Progresses and Problems in Theory of Nonlinear Expectations and - PowerPoint PPT Presentation

Law of Large Numbers and Central Limit Theorem Theorem ( Law of large numbers ) Let { Y i } ∞ i = 1 be a sequence of R d -valued random variables on a sublinear expectation space ( Ω , H , E ) . We assume that d = Y i and Y i + 1 is indep. from { Y 1 , · · · , Y i } for each i = 1 , 2 , · · · . Y i + 1 Then ¯ S n = 1 n ∑ n i = 1 Y i converges in law to a maximal distribution: n → ∞ E [ ϕ ( ¯ lim S n )] = E [ ϕ ( η )] = max µ ≤ v ≤ µ ϕ ( v ) , with µ = E [ X 1 ] , µ = − E [ − X 1 ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Maximal Distribution and G -normal Distribution “worst case risk measure”— maximal distribution . Definition ( maximal distribution ) A random variable η in ( Ω , H , E ) is called maximal distributed, denoted by η d = M ([ µ , µ ]) , if E [ ϕ ( η )] = max µ ≤ y ≤ µ ϕ ( y ) with µ = E [ η ] , µ = − E [ − η ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Remark. In general a maximal distributed η satisfies η d a η + b ¯ = ( a + b ) η for a , b ≥ 0 , where ¯ η is an independent copy of η . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Central limit theorem with zero-mean . Let { X i } ∞ i = 1 be a sequence of R d -valued random variables on a sublinear d ( Ω , H , E ) . We assume that X i + 1 = X i and X i + 1 is independent from { X 1 , · · · , X i } for each i = 1 , 2 , · · · . We further assume that E [ X 1 ] = E [ − X 1 ] = 0 . Then we have convergence in law: n n → ∞ E [ ϕ ( 1 d ∑ = N ( 0 , [ σ 2 , σ 2 ]) lim √ n X i )] = E [ ϕ ( X )] , X i = 1 where σ 2 = E [ X 2 σ 2 = − E [ − X 2 1 ] , 1 ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Definition ( G -normal distribution ) A d -dimensional random variable X on a sublinear ( Ω , H , E ) is called (centralized) G -normal distributed if and � d a 2 + b 2 X aX + b ¯ X = for a , b ≥ 0 , d where ¯ = N ( 0 , [ σ 2 , σ 2 ]) , X is an independent copy of X . We denote X with σ 2 = E [ X 2 ] and σ 2 = − E [ − X ] ( E [ X ] = − E [ − X ] = 0). Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

d = N ( 0 , [ σ 2 , σ 2 ] , then If X For each convex function ϕ , we have � ∞ − ∞ ϕ ( σ 2 y ) exp ( − y 2 1 √ E [ ϕ ( X )] = 2 ) dy . 2 π Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

For each concave function ϕ , we have � ∞ − ∞ ϕ ( σ 2 y ) exp ( − y 2 1 √ E [ ϕ ( X )] = 2 ) dy . 2 π Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Remark. When d = 1, the sequence { ¯ S n } ∞ n = 1 converges in law to N ( 0 , [ σ 2 , σ 2 ]) , where σ 2 = E [ X 2 1 ] and σ 2 = − E [ − X 2 1 ] . In particular, if σ 2 = σ 2 , then it becomes a classical central limit theorem. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Central Limit Theorem with law of large numbers . Let { ( X i , Y i ) } ∞ i = 1 be a sequence of 2 d -valued r.v. in a sublinear expectation space ( Ω , H , E ) . Assume that ( X i + 1 , Y i + 1 ) d = ( X i , Y i ) and ( X i + 1 , Y i + 1 ) is indep. of { ( X 1 , Y 1 ) , · · · , ( X i , Y i ) } . We further assume that E [ X 1 ] = E [ − X 1 ] = 0 . Then n ( X i √ n + Y i ∑ n → ∞ E [ ϕ ( lim n )] = E [ ϕ ( X + η )] , i = 1 for all functions ϕ ∈ C Lip ( R d ) , where the pair ( X , η ) is G -distributed. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Definition The pair ( X , η ) is called G -distributed . It satisfies � X , a 2 η + b 2 ¯ a 2 + b 2 X , ( a 2 + b 2 ) η ) , η ) d ( aX + b ¯ = ( for a , b ≥ 0 , where ( ¯ X , ¯ η ) is an independent copy of ( X , η ) . Thus X is G -normal and η is maximal distributed. G ( p , A ) : = E [ � p , η � + 1 p ∈ R d , 2 � AX , X � ] , A ∈ S ( d ) . S ( d ) is the collection of all d × d symmetric matrices. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Definition Remark. G ( p , X ) satisfies  p , A + ¯ p , ¯ ≤ G ( p , A ) + G ( ¯ G ( p + ¯ A ) A ) ,  G ( λ p , λ A ) = λ G ( p , A ) , ∀ λ ≥ 0 ,  ≥ G ( p , ¯ if A ≥ ¯ G ( p , A ) A ) , A . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The above pair ( X , η ) is characterized via √ x , y ∈ R d , t ≥ 0 , u ( t , x , y ) = E [ ϕ ( x + tX , y + t η )] , by the following parabolic partial differential equation (PDE) defined on [ 0 , ∞ ) × R d × R d : ∂ t u − G ( D y u , D 2 u | t = 0 = ϕ . x u ) = 0 , = N ( 0 , [ σ 2 , σ 2 ]) , η d d = M ([ µ , µ 2 ]) If d = 1, then X Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Corollary If both ( X , η ) and ( ¯ η ) are G-distributed with the same G, i.e., X , ¯ G ( p , A ) : = E [ 1 2 � AX , X � + � p , η � ] = E [ 1 2 � A ¯ X , ¯ X � + � p , ¯ η � ] ∀ ( p , A ) , then ( X , η ) d d = ( ¯ η ) . In particular, X = − X. X , ¯ Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Problem How to estimate [ µ , µ ] and [ σ 2 , σ 2 ] with a given data { x i } N i = 0 ? More generaly, How to determine G ( p , A ) ? How to use the above LLN and CLT to do statistics? [Zengjing Chen 2009]: n c ( lim inf 1 ∑ ˆ Y i < µ ) = 0 , n i = 1 n c ( lim sup 1 ∑ ˆ Y i > µ ) = 0 . n i = 1 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

