arXiv:0912.1515v3 [math.PR] 20 Oct 2012 motnesesfo h atta hyaerltdt ikm risk to related are they that fact the from stems importance The a nrdcdanwnto fnnierepcain h so the expectation, nonlinear of notion new a introduced has h betv ftepeetppri osuytelcltm a time local the study to is the paper for present the of objective The Introduction 1 variation. 30 ilfl,Germany Bielefeld, 33501 Keywords: hr,frgvnprmtr 0 parameters given for where, [ 10 oa ieadTnk oml o the for formula Tanaka and time Local ∗ † ,[ ], orsodn drs:Isiueo ahmtclEconomi Mathematical of Institute address: Corresponding mi address Email oiae yucranypolm,rs esrsadsuper and measures risk problems, uncertainty by Motivated 2 G aoaor eMt´mtqe,CR M 25 nvri´ eBr Universit´e de 6205, UMR Math´ematiques, CNRS de Laboratoire 12 rwinmto sotie n t udai aito spoe.A a As proven. the is to variation respect quadratic Itˆo’s with its generalize formula we and obtained is motion Brownian o the for epcainrpeet pca aeo eea nonlinea general of case special a represents -expectation G ,[ ], nti ae,w td h oino oa ieadoti h Tanak the obtain and time local of notion the study we paper, this In Bona motion. -Brownian 13 G ,[ ], ,aeu itrL ogu S987 93 rs ee ,Fran 3, cedex Brest 29238 93837, CS Gorgeu, Le Victor avenue 6, 1 Bona oin oevr h on otniyo h oa ieo time local the of continuity joint the Moreover, motion. -Brownian colo ahmtc,Sadn nvriy ia 510 China 250100, Jinan University, Shandong Mathematics, of School G [email protected] : 15 -expectation; ,[ ], 16 u ) hc sascae ihtefloignnierha eq heat nonlinear following the with associated is which ]), ∂u (0 ( ∂t x , ,x t, G = ) ( ) α G ≤ = = ) Bona oin oa ie aaafrua quadratic formula; Tanaka time; local motion; -Brownian ϕ σ ( G x ≤ ( 1 2 ) ∂ , ( σ σ inLIN Qian 2 h ulna function sublinear the , u ∂x 2 Abstract α ( ,x t, + 2 − 1 ) ) σ , G 2 α Bona oint ovxfunctions. convex to motion -Brownian 1 ( − , 2 ,x t, ) † α , ) s ilfl nvriy otah100131, Postfach University, Bielefeld cs, ∈ [0 ∈ G , R ∞ ela h aaaformula Tanaka the as well s . Bona motion -Brownian ) G -called × egn nfiac,Peng finance, in hedging sdfie sfollows: as defined is expectations r easures R tgeOccidentale, etagne , G application, n epcain(see -expectation ce. ρ formula a ; nfiac by finance in the f E ˆ G uation - whose ∗ the relation Eˆ[X] = ρ( X), where X runs the class of contingent claims. Although the − G-expectations represent only a special case, their importance inside the class of nonlinear expectations stems from the stochastic analysis which it allows to develop, in particular, such important results as the law of large numbers and the central limit theorem under nonlinear expectations, obtained by Peng [11], [14] and [17]. Together with the notion of G-expectations Peng also introduced the related G-normal distribution and the G-Brownian motion. The G-Brownian motion is a stochastic process with stationary and independent increments and its quadratic variation process is, unlike the classical case of a linear expectation, a non deterministic process. Moreover, an Itˆo calculus for the G-Brownian motion has been developed recently in [10], [12] and [13] and [8]. The fundamental notion of the local time for the classical Brownian motion has been introduced by L´evy in [7], and its existence was established by Trotter [19] in 1958. In virtue of its various applications in stochastic analysis the notion of local time and the related Tanaka formula have been well studied by several authors, and it has also been extended to other classes of stochastic processes. It should be expected that the notion of local time and the very narrowly related Tanaka formula will have the same importance in the G-stochastic analysis on sublinear expectation spaces, which is developed by Peng ([10], [12], [13], [15], [16]). However, unlike the study of the local time for the classical Brownian motion, its investigation with respect to the G-Brownian motion meets several difficulties: firstly, in contrast to the classical Brownian motion the G-Brownian motion is not defined on a given probability space but on a sublinear expectation space. The G-expectation Eˆ can be represented as the upper expectation of a subset of linear expectations E , P , i.e., { P ∈ P}
Eˆ[ ]= sup EP [ ], · P ∈P · where P runs a large class of probability measures , which are mutually singular. Let P us point out that this is a very common situation in financial models under volatility uncertainty (see [1], [4], [9]). Secondly, related with the novelty of the theory of G- expectations, there is short of a lot of tools in the G-stochastic analysis, among them there is, for instance, the dominated convergence theorem. In order to point out the difficulties, in Section 3, ad hoc definition of the local time for the G-Brownian motion will be given independently of the probability measures P , and the limits of the ∈ P classical stochastic analysis in the study of this local time will be indicated. Our paper is organized as follows: Section 2 introduces the necessary notations and preliminaries and it gives a short recall of some elements of the G-stochastic analysis which will be used in what follows. In Sections 3, an intuitive approach to the local time shows its limits in the frame of the classical stochastic analysis and works out the problems to be studied. In Section 4, we define the notion of the local time for the G-Brownian motion and prove the related Tanaka formula, which non-trivially generalizes the classical one. Moreover, by using Kolmogorov’s continuity criterion under the nonlinear expectation the
2 existence of a jointly continuous version of the local time is shown. Finally, Section 5 investigates the quadratic variation of the local time for the G-Brownian motion. As an application of the Local time and the Tanaka formula for the G-Brownian motion obtained in Section 4, Section 6 gives a generalization of the Itˆoformula for the G-Brownian motion to convex functions, which generalizes the corresponding result in [8].
2 Notations and preliminaries
In this section, we introduce some notations and preliminaries of the theory of sublinear expectations and the related G-stochastic analysis, which will be needed in what follows. More details of this section can be found in Peng [10], [12], [13], [15] and [16]. Let Ω be a given nonempty set and a linear space of real valued functions defined H on Ω such that, 1 and X , for all X . ∈H | |∈H ∈H Definition 2.1 A sublinear expectation Eˆ on is a functional Eˆ : R satisfying H H 7→ the following properties: for all X,Y , we have ∈H (i) Monotonicity: If X Y , then Eˆ[X] Eˆ[Y ]. ≥ ≥ (ii) Preservation of constants: Eˆ[c]= c, for all c R. ∈ (iii) Subadditivity: Eˆ[X] Eˆ[Y ] Eˆ[X Y ]. − ≤ − (iv) Positive homogeneity: Eˆ[λX]= λEˆ[X], for all λ 0. ≥ The triple (Ω, , Eˆ) is called a sublinear expectation space. H Remark 2.2 is considered as the space of random variables on Ω. H Let us now consider a space of random variables with the additional property of H stability with respect to bounded Lipschitz functions. More precisely, we suppose, if X , i = 1, , d, then i ∈H · · · ϕ(X , , X ) , for all ϕ C (Rd), 1 · · · d ∈H ∈ b,Lip d d where Cb,Lip(R ) denotes the space of all bounded Lipschitz functions on R .
