Nonlinear Expectations and Stochastic Calculus under Uncertainty - PowerPoint PPT Presentation

Nonlinear Expectations and Stochastic Calculus under Uncertainty —with Robust Central Limit Theorem and G-Brownian Motion ✩ ➣ ④ Shige Peng Shandong University Presented at Spring School of Roscoff Shandong University Presented at Nonlinear Expectations and Stochastic Calculus under Uncertainty —with Robust ✩ ✩ ✩ ➣ ➣ ➣ ④ ④ ④ Shige Peng () / 5

L 0 ( Ω ) : the space of all B ( Ω ) -measurable real functions; B b ( Ω ) : all bounded functions in L 0 ( Ω ) ; C b ( Ω ) : all continuous functions in B b ( Ω ) . All along this section, we consider a given subset P ⊆ M . We denote c ( A ) : = sup P ( A ) , A ∈ B ( Ω ) . P ∈P () March 13, 2010 67 / 186

One can easily verify the following theorem. Theorem The set function c ( · ) is a Choquet capacity, i.e. (see [ ? , ? ]), 1 0 ≤ c ( A ) ≤ 1 , ∀ A ⊂ Ω . 2 If A ⊂ B, then c ( A ) ≤ c ( B ) . 3 If ( A n ) ∞ n = 1 is a sequence in B ( Ω ) , then c ( ∪ A n ) ≤ ∑ c ( A n ) . 4 If ( A n ) ∞ n = 1 is an increasing sequence in B ( Ω ) : A n ↑ A = ∪ A n , then c ( ∪ A n ) = lim n → ∞ c ( A n ) . () March 13, 2010 68 / 186

Furthermore, we have Theorem For each A ∈ B ( Ω ) , we have c ( A ) = sup { c ( K ) : K compact K ⊂ A } . Proof. It is simply because c ( A ) = sup P ( K ) = P ( K ) = c ( K ) . sup sup sup sup P ∈P P ∈P K compact K compact K compact K ⊂ A K ⊂ A K ⊂ A () March 13, 2010 69 / 186

Definition We use the standard capacity-related vocabulary: a set A is polar if c ( A ) = 0 and a property holds “ quasi-surely ” (q.s.)”qs if it holds outside a polar set. () March 13, 2010 70 / 186

We also have in a trivial way a Borel-Cantelli Lemma. Lemma Let ( A n ) n ∈ N be a sequence of Borel sets such that ∞ ∑ c ( A n ) < ∞ . n = 1 Then lim sup n → ∞ A n is polar . Proof. Applying the Borel-Cantelli Lemma under each probability P ∈ P . The following theorem is Prokhorov’s theorem. Theorem P is relatively compact if and only if for each ε > 0 , there exists a compact set K such that c ( K c ) < ε . () March 13, 2010 71 / 186

The following two lemmas can be found in [ ? ]. Lemma P is relatively compact if and only if for each sequence of closed sets F n ↓ ∅ , we have c ( F n ) ↓ 0 . Proof. We outline the proof for the convenience of readers. “ = ⇒ ” part: It follows from Theorem –newth6 that for each fixed ε > 0, there exists a compact set K such that c ( K c ) < ε . Note that F n ∩ K ↓ ∅ , then there exists an N > 0 such that F n ∩ K = ∅ for n ≥ N , which implies lim n c ( F n ) < ε . Since ε can be arbitrarily small, we obtain c ( F n ) ↓ 0. i ) ∞ = ” part: For each ε > 0, let ( A k “ ⇐ i = 1 be a sequence of open balls of i ) c ↓ ∅ , then there exists an radius 1 / k covering Ω . Observe that ( ∪ n i = 1 A k n k such that c (( ∪ n k i ) c ) < ε 2 − k . Set K = ∩ ∞ k = 1 ∪ n k i = 1 A k i = 1 A k i . It is easy to check that K is compact and c ( K c ) < ε . Thus by Theorem –newth6, P is relatively compact. () March 13, 2010 72 / 186