[Peng2009] Data based sublinear mean n 1 ˜ ∑ E [ ψ ( Y )] : = lim sup ψ ( y i ) n n → ∞ i = 1 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

[Peng2009] Data based sublinear mean n 1 ˜ ∑ E [ ψ ( Y )] : = lim sup ψ ( y i ) n n → ∞ i = 1 In risk control: try to use N ( 0 , [ σ 2 , σ 2 ]) -model. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Notes and Comments Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The contents of this chapter are mainly from Peng (2008) [66] (see also Peng (2007) [62]). The notion of G -normal distribution was firstly introduced by Peng (2006) [61] for 1-dimensional case, and Peng (2008) [65] for multi-dimensional case. In the classical situation, a distribution satisfying equation (–ch2e1) is said to be stable (see L´ evy (1925) [43] and (1965) [44]). In this sense, our G -normal distribution can be considered as the most typical stable distribution under the framework of sublinear expectations. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Marinacci (1999) [47] used different notions of distributions and independence via capacity and the corresponding Choquet expectation to obtain a law of large numbers and a central limit theorem for non-additive probabilities (see also Maccheroni and Marinacci (2005) [48] ). But since a sublinear expectation can not be characterized by the corresponding capacity, our results can not be derived from theirs. In fact, our results show that the limit in CLT, under uncertainty, is a G -normal distribution in which the distribution uncertainty is not just the parameter of the classical normal distributions (see Exercise –exxee1). Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The notion of viscosity solutions plays a basic role in the definition and properties of G -normal distribution and maximal distribution. This notion was initially introduced by Crandall and Lions (1983) [15]. This is a fundamentally important notion in the theory of nonlinear parabolic and elliptic PDEs. Readers are referred to Crandall, Ishii and Lions (1992) [16] for rich references of the beautiful and powerful theory of viscosity solutions. For books on the theory of viscosity solutions and the related HJB equations, see Barles (1994) [6], Fleming and Soner (1992) [26] as well as Yong and Zhou (1999) [ ? ]. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We note that, for the case when the uniform elliptic condition holds, the viscosity solution (–e320) becomes a classical C 1 + α 2 , 2 + α -solution (see Krylov (1987) [42] and the recent works in Cabre and Caffarelli (1997) [ ? ] and Wang (1992) [73]). In 1-dimensional situation, when σ 2 > 0, the G -equation becomes the following Barenblatt equation: ∂ t u + γ | ∂ t u | = △ u , | γ | < 1 . This equation was first introduced by Barenblatt (1979) [5] (see also Avellaneda, Levy and Paras (1995) [4]). Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

