Martingale Problem under Nonlinear Expectations Chen Pan USTC, - PowerPoint PPT Presentation

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions Martingale Problem under Nonlinear Expectations Chen Pan USTC, China and UC Berkeley Sixth WCMF 2014, Santa Barbara joint work with Xin Guo (UC Berkeley) & Shige Peng (Shandong University, China) Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions Outline Background 1 Motivation for Martingale Problem 2 Martingale Problem under Nonlinear Expectations 3 Definition of Nonlinear Martingale Problem Existence of Solution to Martingale Problem Related Result – Weak Solution to G -SDE 4 Weak solution Discussions 5 Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions Linearity in probability theory A random variable X is uniquely determined by its probability distribution P , or equivalently its expectation, P ( X ∈ A ) = E [1 A ] 1-1 correspondence between linear expectation and additive probability measure Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions Sublinear Expectation ˜ E Given χ (e.g. all bounded measurable random variables), ˜ E : χ → R is sublinear iff (a) Monotonicity: If X ≤ Y , then ˜ E [ X ] ≤ ˜ E [ Y ] (b) Constant preserving: ˜ E [ X + c ] = ˜ E [ X ] + c (c) Sublinearity: ˜ E [ X + Y ] ≤ ˜ E [ X ] + ˜ E [ Y ]. (d) Positive homogeneity: ˜ E [ λ X ] = λ ˜ E [ X ] for all λ > 0 Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions No more 1-1 correspondence between ˜ E and ˜ P Clearly P ( A ) = ˜ ˜ E [1 A ] = ˜ E f [1 A ] for all f continuous and strictly increasing, f ( x ) = x for x ∈ [0 , 1], where E f [ X ] = f − 1 (˜ ˜ E [ f ( X )]) . Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions Connecting linear and sublinear expectations Denis, Hu and Peng (2011) There exists a weakly compact family of probability measures P on (Ω , B (Ω)) such that ˜ P ∈P E P [ X ] , E [ X ] = max where E P is the linear expectation with respect to P , for a proper class of random process X . Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions Sublinear expectation and model uncertainty Sublinear expectation “measures” the model uncertainty, the bigger the expectation ˜ E , the more the uncertainty. E 1 [ X ] ≤ ˜ ˜ E 2 [ X ] iff P 1 ⊂ P 2 Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions Sublinear expectation and risk measure Let ρ ( X ) = ˜ E [ − X ] Then we get a coherent risk measure ρ : χ → R (a) Monotonicity: If X ≥ Y , then ρ ( X ) ≤ ρ ( Y ). (b) Constant translatability: ρ ( X + c ) = ρ ( X ) − c (c) Convexity: ρ ( α X + (1 − α ) Y ) ≤ αρ ( X ) + (1 − α ) ρ ( Y ), α ∈ [0 , 1]. (d) Positive homogeneity: ρ ( λ X ) = λρ ( X ) for all λ > 0. Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions G − normal distribution and G -heat equation Starting point (Peng 2005) G-normal distribution X , N (0 × [ σ 2 , σ 2 ]) is characterized by the G-heat equation ∂ t u − G ( D 2 u ) = 0 , u | t =0 = φ. Here G ( · ) : R → R is a monotonic, sublinear function, with G ( γ ) = 1 sup γα, 2 α ∈ [ σ 2 ,σ 2 ] where σ 2 ≤ σ 2 are constants. Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Martingale Problem under Nonlinear Expectations Related Result – Weak Solution to G -SDE Discussions An Alternative Way Many difficulties in dealing with the optimization problem in G-expectation framework, other ways are needed to explore to avoid the inconvenience. Peng’s way(sublinear case): G -normal distribution + “independence” ⇒ G -Brownian motion ⇒ general G -(semi)-martingales Our idea: General stochastic processes in a nonlinear expectation space Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Definition of Nonlinear Martingale Problem Martingale Problem under Nonlinear Expectations Existence of Solution to Martingale Problem Related Result – Weak Solution to G -SDE Discussions Martingale Problem under Nonlinear Expectations Definition of martingale problem Find a family of operators { � E t } t ≥ 0 on a nonlinear expectation space (Ω , H ) such that � t � ϕ ( X t ) − G ( X θ , ϕ x ( X θ ) , ϕ xx ( X θ )) d θ, t ≥ 0 0 is an { � E t } -martingale for all ϕ ∈ C ∞ 0 ( R d ). G : R d × R d × S d → R continuous with desirable properties � Ω = C x 0 ([0 , ∞ ); R d ) and X t ( ω ) = ω ( t ) , ω ∈ Ω. Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Definition of Nonlinear Martingale Problem Martingale Problem under Nonlinear Expectations Existence of Solution to Martingale Problem Related Result – Weak Solution to G -SDE Discussions Comparison with classical martingale problems Classical M.P.’s Our M.P.’s ˜ to find a probability measure P to find a nonlinear expectation E on on (Ω , F ) (Ω , H ) X 0 = x in L 1 X 0 = x P -a.s. � � t � t E 0 � ϕ ( X t ) − 0 L θ ϕ ( X θ ) d θ is a P - ϕ ( X t ) − G ( θ, X θ , ϕ x ( X θ ) , ϕ xx ( X θ )) d θ martingale for ∀ ϕ ∈ C ∞ is an � E -martingale for ∀ ϕ ∈ C ∞ 0 0 � a ij ( θ, · ) ∂ 2 � 1 = + Nonlinear PDE associated with : L θ G 2 ∂ x i ∂ x j � b i ( θ, · ) ∂ [0 , ∞ ) × R d × R d × S d → R is a con- ∂ x i is a linear differen- tinuous function with some properties tial operator Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Definition of Nonlinear Martingale Problem Martingale Problem under Nonlinear Expectations Existence of Solution to Martingale Problem Related Result – Weak Solution to G -SDE Discussions The critical step To identify appropriate classes of � G Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Definition of Nonlinear Martingale Problem Martingale Problem under Nonlinear Expectations Existence of Solution to Martingale Problem Related Result – Weak Solution to G -SDE Discussions Recipe and Ingredients PDEs associated with � G For a given � G , the associated state-dependent parabolic PDE � ∂ t u − � G ( x , Du , D 2 u ) = 0 , ( t , x ) ∈ (0 , T ] × R d , ( � P ) u (0 , x ) = ϕ ( x ) , x ∈ R d . Comparison and existence theorems for the PDEs Constructing conditional expectations from solutions of PDEs Finite dimensional distribution + Kolmogorov’s time consistency theorem Properties of the conditional expectations Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Definition of Nonlinear Martingale Problem Martingale Problem under Nonlinear Expectations Existence of Solution to Martingale Problem Related Result – Weak Solution to G -SDE Discussions Class D for � G G : R d × R d × S d → R is of class D , if A continuous function � � G ( x , 0 , 0) ≡ 0 for all x ∈ R d � G is positive homogeneous, (DOM) G ( x , p , A ) − � � G ( x , p ′ , A ′ ) ≤ G ( x , p − p ′ , A − A ′ ) for each x ∈ R d , ( p , A ) , ( p ′ , A ′ ) ∈ R d × S d , and the continuous function G : R d × R d × S d → R satisfies p , A + ¯ p , ¯ A. Subadditivity G ( x , p + ¯ A ) ≤ G ( x , p , A ) + G ( x , ¯ A ); B. Positive Homogeneity G ( x , λ p , λ A ) = λ G ( x , p , A ); C. Monotonicity G ( x , p , A ) ≤ G ( x , p , A + ˜ A ); D. G is uniformly Liptschitz continuous with respect to x . Pan Martingale Problem under Nonlinear Expectations

Background Motivation for Martingale Problem Definition of Nonlinear Martingale Problem Martingale Problem under Nonlinear Expectations Existence of Solution to Martingale Problem Related Result – Weak Solution to G -SDE Discussions Example of G and � G G is sublinear with the form � 1 � G ( x , p , A ) = sup 2 tr [ a ( x , γ ) A ] + ( b ( x , γ ) , p ) , γ ∈ Γ and � G has the form � 1 � � 2 tr [ σ ( x , γ, λ ) σ ′ ( x , γ, λ ) A ] + ( b ( x , γ, λ ) , p ) G ( x , p , A ) = sup inf λ ∈ Λ γ ∈ Γ or � 1 � � 2 tr [ σ ( x , γ, λ ) σ ′ ( x , γ, λ ) A ] + ( b ( x , γ, λ ) , p ) G ( x , p , A ) = inf γ ∈ Γ sup , λ ∈ Λ where Γ , Λ are index sets, and the coefficients satisfy some proper conditions Pan Martingale Problem under Nonlinear Expectations