cid:27) . We consider L T N as the canonical space of paths on the hyperfinite timeline, and X N = ( X Nk ) Nk =0 as the canonical process denoted by X Nk (¯ ω ) = ¯ ω k for ¯ ω ∈ L T N . F N is the internal filtration generated by X N . The linear interpolation operator canbe written as (cid:101) : (cid:98) · ◦ ι − → ∗ Ω , for (cid:103) L T N ⊆ ∗ Ω , where (cid:98) ω ( t ) := ( (cid:98) N t/T (cid:99) + 1 − N t/T ) ω (cid:98) Nt/T (cid:99) + (
N t/T − (cid:98)
N t/T (cid:99) ) ω (cid:98) Nt/T (cid:99) +1 , for ω ∈ L N +1 N and for all t ∈ ∗ [0 , T ]. (cid:98) y (cid:99) denotes the greatest integer less than orequal to y and ι : T → { , · · · , N } for ι : t (cid:55)→ N t/T .For the hyperfinite strong formulation of the volatility uncertainty, fix N ∈ ∗ N \ N . Consider (cid:110) ± √ N (cid:111) T , and let P N be the uniform counting measure on (cid:110) ± √ N (cid:111) T . P N can also be seen as a measure on L T N , concentrated on (cid:110) ± √ N (cid:111) T .Let Ω N = { ω = ( ω , · · · , ω N ); ω i = {± } , i = 1 , · · · , N } , and let Ξ , · · · , Ξ N be a ∗ -independent sequence of {± } -valued random variables on Ω N and the components YPERFINITE CONSTRUCTION OF G -EXPECTATION 9 of Ξ k are orthonormal in L ( P N ). We denote the hyperfinite random walk by X t = 1 √ N Nt/T (cid:88) l =1 Ξ l for all t ∈ T . The hyperfinite-time stochastic integral of some F : T × L T N → ∗ R with respect tothe hyperfinite random walk is given by t (cid:88) s =0 F ( s, X )∆ X s : Ω N → ∗ R , ω ∈ Ω N (cid:55)→ t (cid:88) s =0 F ( s, X ( ω ))∆ X s ( ω ) . Thus, the hyperfinite set of martingale laws can be defined by¯ Q N D (cid:48) N = (cid:8) P N ◦ ( M F, X ) − ; F : T × L T N → (cid:112) D (cid:48) N (cid:9) where D (cid:48) N = ∗ D ∩ (cid:18) N ∗ N (cid:19) and M F, X = (cid:32) t (cid:88) s =0 F ( s, X )∆ X s (cid:33) t ∈ T . Remark 3.1.
Up to scaling, ¯ Q N D (cid:48) N = Q n D (cid:48) n . Results and proofs.Definition 3.3 ((Uniform lifting of ξ )) . Let Ξ : L T N → ∗ R be an internal function,and let ξ : Ω → R be a continuous function. Ξ is said to be a uniform lifting of ξ ifand only if ∀ ¯ ω ∈ L T N (cid:16)(cid:101) ¯ ω ∈ ns ( ∗ Ω) ⇒ ◦ Ξ(¯ ω ) = ξ ( st ( (cid:101) ¯ ω )) (cid:17) , where st ( (cid:101) ¯ ω ) is defined with respect to the topology of uniform convergence on Ω.In order to construct the hyperfinite version of the G -expectation, we need toshow that the ∗ -image of ξ , ∗ ξ , with respect to (cid:101) ¯ ω ∈ ns ( ∗ Ω), is the canonical liftingof ξ with respect to st ( (cid:101) ¯ ω ) ∈ Ω. i.e., for every (cid:101) ¯ ω ∈ ns ( ∗ Ω), ◦ (cid:0) ∗ ξ ( (cid:101) ¯ ω ) (cid:1) = ξ ( st ( (cid:101) ¯ ω )). Todo this, we need to show that ∗ ξ is S-continuous in every nearstandard point (cid:101) ¯ ω .It is easy to prove that there are two equivalent characteristics of S -continuityon ∗ Ω. Remark 3.2.