Definition 2.3 In a sublinear expectation space (Ω, , Eˆ), a random vector Y = (Y , ,Y ), H 1 · · · n Y , is said to be independent under Eˆ from another random vector X = (X , , X ), X i ∈H 1 · · · m i ∈ , if for each test function ϕ C (Rm+n) we have H ∈ b,lip
Eˆ[ϕ(X,Y )] = Eˆ[Eˆ[ϕ(x,Y )]x=X ].
3 Definition 2.4 Let X1 and X2 be two n–dimensional random vectors defined respectively in the sublinear expectation spaces (Ω , , Eˆ ) and (Ω , , Eˆ ). They are called identi- 1 H1 1 2 H2 2 cally distributed, denoted by X X , if 1 ∼ 2 Eˆ [ϕ(X )] = Eˆ [ϕ(X )], for all ϕ C (Rn). 1 1 2 2 ∈ b.Lip After the above basic definition we introduce now the central notion of G-normal distribution.
Definition 2.5 (G-normal distribution) Let be given two reals σ, σ with 0 σ σ. A ≤ ≤ random variable ξ in a sublinear expectation space (Ω, , Eˆ) is called G -normal dis- H σ,σ tributed,denoted by ξ (0, [σ2, σ2]), if for each ϕ C (R), the following function ∼ N ∈ b,lip defined by u(t,x) := Eˆ[ϕ(x + √tξ)], (t,x) [0, ) R, ∈ ∞ × is the unique viscosity solution of the following parabolic partial differential equation :
∂ u(t,x)= G(∂2 u(t,x)), (t,x) [0, ) R, t xx ∈ ∞ × u(0,x)= ϕ(x).
Here G = Gσ,σ is the following sublinear function parameterized by σ and σ: 1 G(α)= (σ2α+ σ2α−), α R 2 − ∈ (recall that α+ = max 0, α and α− = min 0, α ). { } − { } Definition 2.6 A process B = B ,t 0 in a sublinear expectation space (Ω, , Eˆ) { t ≥ }⊂H H is called a G-Brownian motion if the following properties are satisfied:
(i) B0 = 0; (ii) for each t,s 0, the difference B B is (0, [σ2s, σ2s])-distributed and is ≥ t+s − t N independent from (B , ,B ), for all n N and 0 t t t. t1 · · · tn ∈ ≤ 1 ≤···≤ n ≤ + Throughout this paper, we let from now on Ω = C0(R ) be the space of all real valued continuous functions (ωt)t∈R+ with ω0 = 0, equipped with the distance
∞ 1 2 −i 1 2 1 2 ρ(ω ,ω )= 2 (max ωt ωt ) 1 , ωt ,ωt Ω. t∈[0,i] | − | ∧ ∈ Xi=1 h i We denote by (Ω) the Borel σ-algebra on Ω. We also set, for each t [0, ), Ω := B ∈ ∞ t ω : ω Ω and := (Ω ). Moreover, we will work with the following spaces: { ·∧t ∈ } Ft B t L0(Ω): the space of all (Ω)-measurable real valued functions on Ω; • B
4 L0(Ω ): the space of all (Ω )-measurable real valued functions on Ω; • t B t L (Ω) : the space of all bounded elements in L0(Ω); • b L (Ω ): the space of all bounded elements in L0(Ω ). • b t t Lp Eˆ In [10], a G-Brownian motion is constructed on a sublinear expectation space (Ω, G(Ω), ), where Lp (Ω) is the Banach space defined as closure of := ϕ(ω , ,ω ), for all ϕ G H { t1 · · · td ∈ C (Rd), 0 t < Eˆ[ ]= sup EP [ ]. (2.1) · P ∈P · The Choquet capacity: cˆ(A) := sup P (A), A (Ω). P ∈P ∈B Definition 2.7 A set A Ω is called polar if cˆ(A) = 0. A property is said to hold ⊂ “quasi-surely” (q.s.) if it holds outside a polar set. Definition 2.8 A mapping X on Ω with values in a topological space is said to be quasi- continuous if for all ε > 0, there exists an open set O with cˆ(O) < ε such that X c is |O continuous. The family of probability measures allows to characterize the space Lp (Ω) as follows: P G Lp 0 p G(Ω) = X L (Ω) : lim sup EP [ X 1|X|>n] = 0, and X isc ˆ-quasi continuous . ∈ n→∞ P ∈P | | n o We also introduce the following spaces, for all p> 0, p 0 p p := X L (Ω) : Eˆ[ X ]= sup EP [ X ] < ; •L ∈ | | P ∈P | | ∞ n o := X L0(Ω) : X = 0,c ˆ-quasi surely (q.s.). •N ∈ n o Obviously, p and are linear spaces and X L0(Ω) : Eˆ[ X p] = 0 = , for all p> 0. L N ∈ | | N We put Lp := p/ . As usual, we will notn make here the distinctiono between classes and L N their representatives. The following three propositions can be consulted in [3] and [5] . 5 Proposition 2.9 For every monotonically decreasing sequence X ∞ of nonnegative { n}n=1 functions in Cb(Ω), which converges to zero q.s. on Ω, it holds lim Eˆ[Xn] = 0. n→∞ Proposition 2.10 For all p> 0, we have 1 1. Lp is a Banach space with respect to the norm X := Eˆ[ X p] p . k kp | | 2. Lp is the completion of C (Ω) with respect to the norm . G b k·kp We denote by Lp(Ω) the completion of L (Ω) with respect to the norm . ∗ b k·kp ∞ Lp Proposition 2.11 For a given p (0, + ], let Xn n=1 be a sequence converging Lp ∈ ∞ { } ⊂ to X in . Then there exists a subsequence (Xnk ) which converges to X quasi-surely in the sense that it converges to X outside a polar set. We now recall the definition of quadratic variation process of the G-Brownian motion. We use 0= t n−1 t 2 2 2 B := L lim [B n B n ] = B 2 B dB . t G ti+1∧t ti ∧t t s s h i n→∞ − − 0 Xi=0 Z B is continuous and increasing outside a polar set. h i In [8], a generalized Itˆointegral and a generalized Itˆoformula with respect to the G-Brownian motion are discussed as follows. For arbitrarily fixed p 1 and T R , we first consider the following set of step ≥ ∈ + processes: n−1 M (0, T ) = η : η (ω)= ξ (ω)1 (t), 0= t < T 1/p p Eˆ p η M∗ (0,T ) = [ ηt dt] || || 0 | | Z on Mb,0(0, T ). With respect to this norm, Mb,0(0, T ) can be continuously extended to a Banach space. Definition 2.13 For each p 1, we denote by M p(0, T ) the completion of M (0, T ) ≥ ∗ b,0 under the norm T 1/p p Eˆ p η M∗ (0,T ) = [ ηt dt] . || || 0 | | Z Definition 2.14 For every η M (0, T ) of the form (2.2) ∈ b,0 n−1 ηt(ω)= ξj(ω)1[tj ,tj+1)(t), Xj=0 the Itˆointegral T n−1 I(η)= ηsdBs := ξj(Btj+1 Btj ). 