G -Brownian Motion and Itˆ o’s Integral Definition A d -dimensional process ( B t ) t ≥ 0 = ( B 1 t , · · · , B d t ) t ≥ 0 defined on a sublinear expectation space ( Ω , H , E ) is called a G –Brownian motion if t | k ∈ H , for i = 1 , · · · , d and k = 1 , 2 , 3, and | B i (i) B 0 ( ω )= 0; d (ii) For each t , s ≥ 0, B t + s − B t = B s and B t + s − B t is independent from ( B t 1 , B t 2 , · · · , B t n ) , for each n ∈ N and 0 ≤ t 1 ≤ · · · ≤ t n ≤ t . (iii) lim t ↓ 0 E [ | B t | 3 ] t − 1 = 0. (iv) E [ B t ] = E [ − B t ] = 0. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The following theorem gives a characterization of G -Brownian motion’s distribution. Theorem t , · · · , B d Let ( B t ) t ≥ 0 = ( B 1 t ) t ≥ 0 be a d-dimensional process defined on a sublinear expectation space ( Ω , H , E ) . Then B s / √ s d = B 1 is G-normally distributed with G ( A ) : = E [ 1 2 � AB 1 , B 1 � ] , A ∈ S ( d ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Construction of G -Brownian Motion Ω = : C d 0 ( R + ) the space of all R d –valued continuous paths ( ω t ) t ∈ R + , with ω 0 = 0, equipped with ∞ 2 − i [( max ρ ( ω 1 , ω 2 ) : = ∑ t ∈ [ 0 , i ] | ω 1 t − ω 2 t | ) ∧ 1 ] . i = 1 We set Ω T : = { ω ·∧ T : ω ∈ Ω } . We will consider the canonical process B t ( ω ) = ω t , t ∈ [ 0 , ∞ ) , for ω ∈ Ω . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

For each fixed T ∈ [ 0 , ∞ ) , we set L ip ( Ω T ) : = { ϕ ( B t 1 ∧ T , · · · , B t n ∧ T ) : n ∈ N , t 1 , · · · , t n ∈ [ 0 , ∞ ) , ϕ ∈ C Lip ( R d It is clear that L ip ( Ω t ) ⊆ L ip ( Ω T ) , for t ≤ T . We also set ∞ � L ip ( Ω ) : = L ip ( Ω n ) . n = 1 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Let ( ξ i ) ∞ i = 1 ξ i ∈ � H is a sublinear expectation space ( � Ω , � H , � E ) such that ξ i is G -normal distributed for a given function G : , and ξ i + 1 is independent from ( ξ 1 , · · · , ξ i ) for each i = 1 , 2 , · · · . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We now introduce a sublinear expectation ˆ E defined on L ip ( Ω ) via the following procedure: for each X ∈ L ip ( Ω ) with X = ϕ ( B t 1 − B t 0 , B t 2 − B t 1 , · · · , B t n − B t n − 1 ) for some ϕ ∈ C Lip ( R d × n ) and 0 = t 0 < t 1 < · · · < t n < ∞ , we set ˆ E [ ϕ ( B t 1 − B t 0 , B t 2 − B t 1 , · · · , B t n − B t n − 1 )] E [ ϕ ( √ t 1 − t 0 ξ 1 , · · · , √ t n − t n − 1 ξ n )] . : = � Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The related conditional expectation of X = ϕ ( B t 1 , B t 2 − B t 1 , · · · , B t n − B t n − 1 ) under Ω t j is defined by E [ X | Ω t j ] = ˆ ˆ E [ ϕ ( B t 1 , B t 2 − B t 1 , · · · , B t n − B t n − 1 ) | Ω t j ] t j + 1 − t j ξ j + 1 , · · · , √ t n − t n − 1 ξ n )] � : = � E [ ϕ ( x 1 , · · · , x j , x 1 = B t 1 . . . x j = B tj − B tj − 1 It is easy to check that ˆ E [ · ] consistently defines a sublinear expectation on L ip ( Ω ) and ( B t ) t ≥ 0 is a G -Brownian motion. Since L ip ( Ω T ) ⊆ L ip ( Ω ) , ˆ E [ · ] is also a sublinear expectation on L ip ( Ω T ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Definition The sublinear expectation ˆ E [ · ] : L ip ( Ω ) → R defined through the above procedure is called a G –expectation . The corresponding canonical process ( B t ) t ≥ 0 on the sublinear expectation space ( Ω , L ip ( Ω ) , ˆ E ) is called a G –Brownian motion. In the rest of this book, when we talk about G –Brownian motion, we mean that the canonical process ( B t ) t ≥ 0 is under G -expectation. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Proposition. We list the properties of ˆ E [ ·| Ω t ] that hold for each X , Y ∈ L ip ( Ω ) : (i) If X ≥ Y , then ˆ E [ X | Ω t ] ≥ ˆ E [ Y | Ω t ] . ˆ E [ η | Ω t ] = η , for each t ∈ [ 0 , ∞ ) and η ∈ L ip ( Ω t ) . (ii) E [ X | Ω t ] − ˆ ˆ E [ Y | Ω t ] ≤ ˆ (iii) E [ X − Y | Ω t ] . E [ η X | Ω t ] = η + ˆ E [ X | Ω t ] + η − ˆ ˆ E [ − X | Ω t ] for each η ∈ L ip ( Ω t ) . (iv) (v) E [ ˆ ˆ E [ X | Ω t ] | Ω s ] = ˆ E [ X | Ω t ∧ s ] , in particular , ˆ E [ ˆ E [ X | Ω t ]] = ˆ E [ X ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Proposition. (continue) . For each X ∈ L ip ( Ω t ) , ˆ E [ X | Ω t ] = ˆ E [ X ] , where L ip ( Ω t ) is the linear space of random variables with the form ϕ ( B t 2 − B t 1 , B t 3 − B t 2 , · · · , B t n + 1 − B t n ) , n = 1 , 2 , · · · , ϕ ∈ C Lip ( R d × n ) , t 1 , · · · , t n , t n + 1 ∈ [ t , ∞ ) . Remark. (ii) and (iii) imply E [ X + η | Ω t ] = ˆ ˆ E [ X | Ω t ] + η for η ∈ L ip ( Ω t ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We now consider the completion of sublinear expectation space ( Ω , L ip ( Ω ) , ˆ E ) . We denote by L p G ( Ω ) , p ≥ 1, the completion of L ip ( Ω ) under the norm E [ | X | p ]) 1 / p . Similarly, we can define L p G ( Ω T ) , L p � X � p : = ( ˆ G ( Ω t T ) and L p G ( Ω t ) . It is clear that for each 0 ≤ t ≤ T < ∞ , L p G ( Ω t ) ⊆ L p G ( Ω T ) ⊆ L p G ( Ω ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