The following are equivalent for an internal function
Φ : ∗ Ω → ∗ R ; (1) ∀ ω (cid:48) ∈ ∗ Ω (cid:16) ∗ (cid:107) ω − ω (cid:48) (cid:107) ∞ (cid:39) ⇒ ∗ | Φ( ω ) − Φ( ω (cid:48) ) | (cid:39) (cid:17) . (2) ∀ ε (cid:29) , ∃ δ (cid:29) ∀ ω (cid:48) ∈ ∗ Ω (cid:16) ∗ (cid:107) ω − ω (cid:48) (cid:107) ∞ < δ ⇒ ∗ | Φ( ω ) − Φ( ω (cid:48) ) | < ε (cid:17) . (The case of Remark 3.2 where Ω = R is well known and proved in Stroyan andLuxemburg [28, Theorem 5 . . Definition 3.4.
Let Φ : ∗ Ω → ∗ R be an internal function. We say Φ is S -continuous in ω ∈ ∗ Ω, if and only if it satisfies one of the two equivalent conditions of Remark3.2.
Proposition 3.3. If ξ : Ω → R is a continuous function satisfying | ξ ( ω ) | ≤ a (1 + (cid:107) ω (cid:107) ∞ ) b , for a, b > , then, Ξ = ∗ ξ ◦ (cid:101) · is a uniform lifting of ξ . Proof.
Fix ω ∈ Ω. By definition, ξ is continuous on Ω. i.e., for all ω ∈ Ω, and forevery ε (cid:29)
0, there is a δ (cid:29)
0, such that for every ω (cid:48) ∈ Ω, if(19) (cid:107) ω − ω (cid:48) (cid:107) ∞ < δ, then | ξ ( ω ) − ξ ( ω (cid:48) ) | < ε. By the Transfer Principle: For all ω ∈ Ω, and for every ε (cid:29)
0, there is a δ (cid:29) ω (cid:48) ∈ ∗ Ω, (19) becomes,(20) ∗ (cid:107) ∗ ω − ω (cid:48) (cid:107) ∞ < δ, and ∗ | ∗ ξ ( ∗ ω ) − ∗ ξ ( ω (cid:48) ) | < ε. So, ∗ ξ is S -continuous in ∗ ω for all ω ∈ Ω. Applying the equivalent characterizationof S -continuity, Remark 3.2, (20) can be written as ∗ (cid:107) ∗ ω − ω (cid:48) (cid:107) ∞ (cid:39) , and ∗ | ∗ ξ ( ∗ ω ) − ∗ ξ ( ω (cid:48) ) | (cid:39) . We assume (cid:101) ¯ ω to be a nearstandard point. By Definition 3.2, this simply implies,(21) ∀ (cid:101) ¯ ω ∈ ns ( ∗ Ω) , ∃ ω ∈ Ω : ∗ (cid:107) (cid:101) ¯ ω − ∗ ω (cid:107) ∞ (cid:39) . Thus, by S -continuity of ∗ ξ in ∗ ω , ∗ | ∗ ξ ( (cid:101) ¯ ω ) − ∗ ξ ( ∗ ω ) | (cid:39) . Using the triangle inequality, if ω (cid:48) ∈ ∗ Ω with ∗ (cid:107) (cid:101) ¯ ω − ω (cid:48) (cid:107) ∞ (cid:39) ∗ (cid:107) ∗ ω − ω (cid:48) (cid:107) ∞ ≤ ∗ (cid:107) ∗ ω − (cid:101) ¯ ω (cid:107) ∞ + ∗ (cid:107) (cid:101) ¯ ω − ω (cid:48) (cid:107) ∞ (cid:39) S -continuity of ∗ ξ in ∗ ω , ∗ | ∗ ξ ( ∗ ω ) − ∗ ξ ( ω (cid:48) ) | (cid:39) . And so, ∗ | ∗ ξ ( (cid:101) ¯ ω ) − ∗ ξ ( ω (cid:48) ) | ≤ ∗ | ∗ ξ ( (cid:101) ¯ ω ) − ∗ ξ ( ∗ ω ) | + ∗ | ∗ ξ ( ∗ ω ) − ∗ ξ ( ω (cid:48) ) | (cid:39) . Thus, for all (cid:101) ¯ ω ∈ ns ( ∗ Ω) and ω (cid:48) ∈ ∗ Ω, if ∗ (cid:107) (cid:101) ¯ ω − ω (cid:48) (cid:107) ∞ (cid:39)
0, then, ∗ | ∗ ξ ( (cid:101) ¯ ω ) − ∗ ξ ( ω (cid:48) ) | (cid:39) . Hence, ∗ ξ is S-continuous in (cid:101) ¯ ω . Equation (21) also implies (cid:101) ¯ ω ∈ m ( ω ) (cid:16) m ( ω ) = (cid:92) { ∗ O ; O is an open neighbourhood of ω } (cid:17) such that ω is unique, and in this case st ( (cid:101) ¯ ω ) = ω .Therefore, ◦ (cid:16) ∗ ξ ( (cid:101) ¯ ω ) (cid:17) = ξ ( st ( (cid:101) ¯ ω )) . (cid:3) Definition 3.5.
Let ¯ E : ∗ R L T N → ∗ R . We say that ¯ E lifts E G if and only if for every ξ : Ω → R that satisfies | ξ ( ω ) | ≤ a (1 + (cid:107) ω (cid:107) ∞ ) b for some a, b > E ( ∗ ξ ◦ ˜ · ) (cid:39) E G ( ξ ) . Theorem 2. (22) max ¯ Q ∈ ¯ Q N D (cid:48) N E ¯ Q [ · ] lifts E G ( ξ ) . YPERFINITE CONSTRUCTION OF G -EXPECTATION 11 Proof.
From Theorem 1,(23) max Q ∈Q n D (cid:48) n E Q [ ξ ( (cid:98) X n )] → E G ( ξ ) , as n → ∞ . For all N ∈ ∗ N \ N , we know that (23) holds if and only if(24) max Q ∈ ∗ Q N D (cid:48) N E Q [ ∗ ξ ( (cid:98) X N )] (cid:39) E G ( ξ ) , (see Albeverio et al. [1], Proposition 1 . . Q N D (cid:48) N . i.e., to show that max ¯ Q ∈ ¯ Q N D (cid:48) N E ¯ Q [ ∗ ξ ◦ ˜ · ] (cid:39) E G ( ξ ) . To do this, use E Q [ ∗ ξ ◦ ˆ · ] = E Q [ ∗ ξ ◦ ˆ · ◦ ι − ◦ ι ]and E Q [ ∗ ξ ◦ ˆ · ◦ ι − ◦ ι ] = E Q [ ∗ ξ ◦ ˜ · ◦ ι ]= (cid:90) ∗ R N +1 ∗ ξ ◦ ˜ · ◦ ιdQ, (transforming measure)= (cid:90) ∗ R T ∗ ξ ◦ ˜ · d ( Q ◦ j ) , = E Q ◦ j [ ∗ ξ ◦ ˜ · ]for j : ∗ R T → ∗ R N +1 , ( xt ) t ∈ T (cid:55)→ (cid:0) xNtT (cid:1) t ∈ R N +1 . Thus, ¯ Q N D (cid:48) N = { Q ◦ j : Q ∈ ∗ Q N D (cid:48) N } . This implies, max ¯ Q ∈ ¯ Q N D (cid:48) N E ¯ Q [ ∗ ξ ◦ ˜ · ] = max Q ∈ ∗ Q N D (cid:48) N E Q [ ∗ ξ ◦ ˆ · ] . (cid:3) Appendix
Proof of Lemma 2.1.