Lemma Let P be weakly compact. Then for each sequence of closed sets F n ↓ F, we have c ( F n ) ↓ c ( F ) . Proof. We outline the proof for the convenience of readers. For each fixed ε > 0, by the definition of c ( F n ) , there exists a P n ∈ P such that P n ( F n ) ≥ c ( F n ) − ε . Since P is weakly compact, there exist P n k and P ∈ P such that P n k converge weakly to P . Thus P ( F m ) ≥ lim sup P n k ( F m ) ≥ lim sup P n k ( F n k ) ≥ lim n → ∞ c ( F n ) − ε . k → ∞ k → ∞ Letting m → ∞ , we get P ( F ) ≥ lim n → ∞ c ( F n ) − ε , which yields c ( F n ) ↓ c ( F ) . () March 13, 2010 73 / 186

Following [ ? ] (see also [ ? , ? ]) the upper expectation of P is defined as follows: for each X ∈ L 0 ( Ω ) such that E P [ X ] exists for each P ∈ P , E [ X ] = E P [ X ] : = sup E P [ X ] . P ∈P () March 13, 2010 74 / 186

It is easy to verify Theorem The upper expectation E [ · ] of the family P is a sublinear expectation on B b ( Ω ) as well as on C b ( Ω ) , i.e., 1 for all X , Y in B b ( Ω ) , X ≥ Y = ⇒ E [ X ] ≥ E [ Y ] . 2 for all X , Y in B b ( Ω ) , E [ X + Y ] ≤ E [ X ] + E [ Y ] . 3 for all λ ≥ 0 , X ∈ B b ( Ω ) , E [ λ X ] = λ E [ X ] . 4 for all c ∈ R , X ∈ B b ( Ω ) , E [ X + c ] = E [ X ] + c. () March 13, 2010 75 / 186

Moreover, it is also easy to check Theorem We have 1 Let E [ X n ] and E [ ∑ ∞ n = 1 X n ] be finite. Then E [ ∑ ∞ n = 1 X n ] ≤ ∑ ∞ n = 1 E [ X n ] . 2 Let X n ↑ X and E [ X n ] , E [ X ] be finite. Then E [ X n ] ↑ E [ X ] . () March 13, 2010 76 / 186

Definition The functional E [ · ] is said to be regular if for each { X n } ∞ n = 1 in C b ( Ω ) such that X n ↓ 0 on Ω , we have E [ X n ] ↓ 0. Similar to Lemma –Lemma1 we have: Theorem E [ · ] is regular if and only if P is relatively compact. () March 13, 2010 77 / 186

Proof. “ = ⇒ ” part: For each sequence of closed subsets F n ↓ ∅ such that F n , n = 1 , 2 , · · · , are non-empty (otherwise the proof is trivial), there exists { g n } ∞ n = 1 ⊂ C b ( Ω ) satisfying g n = 1 on F n and g n = 0 on { ω ∈ Ω : d ( ω , F n ) ≥ 1 0 ≤ g n ≤ 1 , n } . We set f n = ∧ n i = 1 g i , it is clear that f n ∈ C b ( Ω ) and 1 F n ≤ f n ↓ 0. E [ · ] is regular implies E [ f n ] ↓ 0 and thus c ( F n ) ↓ 0. It follows from Lemma –Lemma1 that P is relatively compact. = ” part: For each { X n } ∞ “ ⇐ n = 1 ⊂ C b ( Ω ) such that X n ↓ 0, we have � ∞ � ∞ E [ X n ] = sup E P [ X n ] = sup P ( { X n ≥ t } ) dt ≤ c ( { X n ≥ t } ) dt . P ∈P P ∈P 0 0 For each fixed t > 0, { X n ≥ t } is a closed subset and { X n ≥ t } ↓ ∅ as n ↑ ∞ . By Lemma –Lemma1, c ( { X n ≥ t } ) ↓ 0 and thus � ∞ 0 c ( { X n ≥ t } ) dt ↓ 0. Consequently E [ X n ] ↓ 0. () March 13, 2010 78 / 186