ˆ E [ · ] can be continuously extended to a sublinear expectation on ( Ω , L 1 G ( Ω )) . For each fixed t ≤ T < ∞ , the conditional G -expectation ˆ E [ ·| Ω t ] : L ip ( Ω T ) → L ip ( Ω t ) is a continuous mapping under �·� . Indeed, we have E [ X | Ω t ] − ˆ ˆ E [ Y | Ω t ] ≤ ˆ E [ X − Y | Ω t ] ≤ ˆ E [ | X − Y || Ω t ] , then | ˆ E [ X | Ω t ] − ˆ E [ Y | Ω t ] | ≤ ˆ E [ | X − Y || Ω t ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We thus obtain � � � ≤ � X − Y � . � ˆ E [ X | Ω t ] − ˆ E [ Y | Ω t ] It follows that ˆ E [ ·| Ω t ] can be also extended as a continuous mapping ˆ E [ ·| Ω t ] : L 1 G ( Ω T ) → L 1 G ( Ω t ) . If the above T is not fixed, then we can obtain ˆ E [ ·| Ω t ] : L 1 G ( Ω ) → L 1 G ( Ω t ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Remark. The above proposition also holds for X , Y ∈ L 1 G ( Ω ) . But in (iv), η ∈ L 1 G ( Ω t ) should be bounded, since X , Y ∈ L 1 G ( Ω ) does not imply X · Y ∈ L 1 G ( Ω ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