0 − Z Xj=0 L2 Lemma 2.15 The mapping I : Mb,0(0, T ) ∗(ΩT ) is a continuous, linear mapping. → 2 L2 Thus it can be continuously extended to I : M∗ (0, T ) ∗(ΩT ). Moreover, for all η 2 → ∈ M∗ (0, T ), we have T Eˆ[ ηsdBs] = 0, (2.3) 0 ZT T Eˆ[( η dB )2] σ2Eˆ[ η2ds]. (2.4) s s ≤ s Z0 Z0 p Definition 2.16 For each fixed p> 0, we denote by η Mω(0, T ) the space of stochastic ∈ processes η = (ηt)t∈[0,T ] for which there exists a sequence of increasing stopping times ∞ σm m=1 (which can depend on the process η), with σm T , quasi-surely, such that { } p ↑ η1[0,σm] M∗ (0, T ) and ∈ T c( η pds < ) = 1. | s| ∞ Z0 The generalized Itˆoformula isb obtained in [8]. 7 Theorem 2.17 Let ϕ C2(R) and ∈ t t t X = X + α ds + η d B + β dB , for all t [0, T ], t 0 s s h is s s ∈ Z0 Z0 Z0 where α, η in M 1(0, T ) and β M 2(0, T ). Then, for each 0 t T , we have ω ∈ ω ≤ ≤ t t ϕ(X ) ϕ(X )= ∂ ϕ(X )β dB + ∂ ϕ(X )α du t − 0 x u u u x u u Z0 Z0 t 1 + [∂ ϕ(X )η + ∂2 ϕ(X )β2]d B . x u u 2 xx u u h iu Z0 Example 2.18 For all ϕ C2(R) and t 0, we have ∈ ≥ t 1 t ϕ(B )= ϕ(0) + ϕ (B )dB + ϕ (B )d B . t x s s 2 xx s h is Z0 Z0 3 An intuitive approach to the local time We consider the G-Brownian motion B. Recall that B is continuous and a P -martingale, for all P (see (2.1)), and its quadratic variation process B is continuous and in- ∈ P h i creasing outside a polar set . Let us give a definition of the local time of B, which is N independent of the underlying probability measure P . For all a R, t [0, T ], and ∈ P ∈ ∈ ω Ω , ∈ \N 1 t 1 t a lim 1(a−ε,a+ε)(Bs(ω))d B s(ω), lim 1(a−ε,a+ε)(Bs(ω))d B s(ω) < ; L (ω)= ε↓0 2ε 0 h i ε↓0 2ε 0 h i ∞ t R 0, otherwiseR . We now study La under each probability P . For this we consider the stochastic ∈ P integral t M P = sgn(B a)dB , t [0, T ], (3.1) t s − s ∈ Z0 under P . Indeed, under P the G-Brownian motion is a continuous square integrable ∈ P martingale, so that the above stochastic integral under P is well-defined. We emphasize that the process M P is defined only P a.s. Let La,P be the local time associated with − B under P , defined by the relation: La,P = B a a M P , t [0, T ]. t | t − | − | |− t ∈ a,P It is well known that Lt admits a P -modification that is continuous in (t, a), and t a,P 1 Lt = lim 1(a−ε,a+ε)(Bs)d B s, P a.s., t [0, T ], ε↓0 2ε h i − ∈ Z0 8 Thus, due to the definition of La, we have La = La,P , P a.s, t [0, T ]. t t − ∈ Consequently, La has a continuous P -modification, for all P . However, P is not ∈ P dominated by a probability measure. Thus the question, if we can find a continuous modification of La or even a jointly continuous modification of (α, t) Lα under Eˆ, 7→ t i.e., with respect to all P , is a nontrivial one which cannot be solved in the frame ∈ P of the classical stochastic analysis. The other question which has its own importance is that of the integrability of sgn(B a) in the framework of G-expectations. Indeed, in · − (3.1) we have considered the stochastic integral of sgn(B a) under P , for each P · − ∈ P separately. However, the main difficulty is that the above family of probability measures is not dominated by one of these probability measures. To overcome this difficulty, we shall show that sgn(B· a) belongs to a suitable space of processes integrable with respect to the − 2 G-Brownian motion B. It should be expected that this space is M∗ (0, T ) (see Definition 2.13). This turns out to be correct if σ > 0, but it is not clear at all in the case of σ = 0. 2 For this reason a larger space M∗ (0, T ) (see Definition 4.1) is introduced in Section 4, 2 and its relationship with M∗ (0, T ) is discussed. To get a jointly continuous modification of (α, t) Lα under Eˆ in the frameworkf of G-expectations, we use, in addition to the 7→ t integrality of sgn(B a) with respect to the G-Brownian motion B, an approximation · − approach which is different from the classical method. 