Conjugate duality in stochastic optimization Ari-Pekka Perkki¨ o, Institute of Mathematics , Aalto University Ph.D. instructor/joint work with Teemu Pennanen , Institute of Mathematics , Aalto University March 15th 2010 1 / 13
Introduction We study convex stochastic optimization problems. Introduction Introduction Introduction ✔ Stochastic LP duality, linear quadratic control and calculus of Introduction variations Introduction Stochastic Optimization ✔ Stochastic problems of Bolza, shadow price of information and Stochastic Optimization inf E [ f ( ω, x ( ω ) , u ( ω ))] optimal stopping Conjugate duality Conjugate duality ✔ Illiquid convex market models (Jouni&Kallal, Kabanov, Conjugate duality in stochastic optimization Schachermayer, Guasoni, Pennanen) Conjugate duality in ✔ Super-hedging and pricing, utility maximization and optimal stochastic optimization Conjugate duality in consumption stochastic optimization Convexity gives rise to dual optimization problems and dual characterisations of the objective functionals. 2 / 13
Introduction Introduction Example (Super-hedging in a liquid market) . Introduction Introduction inf x 0 Introduction 0 , Introduction Stochastic Optimization � C T ≤ x 0 � s.t. 0 + T x t · dS t a.s. Stochastic Optimization inf E [ f ( ω, x ( ω ) , u ( ω ))] Conjugate duality Conjugate duality where x 0 0 is the initial wealth, C T is a claim, x is a predictable process Conjugate duality in stochastic optimization (portfolio of risky assets) and S is a price process. The infimum is over Conjugate duality in stochastic optimization initial wealths x 0 and predictable processes x . Conjugate duality in stochastic optimization 3 / 13
Introduction Introduction Example (Super-hedging in a liquid market) . Introduction Introduction inf x 0 Introduction 0 , Introduction Stochastic Optimization � C T ≤ x 0 � s.t. 0 + T x t · dS t a.s. Stochastic Optimization inf E [ f ( ω, x ( ω ) , u ( ω ))] Conjugate duality Conjugate duality where x 0 0 is the initial wealth, C T is a claim, x is a predictable process Conjugate duality in stochastic optimization (portfolio of risky assets) and S is a price process. The infimum is over Conjugate duality in stochastic optimization initial wealths x 0 and predictable processes x . Conjugate duality in stochastic optimization The dual problem is E Q [ C T ] , sup Q ∈M where the supremum is over martingale measures. 3 / 13
Introduction Introduction Example (Kabanovs model) . Consider a set Introduction Introduction Introduction { x ∈ BV | ( dx/ | dx | ) t ∈ C ( ω, t ) ∀ t } Introduction Stochastic Optimization where C ( ω, t ) ⊂ R d is a convex cone for all ( ω, t ) . C ( ω, t ) is the set of Stochastic Optimization inf E [ f ( ω, x ( ω ) , u ( ω ))] Conjugate duality self-financing trades in the market at time t . A predictable process of Conjugate duality bounded variation is self-financing if Conjugate duality in stochastic optimization Conjugate duality in stochastic optimization ( dx ( ω ) / | dx ( ω ) | ) t ∈ C ( ω, t ) ∀ t a.s. Conjugate duality in stochastic optimization 4 / 13
Introduction Introduction Example (Kabanovs model) . Consider a set Introduction Introduction Introduction { x ∈ BV | ( dx/ | dx | ) t ∈ C ( ω, t ) ∀ t } Introduction Stochastic Optimization where C ( ω, t ) ⊂ R d is a convex cone for all ( ω, t ) . C ( ω, t ) is the set of Stochastic Optimization inf E [ f ( ω, x ( ω ) , u ( ω ))] Conjugate duality self-financing trades in the market at time t . A predictable process of Conjugate duality bounded variation is self-financing if Conjugate duality in stochastic optimization Conjugate duality in stochastic optimization ( dx ( ω ) / | dx ( ω ) | ) t ∈ C ( ω, t ) ∀ t a.s. Conjugate duality in stochastic optimization Example (Linear case) . C ( ω, t ) = { ( x 0 , x 1 ) ∈ R 2 | x 0 + x 1 · S t ( ω ) ≤ 0 } , where S is the price process of a risky asset, x 0 refers to a bank account and x 1 to the risky asset. Portfolio is self-financing if all trades of the risky assets are financed using the bank account. The inequality allows a free disposal of money or assets. 4 / 13
Introduction Introduction Example (Optimal consumption in a convex market model) . Introduction Introduction � Introduction sup E U t ( ω, dc ) , Introduction Stochastic Optimization T Stochastic Optimization � inf E [ f ( ω, x ( ω ) , u ( ω ))] ( d ( x ( ω ) + c ( ω )) / | d ( x ( ω ) + c ( ω )) | ) t ∈ C ( ω, t ) ∀ t a.s. Conjugate duality s.t. Conjugate duality x t ( ω ) ∈ D ( ω, t ) ∀ t a.s. . Conjugate duality in stochastic optimization Conjugate duality in where U t is an utility function for all t almost surely and D ( ω, t ) is the stochastic optimization Conjugate duality in set of allowed portfolio positions at time t . The supremum is over stochastic optimization predictable processes of bounded variation x and c . x is the portfolio process, and c is the consumption process. 5 / 13