In particular, we have the following independence: E [ X | Ω t ] = ˆ ˆ ∀ X ∈ L 1 G ( Ω t ) . E [ X ] , Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We give the following definition similar to the classical one: Definition G ( Ω )) n is said to be independent An n -dimensional random vector Y ∈ ( L 1 from Ω t for some given t if for each ϕ ∈ C b . Lip ( R n ) we have E [ ϕ ( Y ) | Ω t ] = ˆ ˆ E [ ϕ ( Y )] . Remark. Just as in the classical situation, the increments of G –Brownian motion ( B t + s − B t ) s ≥ 0 are independent from Ω t . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The following property is very useful. Proposition. G ( Ω ) be such that ˆ E [ Y | Ω t ] = − ˆ Let X , Y ∈ L 1 E [ − Y | Ω t ] , for some t ∈ [ 0 , T ] . Then we have E [ X + Y | Ω t ] = ˆ ˆ E [ X | Ω t ] + ˆ E [ Y | Ω t ] . In particular, if ˆ E [ Y | Ω t ] = ˆ E G [ − Y | Ω t ] = 0, then E [ X + Y | Ω t ] = ˆ ˆ E [ X | Ω t ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Proof. This follows from the following two inequalities: E [ X + Y | Ω t ] ≤ ˆ ˆ E [ X | Ω t ] + ˆ E [ Y | Ω t ] , E [ X + Y | Ω t ] ≥ ˆ ˆ E [ X | Ω t ] − ˆ E [ − Y | Ω t ] = ˆ E [ X | Ω t ] + ˆ E [ Y | Ω t ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Now, we give the representation of G -expectation. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Theorem ([Denis,Hu,Peng2009], [Hu-Peng2009]) For each continuous monotonic and sublinear function G : S ( d ) → R , let ˆ E be the corresponding G-expectation on ( Ω , L ip ( Ω )) and B t ( ω ) = ω t be the G-BM. Then there exists a weakly compact family of probability measures P on ( Ω , B ( Ω )) such that ˆ E [ X ] = max P ∈P E P [ X ] for X ∈ L ip ( Ω ) . Moreover L p G ( Ω ) is a strict subset P ∈P ( E P [ | X | p ]) 1 / p < ∞ } . L p ( Ω ) = { X ∈ L 0 ( Ω , B ( Ω )) , � X � p = max Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Let p ≥ 1 be fixed. We consider the following type of simple processes: for a given partition π T = { t 0 , · · · , t N } of [ 0 , T ] we set N − 1 ∑ η t ( ω ) = ξ k ( ω ) I [ t k , t k + 1 ) ( t ) , k = 0 where ξ k ∈ L p G ( Ω t k ) , k = 0 , 1 , 2 , · · · , N − 1 are given. The collection of these processes is denoted by M p , 0 G ( 0 , T ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Definition G ( 0 , T ) the completion of M p , 0 We denote by M p G ( 0 , T ) under the norm � � 1 / p � T ˆ | η t | p dt ] � η � M p G ( 0 , T ) : = E [ . 0 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

It is clear that M p G ( 0 , T ) ⊃ M q G ( 0 , T ) for 1 ≤ p ≤ q . We also use M p G ( 0 , T ; R n ) for all n -dimensional stochastic processes t ∈ M p η t = ( η 1 t , · · · , η n t ) , t ≥ 0 with η i G ( 0 , T ) , i = 1 , 2 , · · · , n . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We now give the definition of Itˆ o’s integral. For simplicity, we first introduce Itˆ o’s integral with respect to 1-dimensional G –Brownian motion. Let ( B t ) t ≥ 0 be a 1-dimensional G –Brownian motion with σ 2 α + − σ 2 α − ) , where 0 ≤ σ ≤ ¯ G ( α ) = 1 2 ( ¯ σ < ∞ . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Definition For each η ∈ M 2 , 0 G ( 0 , T ) of the form N − 1 ∑ η t ( ω ) = ξ j ( ω ) I [ t j , t j + 1 ) ( t ) , j = 0 we define � T N − 1 ∑ I ( η ) = η t dB t : = ξ j ( B t j + 1 − B t j ) . 0 j = 0 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Lemma From Example –eee1, for each j, � T � s ˆ E [ η t dB t | Ω s ] = 0 η t dB t , 0 � T � T ˆ η t dB t | Ω 0 ] = ˆ E [ E [ η t dB t ] = 0 , 0 0 � � T � 2 � T σ 2 ˆ ˆ η 2 ] ≤ ¯ E [ η t dB t E [ t dt ] . 0 0 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Definition We define, for a fixed η ∈ M 2 G ( 0 , T ) , the stochastic integral � T η t dB t : = I ( η ) . 0 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Proposition. Let η , θ ∈ M 2 G ( 0 , T ) and let 0 ≤ s ≤ r ≤ t ≤ T . Then we have (i) � t s η u dB u = � r s η u dB u + � t r η u dB u . (ii) � t s ( αη u + θ u ) dB u = α � t s η u dB u + � t s θ u dB u , if α is bounded and in E [ X + � T G ( Ω s ) . (iii) ˆ r η u dB u | Ω s ] = ˆ L 1 for X ∈ L 1 E [ X | Ω s ] G ( Ω ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Similar to 1-dimensional case, we can define Itˆ o’s integral by � T η t dB a for η ∈ M 2 I ( η ) : = t , G ( 0 , T ) . 0 We still have, for each η ∈ M 2 G ( 0 , T ) , � T ˆ η t dB a E [ t ] = 0 , 0 � T � T ˆ η t dB a t ) 2 ] ≤ σ 2 aa T ˆ η 2 E [( E [ t dt ] . 0 0 Furthermore, the above properties still hold for the integra. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

quadratic variation process � t N − 1 j ) 2 = B 2 ∑ � B � t : = ( B t N j + 1 − B t N t − 2 lim 0 B s dB s . µ ( π N t ) → 0 j = 0 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