4 Local time and Tanaka formula for the G-Brownian mo- tion The objective of this section is to study the notion of local time and to obtain Tanaka formula for the G-Brownian motion. First, we shall generalize Itˆointegral with respect to the G-Brownian motion, which plays an important role in what follows. For each η M (0, T ), we introduce the norm ∈ b,0 T 1/p Eˆ p η M p(0,T ) = [ ηt d B t] || ||f∗ 0 | | h i Z on Mb,0(0, T ). With respect to this norm, Mb,0(0, T ) can be continuously extended to a Banach space. Definition 4.1 For each p 1, we denote by M p(0, T ) the completion of M (0, T ) under ≥ ∗ b,0 the norm T 1/p Eˆ f p η M p(0,T ) = [ ηt d B t] . || ||f∗ 0 | | h i Z p Remark 4.2 The elements of M∗ (0, T ) are defined only d B tdP -a.e., for all P . p h i ∈ P While for σ > 0, this means that η M∗ (0, T ) is defined dtdP -a.e., for all P f ∈ ∈ P f9 (and, may, hence, not be well defined with respect to P on Γ Ω, for Γ [0, T ] ∈ P × ⊂ of Lebesgue measure zero). In the case σ = 0, this has the consequence that, if for some P , 0 t T T E [ 1 d = EP0 [ 1{Ns=0}ds] EP0 [ 1{Ns=0}ds] 0 ≤ 0 ∞ Z Z = P N = 0 ds = 0. 0{ s } Z0 Consequently, T EP0 [ 1{Ms=0}d T 1 T 1 s 2 0= EP0 [ 1{Ms=0}d T 1 T 1 Eˆ s 2 [ 1{Bs=0}d B s] = sup EP0 [ 1{R usdBs=0}usds] = 0, s h i s 0 Z0 u∈A Z0 10 1 − p p i.e., s 1{Bs=0} s∈[0,T ] can be identified with the process identically equal to 0 in M∗ (0, T ), and so it belongs to this space. On the other hand, for u = 0,s [0, T ], as the optimal s ∈ control, f T 1 T 1 Eˆ s [ 1{Bs=0}ds] = sup EP0 [ 1{R usdBs=0}ds] s s 0 Z0 u∈A Z0 T 1 = ds =+ . s ∞ Z0 1 − p p Therefore, s 1{Bs=0} s∈[0,T ] cannot belong to M∗ (0, T ). For every η M (0, T ), the Itˆointegral with respect to the G-Brownian motion is ∈ b,0 defined in Definition 2.14. L2 Lemma 4.4 The mapping I : Mb,0(0, T ) ∗(ΩT ) is a continuous and linear mapping. → 2 L2 Thus it can be continuously extended to I : M∗ (0, T ) ∗(ΩT ). Moreover, for all η 2 → ∈ M∗ (0, T ), we have f T f Eˆ[ ηsdBs] = 0, 0 ZT T Eˆ[( η dB )2] Eˆ[ η2d B ]. s s ≤ s h is Z0 Z0 Now we establish the Burkh¨older-Davis-Gundy inequality for the framework of the G-stochastic analysis, which will be needed in what follows. Lemma 4.5 For each p> 0, there exists a constant cp > 0 such that (i) for all η M 2(0, T ), ∈ ∗ t T T Eˆ[ sup η dB 2p] c Eˆ[( η2d B )p] σ2pc Eˆ[( η2ds)p], | s s| ≤ p s h is ≤ p s 0≤t≤T Z0 Z0 Z0 t 1 T σ2p T Eˆ[ sup η dB 2p] Eˆ[( η2d B )p] Eˆ[( η2ds)p]; | s s| ≥ c s h is ≥ c s 0≤t≤T Z0 p Z0 p Z0 (ii) for all η M 2(0, T ), ∈ ∗ 1 T t T Eˆ[(f η2d B )p] Eˆ[ sup η dB 2p] c Eˆ[( η2d B )p]. c s h is ≤ | s s| ≤ p s h is p Z0 0≤t≤T Z0 Z0 11 Proof. We only give the proof of (i), i.e., for η M 2(0, T ). The proof of (ii) is similar. ∈ ∗ Since for each α L (Ω ) we have ∈ b t T Eˆ[α ηsdBs] = 0, Zt then the process · η dB is a P -martingale, for all P . Thus from the classical 0 s s ∈ P Burkh¨older-Davis-Gundy inequality and the relation R σ2t B σ¯2t, q.s, ≤ h it ≤ we have, for all P , ∈ P t T T E [ sup η dB 2p] c E [( η2d B )p] σ2pc E [( η2ds)p], P | s s| ≤ p P s h is ≤ p P s 0≤t≤T Z0 Z0 Z0 and t 1 T σ2p T E [ sup η dB 2p] E [( η2d B )p] E [( η2ds)p.] P | s s| ≥ c P s h is ≥ c P s 0≤t≤T Z0 p Z0 p Z0 We emphasize that the constant cp coming from the classical Burkh¨older-Davis-Gundy inequality, only depends on p but not on the underlaying probability measure P . Conse- quently, by taking the supremum over all P we have ∈ P t T T Eˆ[ sup η dB 2p] c Eˆ[( η2d B )p] σ2pc Eˆ[( η2ds)p], | s s| ≤ p s h is ≤ p s 0≤t≤T Z0 Z0 Z0 and t 1 T σ2p T Eˆ[ sup η dB 2p] Eˆ[( η2d B )p] Eˆ[( η2ds)p]. | s s| ≥ c s h is ≥ c s 0≤t≤T Z0 p Z0 p Z0 The proof is complete. The following proposition is very important for our approach. Proposition 4.6 For any real a, all δ > 0 and t 0, we have ≥ t Eˆ[ 1 (B )d B ] Cδ. [a,a+δ] s h is ≤ Z0 Moreover, if σ > 0, then we also have t Eˆ[ 1 (B )ds] Cδ. [a,a+δ] s ≤ Z0 Here C is a constant which depends on t but not on δ neither on a. 12 Proof. For δ > 0, we define the C2 function ϕ : R R such that ϕ(0) = 0, ϕ′(x) − → | | ≤ δ−1, x ( , a δ], and ∈ ∞ − 0; if x a δ, x a + δ ≤ − − 3 ; if a δ t 1 t ϕ(B )= ϕ′(B )dB + ϕ′′(B )d B . t s s 2 s h is Z0 Z0 Therefore, t t 2 ′′ 1[a,a+δ](Bs)d B s δ ϕ (Bs)d B s 0 h i ≤ 0 h i Z Z t 2 2 ′ = 2δ ϕ(Bt) 2δ ϕ (Bs)dBs − 0 tZ 8δ B 2δ2 ϕ′(B )dB . ≤ | t|− s s Z0 From the above inequalities and Lemma 2.15 it follows that t t ˆ ˆ 2 ′ E[ 1[a,a+δ](Bs)d B s] E[8δ Bt 2δ ϕ (Bs)dBs] 0 h i ≤ | |− 0 Z Z t 8δEˆ[ B ] + 2δ2Eˆ[ ϕ′(B )dB ] ≤ | t| − s s Z0 = 8δEˆ[ B ]= Cδ. | t| From Lemma 4.5 we have t 1 t Eˆ[ 1 (B )ds] Eˆ[ 1 (B )d B ] Cδ. [a,a+δ] s ≤ σ2 [a,a+δ] s h is ≤ Z0 Z0 The proof is complete. From the above proposition and Lemma 4.5, we can derive interesting results as follows: 13 Corollary 4.7 For any real number a and t 0, we have ≥ t 1 (B )d B = 0, q.s. {a} s h is Z0 Moreover, if σ > 0, then we have t 1{a}(Bs)ds = 0, q.s. Z0 The following lemma will play an important role in what follows and its proof will be given after Theorem 4.9. Lemma 4.8 For each a R, the process sgn(B a) M 2(0, T ). ∈ · − ∈ ∗ Now we can state Tanaka formula for the G-Brownianf motion as follows. For this we denote 1; x> 0, sgn(x)= 0; x = 0, 