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Convex Risk Measures on Banach Lattices

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Convex Risk Measures on Banach Lattices Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Lectures for the Course on Foundations of Mathematical Finance First Part: Convex Risk Measures Marco Frittelli Milano University The Fields Institute, Toronto, April 2010 Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Outline Lectures Convex Risk Measures on L1 For a nice dual representation we need σ(L1; L1) lower semicontinuity For Convex Risk Measures on more general spaces what replaces this continuity assumption? Which are these more general spaces and why are they needed? Orlicz spaces associated to a utility function How do we associate a Risk Measure to a utility function Utility maximization with unbounded price process (here Orlicz spaces are needed) and with random endowment Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices ...Outline Lectures Indifference price as a convex risk measure on the Orlicz space associated to the utility function Weaker assumption than convexity: Quasiconvexity Quasiconvex (and convex) dynamic risk measures: their dual representation The conditional version of the Certainty Equivalent as a quasiconvex map The need of (dynamic and stochastic) generalized Orlicz space The dual representation of the Conditional Certainty Equivalent Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Lectures based on the following papers First Part On the extension of the Namioka-Klee theorem and on the Fatou property for risk measures, Joint with S. Biagini (2009), In: Optimality and risk: modern trends in mathematical finance. The Kabanov Festschrift. Second Part 1 A unified framework for utility maximization problems: an Orlicz space approach, Joint with S. Biagini (2008), Annals of Applied Probability. 2 Indifference Price with general semimartingales Joint with S. Biagini and M. Grasselli (2009), Forthcoming on Mathematical Finance Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices ...Lectures based on the following papers Third Part 1 Dual representation of Quasiconvex Conditional maps, Joint with M. Maggis (2009), Preprint 2 Conditional Certainty Equivalent, Joint with M. Maggis (2009), Preprint Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices 1 Definitions and properties of Risk Measures 2 Convex analysis 3 Orlicz spaces 4 Convex risk measures on Banach lattices Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Notations (Ω; F; P) is a probability space. 0 0 L = L (Ω; F; P) is the space of all P.a.s. finite random variables. L1 is the subspace of all essentially bounded random variables. L(Ω) is a linear subspace of L0. We suppose that L1 ⊆ L(Ω) ⊆ L0: We adopt in L(Ω) the P-a.s. usual order relation between random variables. Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Monetary Risk Measure Definition A map ρ : L(Ω) ! R is called a monetary risk measure on L(Ω) if it has the following properties: (CA) Cash additivity : 8 x 2 L(Ω) and 8 c 2 R ρ(x + c) = ρ(x) − c. (M) Monotonicity : ρ(x) ≤ ρ(y) 8 x; y 2 L(Ω) such that x ≥ y. (G) Grounded property : ρ(0) = 0. Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Acceptance set Definition Let ρ : L(Ω) ! R be a monetary risk measure. The set of acceptable positions is defined as Aρ := fx 2 L(Ω) : ρ(x) ≤ 0g: Remark ρ : L(Ω) ! R satisfies cash additivity (CA) if and only if there exists a set A ⊆ L(Ω) : ρ(x) = inffa 2 R : a + x 2 Ag: Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Coherent and Convex Risk Measures Coherent and convex risk measures are monetary risk measures which satisfies some of the following properties: (CO) Convexity : For all λ 2 [0; 1] and for all x; y 2 L(Ω) we have that ρ(λx + (1 − λ)y) ≤ λρ(x) + (1 − λ)ρ(y). (SA) Subadditivity : ρ(x + y) ≤ ρ(x) + ρ(y) 8 x; y 2 L(Ω). (PH) Positive homogeneity : ρ(λx) = λρ(x) 8 x 2 L(Ω) and 8 λ ≥ 0. Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Proposition Let ρ : L(Ω) ! R and Aρ = fx 2 L(Ω) : ρ(x) ≤ 0g. Then the following properties hold true: (a) If ρ satisfies (CO) then Aρ is a convex set. (b) If ρ satisfies (PH) then Aρ is a cone. (c) If ρ satisfies (M) then Aρ is a monotone set, i.e. x 2 Aρ; y ≥ x ) y 2 Aρ. Viceversa, let A ⊆ L(Ω) and ρA(x) := inffa 2 R : a + x 2 Ag. Then the following properties hold true: 0 (a ) If A is a convex set then ρA satisfies (CO). 0 (b ) If A is a cone then ρA satisfies (PH). 0 (c ) If A is a monotone set then ρA satisfies (M). Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Dual System 0 0 Let X and X be two vector lattices and < ·; · >: X × X ! R a bilinear form. This dual system (X ; X 0; < ·; · >) will be denoted simply with (X ; X 0). We consider the locally convex topological vector space (X ; τ); where τ is a topology compatible with the dual system (X ; X 0), so that the topological dual space (X ; τ)0 coincides with X 0, i.e: (X ; τ)0 = X 0 Examples: (L1; σ(L1; L1))0 = L1; < x; x 0 >= E[xx 0] 1 0 0 0 (L ; k · k1) = ba; < x; x >= x (x) p 0 q 0 0 (L ; k · kp) = L ; p 2 [1; 1); < x; x >= E[xx ] Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Convex conjugate The convex conjugate f ∗ and the convex bi-conjugate f ∗∗ of a convex function f : X ! R [ f1g with Dom(f ) := fx 2 X : f (x) < +1g 6= ; are given by ∗ 0 f : X ! R [ f1g f ∗(x0) := supfx0(x) − f (x)g = sup fx0(x) − f (x)g x2X x2 Dom(f ) ∗∗ f : X ! R [ f1g f ∗∗(x) := sup fx0(x) − f ∗(x0)g = sup fx0(x) − f ∗(x0)g x02X 0 x02 Dom(f ∗) Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices One of the main results about convex conjugates is the Fenchel-Moreau theorem. Theorem (Fenchel-Moreau) Let f : X ! R [ f1g be a convex lower semi-continuous function with Dom(f ) 6= ;. Then f (x) = f ∗∗(x); 8 x 2 X ; f (x) = sup fx0(x) − f ∗(x0)g 8 x 2 X : x02X 0 A simple application of this theorem gives the dual representation of risk measures. Set first: 0 0 0 Z := fx 2 X+ : x (1) = 1g Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Proposition Let f : X ! R [ f1g be a convex lower semi-continuous function with Dom(f ) 6= ;. Then there exists a convex, lower 0 semi-continuous function α : X ! R [ f1g with D := Dom(α) 6= ; such that f (x) = sup fx0(x) − α(x0)g 8 x 2 X : x02D 0 (M) If f is monotone (i.e. x ≤ y =) f (x) ≤ f (y)) then D ⊆ X+. (CA) Iff (x + c) = f (x) + c 8 c 2 R, then D ⊆ fx0 2 X 0 : x0(1) = 1g. (M + CA) If f satisfies both (M) and (CA) then D ⊆ Z. 0 x0 2 D (PH) If f (λx) = λf (x) 8 λ ≥ 0 then α(x0) = +1 x0 2= D Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Representation of convex risk measures on L1 A monetary risk measure on L1 = L1(Ω; F; P) is norm continuous (L1; k · k)0 = ba := ba(Ω; F; P), Therefore, applying the previous Proposition to a convex risk measure (a monotone decreasing, cash additive, convex map) 1 ρ : L (Ω; F; P) ! R we immediately deduce the following Proposition: Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Representation of convex risk measures on L1 Proposition 1 A convex risk measure ρ : L (Ω; F; P) ! R admits the representation ρ(x) = sup fµ(−x) − α(µ)g µ2ba+(1) where ba+(1) = fµ 2 ba j µ ≥ 0, µ(1Ω) = 1g ; If in addition ρ is σ(L1; L1) -LSC then ρ(x) = sup fEQ [−x] − α(Q)g Q<<P Marco Frittelli, First Part Risk Measures Outline First Part Definitions and properties of Risk Measures Convex analysis Orlicz spaces Convex risk measures on Banach lattices Generalization We will need a representation result for convex maps defined on more general spaces: Examples of these spaces are the Orlicz spaces Therefore, we need to understand under which conditions we may deduce a representation of the type: ρ(x) = sup fEQ [−x] − α(Q)g Q We will need a sort of LSC condition For this we need to introduce some terminology and results from Banach lattices theory.
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