By the above construction, ( � B � t ) t ≥ 0 is an increasing process with � B � 0 = 0. We call it the of the G –Brownian motion B . It characterizes the part of statistic uncertainty of G –Brownian motion. It is important to keep in mind that � B � t is not a deterministic process unless σ = ¯ σ , i.e., when ( B t ) t ≥ 0 is a classical Brownian motion. In fact we have the following lemma. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Lemma For each 0 ≤ s ≤ t < ∞ , we have ˆ σ 2 ( t − s ) , E [ � B � t − � B � s | Ω s ] = ¯ E [ − ( � B � t − � B � s ) | Ω s ] = − σ 2 ( t − s ) . ˆ Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

A very interesting point of the quadratic variation process � B � is, just like the G –Brownian motion B itself, the increment � B � s + t − � B � s is independent from Ω s and identically distributed with � B � t . In fact we have Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Lemma For each fixed s , t ≥ 0 , � B � s + t − � B � s is identically distributed with � B � t and independent from � B � t 1 , · · · , � B � t n , for t 1 , · · · ,t n ∈ [ 0 , T ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Lemma We have E [ � B � 2 ˆ σ 4 t 2 . t ] ≤ 10 ¯ Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Proposition. Let ( b t ) t ≥ 0 be a process on a sublinear expectation space ( Ω , H , ˆ E ) such that (i) b 0 = 0; (ii) For each t , s ≥ 0, b t + s − b t is identically distributed with b s and independent from ( b t 1 , b t 2 , · · · , b t n ) for each n ∈ N and 0 ≤ t 1 , · · · , t n ≤ t ; t ] t − 1 = 0. (iii) lim t ↓ 0 ˆ E [ b 2 Then b t is maximal distributed with ˆ E [ ϕ ( b t )] = max µ ≤ v ≤ µ ϕ ( vt ) µ = ˆ E [ b 1 ] and µ = − ˆ E [ − b 1 ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Theorem � B � t is M ([ σ 2 t , ¯ σ 2 t ]) -distributed, i.e., for each ϕ ∈ C Lip ( R ) , ˆ E [ ϕ ( � B � t )] = sup σ 2 ϕ ( vt ) . σ 2 ≤ v ≤ ¯ Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Corollary For each 0 ≤ t ≤ T < ∞ , we have σ 2 ( T − t ) ≤ � B � T − � B � t ≤ ¯ σ 2 ( T − t ) in L 1 G ( Ω ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Corollary We have, for each t , s ≥ 0 , n ∈ N , E [ � B � n E [( � B � t + s − � B � s ) n | Ω s ] = ˆ ˆ σ 2 n t n t ] = ¯ and E [ − � B � n E [ − ( � B � t + s − � B � s ) n | Ω s ] = ˆ ˆ t ] = − σ 2 n t n . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We now consider a general form of G –Itˆ o’s formula. Consider � t � t � t 0 η s d � B � s + X t = X 0 + 0 α s ds + 0 β s dB s . Theorem Let Φ be a C 2 -function on R such that ∂ 2 xx Φ satisfy polynomial growth condition for µ , ν = 1 , · · · , n. Let α , β and η be bounded processes in M 2 G ( 0 , T ) . Then for each t ≥ 0 we have in L 2 G ( Ω t ) � t � t Φ ( X t ) − Φ ( X s ) = s ∂ x ν Φ ( X u ) β u dB u + s ∂ x Φ ( X u ) α u du � t � B i , B j � s [ ∂ x Φ ( X u ) η u + 1 2 ∂ 2 xx Φ ( X u ) β 2 + u ] d u . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Problem How to work with τ -technique with stopping times τ ? [Li-X.-P.2010], [Song2010]. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Generalized G -Brownian Motion Let G : R d × S ( d ) → R be a given continuous sublinear function monotonic in A ∈ S ( d ) . Then there exists a bounded, convex and closed subset Σ ⊂ R d × S + ( d ) such that [ 1 for ( p , A ) ∈ R d × S ( d ) . 2 tr [ AB ] + � p , q � ] G ( p , A ) = sup ( q , B ) ∈ Σ There exists a pair of d -dimensional random vectors ( X , Y ) which is G -distributed. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We now give the definition of the generalized G -Brownian motion. Definition A d -dimensional process ( B t ) t ≥ 0 on a sublinear expectation space ( Ω , H , ˆ E ) is called a generalized G -Brownian motion if the following properties are satisfied: (i) B 0 ( ω ) = 0; (ii) For each t , s ≥ 0, the increment B t + s − B t identically distributed with √ sX + sY and is independent from ( B t 1 , B t 2 , · · · , B t n ) , for each n ∈ N and 0 ≤ t 1 ≤ · · · ≤ t n ≤ t , where ( X , Y ) is G -distributed. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The following theorem gives a characterization of the generalized G -Brownian motion. Theorem Let ( B t ) t ≥ 0 be a d-dimensional process defined on a sublinear expectation space ( Ω , H , ˆ E ) such that (i) B 0 ( ω )= 0 ; (ii) For each t , s ≥ 0 , B t + s − B t and B s are identically distributed and B t + s − B t is independent from ( B t 1 , B t 2 , · · · , B t n ) , for each n ∈ N and 0 ≤ t 1 ≤ · · · ≤ t n ≤ t. E [ | B t | 3 ] t − 1 = 0 . (iii) lim t ↓ 0 ˆ Then ( B t ) t ≥ 0 is a generalized G-Brownian motion with 2 � AB δ , B δ � ] δ − 1 for ( p , A ) ∈ R d × S ( d ) . G ( p , A ) = lim δ ↓ 0 ˆ E [ � p , B δ � + 1 Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