1; x< 0. − Theorem 4.9 For any real number a and all t 0, we have ≥ t B a = a + sgn(B a)dB + La, | t − | | | s − s t Z0 where t a 1 L2 Lt = lim 1(a−ε,a+ε)(Bs)d B s, (lim in ), ε→0 2ε h i Z0 and La is an increasing process. La is called the local time for G-Brownian motion at a. Proof. Without loss of generality, we assume that a = 0. Let us define η C∞(R) by ∈ putting 1 C exp ; if x < 1, η(x)= x 2 1 | | 0; | | − if x 1, | |≥ where C is the positive constant satisfying R η(x)dx = 1. For every n N, we put ∈ R η (x) := nη(nx),x R. n ∈ Then η C∞(R) and η (x)dx = 1. n ∈ R n R 14 For any ε> 0, we set 1 x2 ε + ); if x < ε, ϕε(x)= 2 ε | | x ; if x ε. | | | |≥ Then, obviously x ; if x ε, ′ ε | |≤ ϕε(x)= 1; if x > ε, 1; if x< ε, − − and 1 ′′ ; if x < ε, ϕε (x)= ε | | ( 0; if x > ε | | (ϕ′′ is not defined at ε and ε). − Let us still introduce ϕ := ϕ η , i.e., n ε ∗ n ϕn(x)= ηn(x y)ϕε(y)dy,x R. R − ∈ Z 1 Then ϕ (x) C∞(R), 0 ϕ′′ , ϕ ϕ , ϕ′ ϕ′ uniformly in R, and ϕ′′ ϕ′′ n ∈ ≤ n ≤ ε n → ε n → ε n → ε pointwise (except at ε and ε), as n . − →∞ By applying the generalized Itˆoformula for the G-Brownian motion (see Theorem 2.17) to ϕn(Bt), we deduce that t 1 t ϕ (B )= ϕ (0) + ϕ′ (B )dB + ϕ′′ (B )d B . (4.1) n t n n s s 2 n s h is Z0 Z0 From ϕ ϕ , ϕ′ ϕ′ , uniformly in R, it follows that n → ε n → ε ϕ (B ) ϕ (B ), ϕ (0) ϕ (0), n t → ε t n → ε and t t ϕ′ (B )dB ϕ′ (B )dB n s s → ε s s Z0 Z0 in L2, as n . →∞ Setting 1 1 1 1 A := ( ε , ε + ) (ε , ε + ), n,ε − − n − n − n n ′′ ′′ c [ we observe that ϕn = ϕε on An,ε. From Corollary 4.7 we know that t t t 1 (B )d B = 1 (B )d B + 1 (B )d B = 0, q.s. {−ε,ε} s h is {−ε} s h is {ε} s h is Z0 Z0 Z0 15 ′′ Let us put ϕε (x) = 0,x = ε. 1 ± 2 Since 0 ϕ′′ , we have ϕ′′ ϕ′′ on A . Therefore, we have ≤ n ≤ ε | n − ε |≤ ε n,ε t t Eˆ ′′ ′′ [ ϕn(Bs)d B s ϕε (Bs)d B s ] | 0 h i − 0 h i | Zt Z Eˆ ′′ ′′ [ ϕn(Bs) ϕε (Bs) d B s] ≤ 0 | − | h i Z t 2Eˆ [ 1An,ε (Bs)d B s] ≤ ε 0 h i Z t t 2Eˆ 2Eˆ [ 1(−ε− 1 ,−ε+ 1 )(Bs)d B s]+ [ 1(ε− 1 ,ε+ 1 )(Bs)d B s]. ≤ ε n n h i ε n n h i Z0 Z0 From Proposition 4.6 we conclude that t t C Eˆ[ ϕ′′ (B )d B ϕ′′(B )d B ] 0, as n . | n s h is − ε s h is| ≤ εn → →∞ Z0 Z0 Therefore, letting n in (4.1), we conclude that →∞ 1 t 1 t ϕ (B )= ε + ϕ′ (B )dB + 1 (B )d B . (4.2) ε t 2 ε s s 2ε (−ε,ε) s h is Z0 Z0 From the definition of ϕε it follows that Eˆ[(ϕ (B ) B )2] ε t − | t| Eˆ[(ϕ (B ) B )21 ]+ Eˆ[(ϕ (B ) B )21 ] ≤ ε t − | t| |Bt|≥ε ε t − | t| |Bt|<ε = Eˆ[(ϕ (B ) B )21 ] ε2 0, as ε 0. ε t − | t| |Bt|<ε ≤ → → By virtue of Lemma 4.5, we know that t t Eˆ ′ 2 [ ϕε(Bs)dBs sgn(Bs)dBs ] | 0 − 0 | Z t Z Eˆ ′ 2 C [ (ϕε(Bs) sgn(Bs)) d B s] ≤ 0 − h i Z t CEˆ[ 1 (B )d B ], ≤ (−ε,ε) s h is Z0 and Proposition 4.6 allows to conclude that t t Eˆ[ ϕ′ (B )dB sgn(B )dB 2] 0, as ε 0. | ε s s − s s| → → Z0 Z0 Finally, from equation (4.2) it follows that t B = sgn(B )dB + L0. | t| s s t Z0 16 The proof is complete. 2 Proof of Lemma 4.8: Now we prove that sgn(B·) M∗ (0, T ). Let π = 0 = t0 < ∈ n−1 { n t1 t Eˆ ′ 2 [ (ϕε(Bs) sgn(Bs)) d B s] 0 − h i Z t CEˆ[ 1 (B )d B ] 0, as ε 0. ≤ (−ε,ε) s h is → → Z0 Consequently, from the above estimates t ′ n 2 lim Eˆ[ (sgn(Bs) ϕ (B )) d B s] 0, as ε 0. n→∞ − n s h i → → Z0 2 Remark 4.10 Similar to the proof of the above theorem, we can obtain 1{B.>a} M∗ (0, T ) 2 ∈ and 1{B.≤a} M∗ (0, T ). ∈ f Remark 4.11fIn analogy to Theorem 4.9, we obtain two other forms of Tanaka formula: t 1 (B a)+ = ( a)+ + 1 dB + La, t − − {Bs>a} s 2 t Z0 and t 1 (B a)− = ( a)− + 1 dB + La. t − − {Bs≤a} s 2 t Z0 Denis, Hu and Peng [3] obtained an extension of Kolmogorov continuity criterion to the framework of nonlinear expectation spaces. It will be needed for the study of joint a continuity of local time for the G-Brownian motion Lt in (t, a). 17 p Lemma 4.12 Let d 1, p > 0 and (X ) d L be such that there exist positive ≥ t t∈[0,T ] ⊆ constants C and ε> 0 such that Eˆ[ X X p] C t s d+ε, for all t,s [0, T ]. | t − s| ≤ | − | ∈ ˜ ˜ Then the field (Xt)t∈[0,T ]d admits a continuous modification (X)t∈[0,T ]d (i.e. Xt = Xt, q.s., for all t [0, T ]) such that ∈ ˜ ˜ Eˆ Xt Xs p (sup | − α |) < + , t6=s t s ∞ h | − | i for every α [0, ε/p[. As a consequence, paths of X˜ are quasi-surely H¨older continuous of ∈ order α, for every α < ε/p. We will show that the local time for the G-Brownian motion has a jointly continuous modification. For the classical case, we refer to [2], [6] and [18]. For the proof we use an approximation method here, which is different from the classical case. Theorem 4.13 For all t [0, T ], there exists a jointly continuous modification of (a, t) ∈ 7→ La. Moreover, (a, t) La is H¨older continuous of order γ for all γ < 1 . t 7→ t 2 Proof. From Lemma 4.5 we know that, for all s,t 0 and a, b R, ≥ ∈ Eˆ[( B a B b )2p] | t − | − | s − | C Eˆ[ B B 2p]+ C a b 2p ≤ p | t − s| p| − | C t s p + C a b 2p (4.3) ≤ p| − | p| − | The constant Cp only depends on p. By choosing p > 2, we see from the generalized Kolmogorov continuity criterion (see Lemma 4.12) that B a has a jointly continuous | t − | modification in (a, t). Let us now prove that also the integral t sgn(B a)dB has a jointly continuous 0 s − s modification. For this end we let δ > 0 and p> 0. Then R t t 2p Eˆ[ sgn(Bs a)dBs sgn(Bs a δ)dBs ] | 0 − − 0 − − | Z t Z 22pEˆ[ 1 (B )dB 2p], ≤ | [a,a+δ] s s| Z0 and from Lemma 4.5 it follows that t t 2p Eˆ[ sgn(Bs a)dBs sgn(Bs a δ)dBs ] | 0 − − 0 − − | Z t Z C Eˆ[ 1 (B )d B p]. (4.4) ≤ p | [a,a+δ] s h is| Z0 18 We denote ϕ the same as in the proof of Proposition 4.6. By applying the generalized Itˆoformula for the G-Brownian motion (see Theorem 2.17) to ϕ(Bt), we deduce that t 1 t ϕ(B )= ϕ′(B )dB + ϕ′′(B )d B . t s s 2 s h is Z0 Z0 Therefore, t t 2 ′′ 1[a,a+δ](Bs)d B s δ ϕ (Bs)d B s 0 h i ≤ 0 h i Z Z t 2 2 ′ = 2δ ϕ(Bt) 2δ ϕ (Bs)dBs − 0 tZ 8δ B + 2δ2 ϕ′(B )dB . (4.5) ≤ | t| | s s| Z0 Hereafter, Cp may be different from line to line, but only depends on p. From the inequal- ities (4.4), (4.5) and Lemma 4.5 it follows that t t 2p Eˆ[ sgn(Bs a)dBs sgn(Bs a δ)dBs ] | 0 − − 0 − − | Z t Z 2 ′ p CpEˆ[(8δ Bt + 2δ ϕ (Bs)dBs ) ] ≤ | | | 0 | Z t C δpEˆ[ B p]+ C δ2pEˆ[ ϕ′(B )dB p] ≤ p | t| p | s s| Z0 t p pEˆ p 2pEˆ ′ 2 2 Cpδ [ Bt ]+ Cpδ [( (ϕ (Bs)) d B s) ] ≤ | | 0 h i Z p pEˆ p pEˆ 2 Cpδ [ Bt ]+ Cpδ [ B t ] ≤ | | h i p pEˆ p pEˆ 2 Cpδ [ BT ]+ Cpδ [ B T ] ≤ p | | h i = Cpδ . (4.6) Therefore, for all 0 r t, ≤ ≤ t r 2p Eˆ[ sgn(Bs a)dBs sgn(Bs a δ)dBs ] | 0 − − 0 − − | Z r Z r 2p CpEˆ[ sgn(Bs a)dBs sgn(Bs a δ)dBs ] ≤ | 0 − − 0 − − | Z t Z 2p +CpEˆ[ sgn(Bs a)dBs ] | r − | Z t C δp + C Eˆ[( (sgn(B a))2d B )p] ≤ p p s − h is Zr C δp + C t r p. (4.7) ≤ p p| − | 19 We choose p> 2, then by the generalized Kolmogorov continuity criterion (Lemma 4.12), we obtain the existence of a jointly continuous modification of (α, t) Lα. The proof is 7→ t complete. Corollary 4.14 For all t [0, T ], a R, we have ∈ ∈ t t t sgn(B a)dB , 1 (B )dB , 1 (B )dB s − s [a,∞) s s (−∞,a) s s Z0 Z0 Z0 have a jointly continuous modification. 5 Quadratic variation of local time for the G-Brownian mo- tion The objective of this section is to study the quadratic variation of the local time for the G-Brownian motion. For this end, we begin with the following Lemma: Lemma 5.1 If f : R [0, ) is a continuous function with compact support, then, for → ∞ all t 0, we have ≥ ∞ t t ∞ f(a) sgn(Bs a)dBs da = f(a)sgn(Bs a)da dBs, q.s., −∞ 0 − 0 −∞ − Z Z Z Z and ∞ t t ∞ f(a) 1[a,∞)(Bs)dBs da = f(a)1[a,∞)(Bs)da dBs, q.s. −∞ 0 0 −∞ Z Z Z Z Proof. We only prove the fist equality, the second equality can be proved in a similar way. Without loss of generality we can assume that f has its support in [0, 1]. Let 2n−1 1 k k ϕ (x)= f( )sgn(x ),x R,n 1. n 2n 2n − 2n ∈ ≥ Xk=0 Then 2n−1 t 1 k t k ϕn(Bs)dBs = n f( n ) sgn(Bs n )dBs. 0 2 2 0 − 2 Z Xk=0 Z By the proof Theorem 4.13 we know that the integral t sgn(B a)dB is jointly contin- 0 s − s uous in (a, t). Therefore, R 1 t t f(a)sgn(Bs a)dBs da = lim ϕn(Bs)dBs, q.s. (5.1) 0 0 − n→∞ 0 Z Z Z 20 From Lemma 4.5 and the convergence of ϕ (x) to 1 f(a)sgn(x a)da, uniformly w.r.t n 0 − x R, we obtain ∈ R t t 1 Eˆ[ ϕ (B )dB f(a)sgn(B a)da dB 2] | n s s − s − s| Z0 Z0 Z0 t 1 2 CEˆ[ ϕ (B ) f(a)sgn(B a)da ds] ≤ n s − s − Z0 Z0 0, as n . → →∞ Using Proposition 2.11 we can deduce the existence of a subsequence ϕ ∞ such that { nk }k=1 t 1 t f(a)sgn(Bs a)da dBs = lim ϕnk (Bs)dBs, q.s. (5.2) 0 0 − k→∞ 0 Z Z Z Finally, the relations (5.1) and (5.2) yield 1 t t 1 f(a) sgn(Bs a)dBs da = f(a)sgn(Bs a)da dBs, q.s. 0 0 − 0 0 − Z Z Z Z The proof is complete. We now establish an occupation time formula. For the classical case, we refer to [2], [6] and [18]. Theorem 5.2 For all t 0 and all reals a b, we have ≥ ≤ t b 1 (B )d B = Lxdx, q.s. (a,b) s h is t Z0 Za Proof. Put ϕ (y) := ϕ (y x), where ϕ is the function in the proof of Theorem 4.9. x,ε ε − ε In analogy to equality (4.2) we have t 1 t ϕ (B )= ϕ (B )+ ϕ′ (B )dB + 1 (B )d B , (5.3) x,ε t x,ε 0 x,ε s s 2ε (x−ε,x+ε) s h is Z0 Z0 where 1 x+ε ϕ′ (z)= 1 (z)dy 1. x,ε ε [y,∞) − Zx−ε Thanks to Lemma 5.1, we have t 1 t x+ε ϕ′ (B )dB = 1 (B )dydB B x,ε s s ε [y,∞) s s − t Z0 Z0 Zx−ε 1 x+ε t = 1 (B )dB dy B . ε [y,∞) s s − t Zx−ε Z0 21 Therefore, by equality (5.3) we have b 1 x+ε t ϕx,ε(Bt) ϕx,ε(B0) 1[y,∞)(Bs)dBsdy + Bt dx a − − ε x−ε 0 Z Z Z 1 b t = 1 (B )d B dx. (5.4) 2ε (x−ε,x+ε) s h is Za Z0 According to Corollary 4.14, we obtain that b 1 x+ε t lim ϕx,ε(Bt) ϕx,ε(B0) 1[y,∞)(Bs)dBsdy + Bt dx ε↓0 − − ε Za Zx−ε Z0 b t = B x x 2 1 (B )dB + B dx | t − | − | |− [x,∞) s s t Za Z0 b t = Bt x x sgn(Bs x)dBs dx, q.s. (5.5) a | − | − | |− 0 − Z Z For z R, we have ∈ 1 b 1 1 lim 