Introduction Introduction Example (Optimal consumption in a convex market model) . A dual Introduction problem is Introduction Introduction Introduction � Stochastic Optimization U ∗ inf E − t ( y t ) dt, Stochastic Optimization inf E [ f ( ω, x ( ω ) , u ( ω ))] T Conjugate duality � y t ( ω ) ∈ C ∗ ( ω, t ) ∀ t a.s. Conjugate duality s.t. Conjugate duality in ( da ( ω ) / | da ( ω ) | ) t ∈ D ∗ ( ω, t ) ∀ t a.s. . stochastic optimization Conjugate duality in stochastic optimization Conjugate duality in where U ∗ t is the concave conjugate of the utility function, C ∗ ( ω, t ) is the stochastic optimization polar of C ( ω, t ) ( y is a consistent price system ), D ∗ ( ω, t ) is the polar of D ( ω, t ) , and the supremum is over semimartingales with the canonical decomposition y = m + a . 6 / 13
Introduction Introduction Example (Optimal consumption in a convex market model) . A dual Introduction problem is Introduction Introduction Introduction � Stochastic Optimization U ∗ inf E − t ( y t ) dt, Stochastic Optimization inf E [ f ( ω, x ( ω ) , u ( ω ))] T Conjugate duality � y t ( ω ) ∈ C ∗ ( ω, t ) ∀ t a.s. Conjugate duality s.t. Conjugate duality in ( da ( ω ) / | da ( ω ) | ) t ∈ D ∗ ( ω, t ) ∀ t a.s. . stochastic optimization Conjugate duality in stochastic optimization Conjugate duality in where U ∗ t is the concave conjugate of the utility function, C ∗ ( ω, t ) is the stochastic optimization polar of C ( ω, t ) ( y is a consistent price system ), D ∗ ( ω, t ) is the polar of D ( ω, t ) , and the supremum is over semimartingales with the canonical decomposition y = m + a . The aim is to formulate problems like this in a general framework and deduce the dual problems by general methods. 6 / 13
Stochastic Optimization Introduction Let (Ω , F , F , P ) be a complete filtered probability space, Let N be the Introduction set of predictable processes of bounded variation. Let U be a separable Introduction Introduction Banach (or its dual). Introduction Stochastic Optimization Stochastic Optimization inf E [ f ( ω, x ( ω ) , u ( ω ))] Conjugate duality Conjugate duality Conjugate duality in stochastic optimization Conjugate duality in stochastic optimization Conjugate duality in stochastic optimization 7 / 13
Stochastic Optimization Introduction Let (Ω , F , F , P ) be a complete filtered probability space, Let N be the Introduction set of predictable processes of bounded variation. Let U be a separable Introduction Introduction Banach (or its dual). Define F : N × L p (Ω; U ) → R ∪ { + ∞} by Introduction Stochastic Optimization Stochastic Optimization F ( x, u ) = E [ f ( ω, x ( ω ) , u ( ω ))] , inf E [ f ( ω, x ( ω ) , u ( ω ))] Conjugate duality Conjugate duality where f is a normal-integrand. The value function is Conjugate duality in stochastic optimization Conjugate duality in stochastic optimization φ ( u ) = inf x ∈N F ( x, u ) = inf x ∈N E [ f ( ω, x ( ω ) , u ( ω ))] . Conjugate duality in stochastic optimization 7 / 13
Stochastic Optimization Introduction Let (Ω , F , F , P ) be a complete filtered probability space, Let N be the Introduction set of predictable processes of bounded variation. Let U be a separable Introduction Introduction Banach (or its dual). Define F : N × L p (Ω; U ) → R ∪ { + ∞} by Introduction Stochastic Optimization Stochastic Optimization F ( x, u ) = E [ f ( ω, x ( ω ) , u ( ω ))] , inf E [ f ( ω, x ( ω ) , u ( ω ))] Conjugate duality Conjugate duality where f is a normal-integrand. The value function is Conjugate duality in stochastic optimization Conjugate duality in stochastic optimization φ ( u ) = inf x ∈N F ( x, u ) = inf x ∈N E [ f ( ω, x ( ω ) , u ( ω ))] . Conjugate duality in stochastic optimization A function f : Ω × ( X × U ) → R ∪ { + ∞} is a normal integrand if the epigraph epi f ⊂ Ω × X × U × R is measurable and ω -sections are closed. In particular ω �→ f ( ω, x ( ω ) , u ( ω )) is measurable when x ∈ L 0 (Ω; X ) and u ∈ L 0 (Ω; U ) , and for fixed ω , ( x, u ) �→ f ( ω, x, u ) is lower semicontinuous. Moreover, F is convex if f ( ω, · , · ) is convex. 7 / 13
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