� G -Brownian Motion under a Nonlinear Expectation We can also define a G -Brownian motion on a nonlinear expectation space ( Ω , H , � E ) . Definition A d -dimensional process ( B t ) t ≥ 0 on a nonlinear expectation space ( Ω , H , � E ) is called a (nonlinear) � G -Brownian motion if the following properties are satisfied: (i) B 0 ( ω ) = 0; (ii) For each t , s ≥ 0, the increment B t + s − B t identically distributed with B s and is independent from ( B t 1 , B t 2 , · · · , B t n ) , for each n ∈ N and 0 ≤ t 1 ≤ · · · ≤ t n ≤ t ; E [ | B t | 3 ] t − 1 = 0. (iii) lim t ↓ 0 ˆ Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The following theorem gives a characterization of the nonlinear G -Brownian motion, and give us the generator � � G of our � G -Brownian motion. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Theorem Let � E be a nonlinear expectation and ˆ E be a sublinear expectation defined on ( Ω , H ) . Let � E [ X ] − � E [ Y ] ≤ ˆ E [ X − Y ] , X , Y ∈ H . Let ( B t , b t ) t ≥ 0 be a given � G-Brownian motion on ( Ω , H , � E ) such that E [ B t ] = ˆ ˆ E [ − B t ] = 0 and lim t → 0 ˆ E [ | b t | 2 ] / t = 0 . Then, for ϕ ∈ C b . Lip ( R 2 d ) , the function u ( t , x , y ) : = � ( t , x , y ) ∈ [ 0 , ∞ ) × R 2 d ˜ E [ ϕ ( x + B t , y + b t )] , solves the PDE: u − � u , D 2 u | t = 0 = ϕ . ∂ t ˜ G ( D y ˜ x ˜ u ) = 0 , where E [ � p , b 1 � + 1 2 � AB 1 , B 1 � ] , ( p , A ) ∈ R d × S ( d ) . G ( p , A ) = � � Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