We set, for p > 0, L p : = { X ∈ L 0 ( Ω ) : E [ | X | p ] = sup P ∈P E P [ | X | p ] < ∞ } ; N p : = { X ∈ L 0 ( Ω ) : E [ | X | p ] = 0 } ; N : = { X ∈ L 0 ( Ω ) : X = 0, c -q.s. } . It is seen that L p and N p are linear spaces and N p = N , for each p > 0. We denote L p : = L p / N . As usual, we do not take care about the distinction between classes and their representatives. () March 13, 2010 79 / 186

Lemma Let X ∈ L p . Then for each α > 0 c ( {| X | > α } ) ≤ E [ | X | p ] . α p () March 13, 2010 80 / 186

Proof. Just apply Markov inequality under each P ∈ P . Similar to the classical results, we get the following proposition and the proof is omitted which is similar to the classical arguments. Proposition. We have 1 For each p ≥ 1, L p is a Banach space under the norm 1 p . � X � p : = ( E [ | X | p ]) 2 For each p < 1, L p is a complete metric space under the distance d ( X , Y ) : = E [ | X − Y | p ] . () March 13, 2010 81 / 186

We set L ∞ : = { X ∈ L 0 ( Ω ) : ∃ a constant M , s.t. | X | ≤ M , q.s. } ; L ∞ : = L ∞ / N . Proposition. Under the norm � X � ∞ : = inf { M ≥ 0 : | X | ≤ M , q.s. } , L ∞ is a Banach space. Proof. From {| X | > � X � ∞ } = ∪ ∞ � | X | ≥ � X � ∞ + 1 � we know that n = 1 n | X | ≤ � X � ∞ , q.s., then it is easy to check that �·� ∞ is a norm. The proof of the completeness of L ∞ is similar to the classical result. () March 13, 2010 82 / 186

With respect to the distance defined on L p , p > 0, we denote by L p b the completion of B b ( Ω ) . L p c the completion of C b ( Ω ) . By Proposition –Prop3, we have L p c ⊂ L p b ⊂ L p , p > 0 . () March 13, 2010 83 / 186

The following Proposition is obvious and the proof is left to the reader. Proposition. We have q = 1. Then X ∈ L p and Y ∈ L q implies 1 Let p , q > 1, 1 p + 1 XY ∈ L 1 and E [ | XY | ] ≤ ( E [ | X | p ]) 1 1 p ( E [ | Y | q ]) q ; Moreover X ∈ L p c and Y ∈ L q c implies XY ∈ L 1 c ; 2 L p 1 ⊂ L p 2 , L p 1 b ⊂ L p 2 b , L p 1 c ⊂ L p 2 c , 0 < p 2 ≤ p 1 ≤ ∞ ; 3 � X � p ↑ � X � ∞ , for each X ∈ L ∞ . Proposition. Let p ∈ ( 0 , ∞ ] and ( X n ) be a sequence in L p which converges to X in L p . Then there exists a subsequence ( X n k ) which converges to X quasi-surely in the sense that it converges to X outside a polar set. () March 13, 2010 84 / 186

Proof. Let us assume p ∈ ( 0 , ∞ ) , the case p = ∞ is obvious since the convergence in L ∞ implies the convergence in L p for all p . One can extract a subsequence ( X n k ) such that E [ | X − X n k | p ] ≤ 1 / k p + 2 , k ∈ N . We set for all k A k = {| X − X n k | > 1 / k } , then as a consequence of the Markov property (Lemma –markov) and the Borel-Cantelli Lemma –BorelC, c ( lim k → ∞ A k ) = 0. As it is clear that on ( lim k → ∞ A k ) c , ( X n k ) converges to X , the proposition is proved. () March 13, 2010 85 / 186

We now give a description of L p b . Proposition. ”Prop5For each p > 0, b = { X ∈ L p : lim L p n → ∞ E [ | X | p 1 {| X | > n } ] = 0 } . () March 13, 2010 86 / 186