1(x−ε,x+ε)(z)dx = 1(a,b)(z)+ 1{a}(z)+ 1{b}(z). ε↓0 2ε 2 2 Za Therefore, 1 b t lim 1(x−ε,x+ε)(Bs)d B sdx ε↓0 2ε h i Za Z0 1 t b = lim 1(x−ε,x+ε)(Bs)dxd B s ε↓0 2ε h i Z0 Za t 1 t = 1 (B )d B + 1 (B )d B , q.s. (5.6) (a,b) s h is 2 {a,b} s h is Z0 Z0 From Corollary 4.7 we know that t 1 (B )d B = 0, q.s. (5.7) {a,b} s h is Z0 Consequently, from (6.3), (6.4), (6.5) and (6.6) it follows that b t t Bt x x sgn(Bs x)dBs dx = 1(a,b)(Bs)d B s, q.s. a | − | − | |− 0 − 0 h i Z Z Z Finally, thanks to Theorem 4.9, we have b t Lxdx = 1 (B )d B , q.s. t (a,b) s h is Za Z0 22 The proof is complete. We now study the quadratic variation of the local time for the G-Brownian motion. For simplicity, we denote, for t 0 and x R, ≥ ∈ t Y x = sgn(B x)dB , t s − s Z0 N x = B x x . t | t − | − | | Then, from Theorem 4.9 we have Lx = N x Y x. t t − t We first give the following lemma, which is an immediate consequence of the proof of Theorem 4.13. Lemma 5.3 For every p 1, there exists a constant C = C(p) > 0 such that ≥ x y p p p Eˆ[ N N ] C( x y + t s 2 ), | t − s | ≤ | − | | − | x y p p Eˆ[ Y Y ] C x y 2 , | t − t | ≤ | − | for all t,s [0, T ] and x,y R. ∈ ∈ Now we give the main result in this section. Theorem 5.4 Let σ > 0. Then for all p 1 and a b, we have along the sequence of i(b a) ≥ ≤ partitions of π = an = a + − , i = 0, 1, , 2n ,n 1, of the interval of [a, b] the n { i 2n · · · } ≥ following convergence n 2 −1 b an n i+1 ai 2 x Lp lim (Lt Lt ) = 4 Lt dx, in , n→∞ − a Xi=0 Z uniformly with respect to t [0, T ]. ∈ Proof. By applying the generalized Itˆoformula for the G-Brownian motion (cf. Theorem 2.17) we obtain 2n−1 n n b ai+1 ai 2 x (Lt Lt ) 4 Lt dx − − a Xi=0 Z 2n−1 n n n n b ai+1 ai ai+1 ai 2 x = (Nt Nt Yt + Yt ) 4 Lt dx − − − a Xi=0 Z 23 2n−1 an an an an an an = (N i+1 N i )2 2(N i+1 N i )(Y i+1 Y i ) t − t − t − t t − t i=0 X b an an +(Y i+1 Y i )2 4 Lxdx t − t − t Za 2n−1 an an an an an = (N i+1 N i )2 2(N ai+1 N i )(Y i+1 Y i ) t − t − t − t t − t Xi=0 t n n t b ai+1 ai x +4 (Y Y )1 n n (B )dB + 4 1 n n (B )d B 4 L dx. s s (ai ,ai+1) s s (ai ,ai+1) s s t 0 − 0 h i − a Z Z Z Thus, thanks to Theorem 5.2, we get 2n−1 n n b ai+1 ai 2 x (Lt Lt ) 4 Lt dx − − a Xi=0 Z 2n−1 2n−1 an an an an an an = (N i+1 N i )2 2 (N i+1 N i )(Y i+1 Y i ) t − t − t − t t − t Xi=0 Xi=0 2n−1 t n n ai+1 ai +4 (Y Y )1 n n (B )dB . (5.8) s s (ai ,ai+1) s s 0 − Xi=0 Z Then, from Lemma 5.3 and the subadditivity of the sublinear expectation Eˆ[ ] it follows · that 2n−1 2n−1 an an p an an Eˆ[ (N i+1 N i )2 ] 2n(p−1) Eˆ[(N i+1 N i )2p] t − t ≤ t − t Xi=0 Xi=0 C2n(p−1)2n2−2np = C2−np. (5.9) ≤ Moreover, by virtue of Lemma 5.3, the subadditivity of the sublinear expectation Eˆ[ ] and · H¨older inequality for the sublinear expectation (see [13]) we can estimate the second term at the right hand of (5.8) as follows: 2n−1 an an an an p Eˆ[ (N i+1 N i )(Y i+1 Y i ) ] t − t t − t Xi=0 2n−1 an an an an 2n(p−1) Eˆ[ (N i+1 N i )(Y i+1 Y i ) p] ≤ | t − t t − t | i=0 Xn 2 −1 1 1 n n n n a a 2 a a 2 2n(p−1) Eˆ[ N i+1 N i 2p] Eˆ[ Y i+1 Y i 2p] ≤ | t − t | | t − t | Xi=0 n(p−1) n −np − n p − np C2 2 2 2 2 = C2 2 . (5.10) ≤ 24 Here, for the latter estimate we have used Lemma 5.3. Finally, let us estimate the third term at the right hand of (5.8). Using again the subadditivity of the sublinear expectation Eˆ[ ] and H¨older inequality for the sublinear · expectation as well as Lemma 4.5 we have n 2 −1 t an n p Eˆ i+1 ai [ (Y Y )1(an,an )(Bs)dBs ] t − t i i+1 i=0 Z0 X n 2 −1 p t n n ai+1 ai 2 2 CEˆ[ (Y Y )1 n n (B ) ds ] t t (ai ,ai+1) s ≤ 0 − Z i=0 nX t 2 −1 an n p Eˆ i+1 ai C [ Y Y 1(an,an )(Bs) ds] ≤ | t − t | i i+1 Z0 i=0 n X t 2 −1 an n Eˆ i+1 ai p = C [ Y Y 1(an,an )(Bs)ds] | t − t | i i+1 Z0 i=0 n X 2 −1 1 1 t n n t ai+1 ai 2p 2 2 C Eˆ[ Y Y ds] Eˆ[ 1 n n (B )ds] . t t (ai ,ai+1) s ≤ 0 | − | 0 Xi=0 Z Z Consequently, thanks to Lemma 5.3 and Proposition 4.6, we have 2n−1 t n n p ai+1 ai Eˆ[ (Y Y )1 n n (B )dB ] t t (ai ,ai+1) s s 0 − Xi=0 Z n − np − n − n(p−1) C2 2 2 2 2 = C2 2 . (5.11) ≤ Finally, by substituting the estimates (5.9),(5.10) and (5.11) for the right hand of (5.8), we obtain 2n−1 b p Eˆ ai+1 ai 2 x [ (Lt Lt ) 4 Lt dx ] − − a Xi=0 Z −np − np − n(p−1) C(2 + 2 2 + 2 2 ) 0, as n , ≤ → →∞ uniformly with respect to t [0, T ]. The proof is complete. ∈ 6 A generalized Itˆoformula for convex functions In this section, our objective is to obtain a generalization of Itˆo’s formula for the G- Brownian motion with help of the Tanaka formula for the G-Brownian motion, which generalizes the corresponding result in [8]. 