From the above equation, we can say that ∆ M fk = f ( k, X ) ξ k .And by the orthonormality property of ξ k , we have E P n [ f ( k, X ) ξ k |F nk ] = E P n [ f ( k, X ) |F nk ] ≤ E P n [( (cid:112) R D ) |F nk ] = R D P n a.s. , as | ξ k | = 1, f ( · · · ) ∈ D implies | (∆ M fk ) | = | f ( k, X ) | ∈ [ r D , R D ] P n a.s . (cid:3) Density argument verification.
Let f : ξ (cid:55)→ sup P ∈Q D E P [ ξ ]and g : ξ (cid:55)→ lim n →∞ sup Q ∈Q n D (cid:48) n/n E Q [ ξ ( (cid:98) X n )] . From (15), we know that for all ξ ∈ C b (Ω , R ) , f ( ξ ) = g ( ξ ) . Since L ∗ is the com-pletion of C b (Ω , R ) under the norm (cid:107) · (cid:107) ∗ , C b (Ω , R ) is dense in L ∗ ; and we want toprove for all ξ ∈ L ∗ , f ( ξ ) = g ( ξ ) . To prove this, it is sufficient to show that f and g are continuous with respect to the norm (cid:107) · (cid:107) ∗ . For continuity of f : For all P ∈ Q D and ξ, ξ (cid:48) ∈ L ∗ ,sup P ∈Q D E P [ ξ ] − sup P ∈Q D E P [ ξ (cid:48) ] ≤ sup P ∈Q D E P [ | ξ − ξ (cid:48) | ] . Since, Q D ⊆ Q ,(25) sup P ∈Q D E P [ ξ ] − sup P ∈Q D E P [ ξ (cid:48) ] ≤ (cid:107) ξ − ξ (cid:48) (cid:107) ∗ . Interchanging ξ and ξ (cid:48) ,(26) sup P ∈Q D E P [ ξ (cid:48) ] − sup P ∈Q D E P [ ξ ] ≤ (cid:107) ξ (cid:48) − ξ (cid:107) ∗ . Adding (25) and (26), we have | f ( ξ ) − f ( ξ (cid:48) ) | ≤ (cid:107) ξ − ξ (cid:48) (cid:107) ∗ . For continuity of g : We follow the same argument as above.
Proof of Remark 3.2.
Let Φ be an internal function such that condition (1) holds.To show that (1) ⇒ (2), fix ε (cid:29)
0. We shall show there exists a δ for this ε as incondition (2). Since Φ is internal, the set I = (cid:110) δ ∈ ∗ R > : ∀ ω (cid:48) ∈ ∗ Ω ( ∗ (cid:107) ω − ω (cid:48) (cid:107) ∞ < δ ⇒ ∗ | Φ( ω ) − Φ( ω (cid:48) ) | < ε ) (cid:111) , is internal by the Internal Definition Principle and also contains every positiveinfinitesimal. By Overspill (cf. Albeverio et al. [1, Proposition 1 . I must thencontain some positive δ ∈ R .Conversely, suppose condition (1) does not hold, that is, there exists some ω (cid:48) ∈ ∗ Ωsuch that ∗ (cid:107) ω − ω (cid:48) (cid:107) ∞ (cid:39) ∗ | Φ( ω ) − Φ( ω (cid:48) ) | is not infinitesimal . If ε = min(1 , ∗ | Φ( ω ) − Φ( ω (cid:48) ) | / , we know that for each standard δ > , there is apoint ω (cid:48) within δ of ω at which Φ( ω (cid:48) ) is farther than ε from Φ( ω ). This shows thatcondition (2) cannot hold either. (cid:3) YPERFINITE CONSTRUCTION OF G -EXPECTATION 13 References [1] Albeverio, S., R. Høegh-Krohn, J. Fenstad, and T. Lindstrøm (1986).
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