It is easy to check that ˆ E [ · ] (resp. � E ) consistently defines a sublinear (resp. nonlinear) expectation and � E [ · ] on ( Ω , L ip ( Ω )) . Moreover ( B t , b t ) t ≥ 0 is a G -Brownian motion under ˆ E and a � G -Brownian motion under � E . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Proposition. For each X , Y ∈ L ip ( Ω ) : (i) If X ≥ Y , then � E [ X | Ω t ] ≥ � E [ Y | Ω t ] . (ii) � E [ X + η | Ω t ] = � E [ X | Ω t ] + η , for each t ≥ 0 and η ∈ L ip ( Ω t ) . (iii) � E [ X | Ω t ] − � E [ Y | Ω t ] ≤ ˆ E [ X − Y | Ω t ] . (iv) � E [ � E [ X | Ω t ] | Ω s ] = � E [ X | Ω t ∧ s ] , in particular , � E [ � E [ X | Ω t ]] = � E [ X ] . (v) For each X ∈ L ip ( Ω t ) , � E [ X | Ω t ] = � E [ X ] , where L ip ( Ω t ) is the linear space of random variables with the form ϕ ( W t 2 − W t 1 , W t 3 − W t 2 , · · · , W t n − W t n − 1 ) , t 1 , · · · , t n ∈ [ t , ∞ ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Since ˆ E can be considered as a special nonlinear expectation of � E dominated by its self, thus ˆ E [ ·| Ω t ] also satisfies above properties (i)–(v). Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Moreover Proposition. The conditional sublinear expectation ˆ E [ ·| Ω t ] satisfies (i)-(v). Moreover ˆ E [ ·| Ω t ] itself is sublinear, i.e., E [ X | Ω t ] − ˆ ˆ E [ Y | Ω t ] ≤ ˆ E [ X − Y | Ω t ] , (vi) . E [ η X | Ω t ] = η + ˆ E [ X | Ω t ] + η − ˆ ˆ (vii) E [ − X | Ω t ] for each η ∈ L ip ( Ω t ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We now consider the completion of sublinear expectation space ( Ω , L ip ( Ω ) , ˆ E ) . We denote by L p G ( Ω ) , p ≥ 1, the completion of L ip ( Ω ) under the norm E [ | X | p ]) 1 / p . Similarly, we can define L p G ( Ω T ) , L p � X � p : = ( ˆ G ( Ω t T ) and L p G ( Ω t ) . It is clear that for each 0 ≤ t ≤ T < ∞ , L p G ( Ω t ) ⊆ L p G ( Ω T ) ⊆ L p G ( Ω ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

ˆ G ( Ω )) . Moreover, since � E [ · ] can be continuously extended to ( Ω , L 1 E is dominated by ˆ G ( Ω ) , ˆ E , thus ( Ω , L 1 E ) forms a sublinear expectation space G ( Ω ) , � and ( Ω , L 1 E ) forms a nonlinear expectation space. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We now consider the extension of conditional G -expectation. For each fixed t ≤ T , the conditional G -expectation ˆ E [ ·| Ω t ] : L ip ( Ω T ) → L ip ( Ω t ) is a continuous mapping under �·� . Indeed, we have E [ X | Ω t ] − � � E [ Y | Ω t ] ≤ ˆ E [ X − Y | Ω t ] ≤ ˆ E [ | X − Y || Ω t ] , then | � E [ X | Ω t ] − � E [ Y | Ω t ] | ≤ ˆ E [ | X − Y || Ω t ] . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

We thus obtain � � � � �� E [ X | Ω t ] − � E [ Y | Ω t ] � ≤ � X − Y � . It follows that � E [ ·| Ω t ] can be also extended as a continuous mapping � E [ ·| Ω t ] : L 1 G ( Ω T ) → L 1 G ( Ω t ) . If the above T is not fixed, then we can obtain � E [ ·| Ω t ] : L 1 G ( Ω ) → L 1 G ( Ω t ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Remark. The above proposition also holds for X , Y ∈ L 1 G ( Ω ) . But in (iv), η ∈ L 1 G ( Ω t ) should be bounded, since X , Y ∈ L 1 G ( Ω ) does not imply X · Y ∈ L 1 G ( Ω ) . Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Notes and Comments Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

Bachelier (1900) [ ? ] proposed Brownian motion as a model for fluctuations of the stock market, Einstein (1905) [ ? ] used Brownian motion to give experimental confirmation of the atomic theory, and Wiener (1923) [ ? ] gave a mathematically rigorous construction of Brownian motion. Here we follow Kolmogorov’s idea (1956) [40] to construct G -Brownian motion by introducing infinite dimensional function space and the corresponding family of infinite dimensional sublinear distributions, instead of linear distributions in [40]. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156

The notions of G -Brownian motion and the related stochastic calculus of Itˆ o’s type were firstly introduced by Peng (2006) [61] for 1-dimensional case and then in (2008) [65] for multi-dimensional situation. It is very interesting that Denis and Martini (2006) [23] studied super-pricing of contingent claims under model uncertainty of volatility. They have introduced a norm on the space of continuous paths Ω = C ([ 0 , T ]) which corresponds to our L 2 G -norm and developed a stochastic integral. There is no notion of nonlinear expectation and the related nonlinear distribution, such as G -expectation, conditional G -expectation, the related G -normal distribution and the notion of independence in their paper. But on the other hand, powerful tools in capacity theory enable them to obtain pathwise results for random variables and stochastic processes through the language of “quasi-surely” (see e.g. Dellacherie (1972) [18], Dellacherie and Meyer (1978 and 1982) [19], Feyel and de La Pradelle (1989) [25]) in place of “almost surely” in classical probability theory. Shandong University Presented at Shige Peng () Progresses and Problems in Theory of Nonlinear Expectations and Applications to Finance / 156