25 Given a convex function f : R R, we associate with the measure µ on (R) which → B is defined as follows: µ[a, b)= f ′ (b) f ′ (a), for all a < b. − − − Then, for ϕ C2 (R), we have ∈ K ϕ′′(x)f(x)dx = ϕ(x)µ(dx). R R Z Z This relation allows to identify the generalized second derivative of f with the measure µ. Now we establish the main result in this section. Theorem 6.1 Let f : R R be a convex function and assume that its right derivative → f ′ is bounded and f ′ (B ) M 2(0, T ). Then, for all t 0, + − · ∈ ω ≥ t ∞ ′ a ′′ f(Bt)= f(0) + f−(Bs)dBs + Lt f (da), Z0 Z−∞ where f ′′ = µ is the measure introduced above. Before giving the proof of Theorem 6.1, we establish the following lemmas, which will be needed later. Lemma 6.2 If f : R [0, ) is a convex function and ϕ C1(R), then for all t 0, we → ∞ ∈ b ≥ have ∞ t t ∞ ′′ ′′ ϕ(Bs a)dBs f (da)= ϕ(Bs a)f (da) dBs. −∞ 0 − 0 −∞ − Z Z Z Z Proof. We put x Φ(x)= ϕ(y)dy,x R. ∈ Z0 Then, by applying the generalized Itˆoformula for the G-Brownian motion (cf. Theorem 2.17) to Φ(B a), we deduce that t − t 1 t Φ(B a)=Φ( a)+ ϕ(B a)dB + ϕ′(B a)d B . t − − s − s 2 s − h is Z0 Z0 Therefore, ∞ t ϕ(B a)dB f ′′(da) s − s Z−∞ Z0 ∞ ∞ ∞ t ′′ ′′ 1 ′ ′′ = Φ(Bt a)f (da) Φ( a) f (da) ϕ (Bs a)d B s f (da) −∞ − − − −∞ − 2 −∞ 0 − h i Z Z Z Z 26 ∞ ∞ = Φ(B a)f ′′(da) Φ( a) f ′′(da) t − − − Z−∞ Z−∞ t ∞ 1 ′ ′′ ϕ (Bs a)f (da) d B s . (6.1) −2 0 −∞ − h i Z Z Let ∞ g(x)= Φ(x a)f ′′(da). − Z−∞ Then, using again the generalized Itˆoformula for the G-Brownian motion (see Theorem 2.17) to g(Bt), we deduce that t 1 t g(B )= g(0) + g′(B )dB + g′′(B )d B . t s s 2 s h is Z0 Z0 Consequently, t ∞ ∞ ∞ ϕ(B a)f ′′(da) dB = Φ(B a)f ′′(da) Φ( a) f ′′(da) s − s t − − − Z0 Z−∞ Z−∞ Z−∞ t ∞ 1 ′ ′′ ϕ (Bs a)f (da) d B s . (6.2) − 2 0 −∞ − h i Z Z Finally, (6.1) and (6.2) allow to conclude that ∞ t t ∞ ′′ ′′ ϕ(Bs a)dBs f (da)= ϕ(Bs a)f (da) dBs. −∞ 0 − 0 −∞ − Z Z Z Z The proof is complete. Lemma 6.3 If f : R [0, ) is a convex function and its right derivative f ′ is bounded, → ∞ + then for all t 0, we have ≥ ∞ t t ∞ ′′ ′′ sgn(Bs a)dBs f (da)= sgn(Bs a)f (da) dBs, q.s. −∞ 0 − 0 −∞ − Z Z Z Z Proof. For ε> 0, we define C1 function ϕ : R R such that ϕ 1, and − ε → | ε|≤ 1; if x ε, ϕ (x)= − ≤− ε 1; if x ε. ≥ Then, we have ϕ (x) sgn(x) 21 (x), for x R. From Lemma 6.2 we have | ε − |≤ (−ε,ε) ∈ ∞ t t ∞ ′′ ′′ ϕε(Bs a)dBs f (da)= ϕε(Bs a)f (da) dBs. −∞ 0 − 0 −∞ − Z Z Z Z 27 From Lemma 4.5 and Proposition 4.6 it then follows that t ∞ t ∞ Eˆ[ ϕ (B a)f ′′(da) dB sgn(B a)f ′′(da) dB ] | ε s − s − s − s| Z0 Z−∞ Z0 Z−∞ t ∞ ∞ 2 ′′ ′′ 1 Eˆ[ ϕ (B a)f (da) sgn(B a)f (da) d B ] 2 ≤ ε s − − s − h is Z0 Z−∞ Z−∞ t ∞ 2 ′′ 1 Eˆ[ ϕ (B a) sgn(B a) f (da) d B ] 2 ≤ | ε s − − s − | h is Z0 Z−∞ t ∞ 2 ′′ 1 CEˆ[ 1 (B a)f (da) d B ] 2 ≤ {−ε,ε} s − h is Z0 Z−∞ ∞ 1 ∞ t 1 ′′ 2 Eˆ ′′ 2 C f (da) [ 1{−ε,ε}(Bs a)d B s]f (da) ≤ −∞ −∞ 0 − h i Z ∞ Z Z 1 ′′ Cε 2 f (da) 0, as ε 0, ≤ → → Z−∞ and ∞ t ∞ t Eˆ[ ϕ (B a)dB f ′′(da) sgn(B a)dB f ′′(da) ] | ε s − s − s − s | Z−∞ Z0 Z−∞ Z0 ∞ t Eˆ[ ϕ (B a) sgn(B a) dB ]f ′′(da) ≤ | ε s − − s − s| Z−∞ Z0 ∞ t 2 1 ′′ Eˆ[ ϕ (B a) sgn(B a) d B ] 2 f (da) ≤ | ε s − − s − h is | Z−∞ Z0 ∞ t 1 Eˆ 2 ′′ C [ 1{−ε,ε}(Bs a)d B s] f (da) ≤ −∞ 0 − h i Z ∞ Z 1 ′′ Cε 2 f (da) 0, as ε 0, ≤ → → Z−∞ Consequently, ∞ t t ∞ ′′ ′′ sgn(Bs a)dBs f (da)= sgn(Bs a)f (da) dBs, q.s. −∞ 0 − 0 −∞ − Z Z Z Z The proof is complete. Now we give the proof of Theorem 6.1. ′ ′′ Proof. Since the function f+ is bounded, the measure f has its support in a compact interval I. From Section 3 in Appendix in [18], we know that 1 f(x)= x a f ′′(a)da + α x + β , (6.3) 2 | − | I I ZI 28 where αI and βI are constants. Moreover, for all x outside of a countable set Γ in R, 1 f ′ (x)= sgn(x a) f ′′(da)+ α . (6.4) − 2 − | I ZI However, due to Corollary 4.7, t 1 (B )d B = 0, q.s.,t 0. Γ s h is ≥ Z0 Consequently, thanks to the equalities (6.3) and (6.4), for all t 0, we have ≥ 1 f(B )= B a f ′′(da)+ α B + β , q.s., (6.5) t 2 | t − | I t I ZI and 1 f ′ (B )= sgn(B a) f ′′(da)+ α , q.s. (6.6) − t 2 t − | I ZI On the other hand, from Theorem 4.9 and equality (6.5) we deduce that t 1 a ′′ f(Bt) = a + sgn(Bs a)dBs + Lt f (da)+ αI Bt + βI 2 I | | 0 − Z Z t 1 ′′ 1 a ′′ = f(0) + sgn(Bs a)dBs f (da)+ Lt f (da)+ αI Bt. 2 I 0 − 2 I Z Z Z Finally, thanks to Lemma 6.3 and equality (6.6), we have t 1 ′′ 1 a ′′ f(Bt) = f(0) + sgn(Bs a)f (da) dBs + Lt f (da)+ αI Bt 2 0 I − 2 I Zt Z Z ′ a ′′ = f(0) + f−(Bs)dBs + Lt f (da). Z0 ZI The proof is complete. Acknowledgements. The author thanks Prof. Rainer Buckdahn for his careful reading and helpful sug- gestions. The author also thanks the editor and anonymous referees for their helpful suggestions. This work is supported by the Young Scholar Award for Doctoral Students of the Ministry of Education of China and the Marie Curie Initial Training Network (PITN- GA-2008-213841). 29 References [1] M. Avellaneda, A. Levy and A. Paras, Pricing and hedging derivative securities in markets with uncertain volatilities, Appl. Math. 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