On the dual problem of utility maximization Yiqing LIN Joint work - PowerPoint PPT Presentation

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment On the dual problem of utility maximization Yiqing LIN Joint work with L. GU and J. YANG University of Vienna Sept. 2nd 2015 Workshop “Advanced methods in financial mathematics” Angers

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Introduction 1 Basic settings Literature review on convex duality methods Utility maximization without random endowment 2 Duality method The dual optimizer: existing result The dual optimizer: alternative method Utility maximization with bounded random endowment 3 Duality method The dual optimizer Remarks

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Basic settings The market model Consider a filtered probability space (Ω , F , ( F t ) 0 ≤ t ≤ T , P ) satisfying the usual conditions, where T is a finite horizon. The market consists of a bond and a stock, where the bond is of zero interest rate and the stock-price process S is a strictly positive semimartingale. “No arbitrage condition”: M e ( S ) � = ∅ . For an initial value x and a predictable S -integrable trading strategy H , the value process X = ( X t ) 0 ≤ t ≤ T is given by X t = x + ( H · S ) t , 0 ≤ t ≤ T. We call H admissible if for 0 ≤ t ≤ T , the associated terminal value X t ≥ − M , for some positive M . The agent receives an exogenous random endowment e T ∈ F T at time T , satisfying ρ := � e T � ∞ < ∞ .

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Basic settings Utility maximization on the positive half line A utility function U : (0 , ∞ ) → R represents the agent’s preferences over the terminal wealth. The function U is assumed to be strictly concave, strictly increasing and continuously differentiable satisfying the Inada conditions: U ′ (0) := lim x → 0 U ′ ( x ) = ∞ , U ′ ( ∞ ) := lim x →∞ U ′ ( x ) = 0 . and the condition of reasonable asymptotic elasticity (RAE): xU ′ ( x ) AE ( U ) = lim sup < 1 . U ( x ) x →∞ The aim of the agent is to maximize the expected utility from the terminal wealth: E [ U ( x + ( H · S ) T + e T )] , u ( x ) := sup x > 0 , H adm

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Literature review on convex duality methods The Itˆ o framework: [Karatzas-Lehoczky-Shreve-Xu, 1991], [Cvitani´ c-Karatzas, 1992], etc...; The general semimartingale framework: U : (0 , ∞ ) → R e T = 0 [Kramkov-Schachermayer, 1999] bounded e T [Cvitani´ c-Schachermayer-Wang, 2001] unbounded e T [Hugonnier-Kramkov, 2004] U : R → R : - locally bounded semimartingale models: [Schachermayer, 2001], [Owen, 2002], [Owen-ˇ Zitkovi´ c, 2009]; - general semimartingale models: [Biagini-Frittelli, 2008], [Biagini-Frittelli-Grasselli, 2011]. Optimal consumption, with constraints, etc...

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Introduction 1 Basic settings Literature review on convex duality methods Utility maximization without random endowment 2 Duality method The dual optimizer: existing result The dual optimizer: alternative method Utility maximization with bounded random endowment 3 Duality method The dual optimizer Remarks

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method Duality method: e T = 0 Define X ( x ) := { X : X = x + ( H · S ) , H adm , X T ≥ 0 } and � � g ∈ L 0 C ( x ) := + ( F T ) : 0 ≤ g ≤ X T , for some X ∈ X ( x ) , where the latter is the set of positive terminal values, which can be dominated by some admissible strategies initialed from x > 0 . Then, the maximization problem (primal problem) can be rewritten into u ( x ) := sup E [ U ( x + ( H · S ) T )] = sup E [ U ( g )] . H adm g ∈C ( x )

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method Definition (Supermartingale deflators) We call a positive semimartingale Y a supermartingale deflator, if for each X ∈ X (1) , XY is a supermartingale. Moreover, we denote by Y ( y ) the collection of all such processes starting from y , namely, � � Y ( y ) := Y ≥ 0 : Y 0 = y, XY is a supermartingale , ∀ X ∈ X (1) . Note D ( y ) := { h ∈ L 0 + ( F T ) : 0 ≤ h ≤ Y T , for some Y ∈ Y ( y ) } . Then, the dual problem is formulated as � � � � v ( y ) = inf V ( Y T ) = inf V ( h ) , Y ∈Y ( y ) E h ∈D ( y ) E where V ( y ) := sup { U ( x ) − xy } , y > 0 . x> 0

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method Theorem ([Kramkov-Schachermayer, 1999]) Assume “No arbitrage condition”: M e ( S ) � = ∅ , U satisfies the Inada conditions and RAE, u ( x ) < ∞ , for some x > 0 . Then The value functions u, v have the same properties as U and V . For any y > 0 , there exists a unique dual optimizer � h ∈ D ( y ) . y := u ′ ( x ) , then there exists a unique primal solution Let � g := ( U ′ ) − 1 ( � � g ∈ C ( x ) , which is defined by � h � y ) . g � E [ � h ] = x � y .

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: existing result In general, the optimal element � Y from Y ( � y ) associated with the dual optimizer � h � y is not a martingale. An example can be found in [Kramkov-Schachermayer, 1999] even with an one-period model. However, under certain condition, � Y is proved a local martingale. Theorem ([Larson-ˇ Zitkovi´ c, 2007]) In addition to the conditions for the above theorem, suppose that S is continuous, then the dual optimizer � h � y is attained by a local martingale from the set of supermartingale deflators. Outline of the proof: That S is continuous and M e ( S ) � = ∅ implies the following representation ([Delbean-Schachermayer, 1995]): � t λ u d � M � u , 0 ≤ t ≤ T, S t = 1 + M t + 0 where M is a local martingale and λ is a predicable M -integrable process.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: existing result Proposition For any Y ∈ Y ( y ) , we have the following multiplicative decomposition Y = y E ( − λ · M ) E ( L ) D, ag local martingale satisfying � M, L � ≡ 0 , and D where L is a c` adl` is a predictable, non-increasing, strictly positive, c` adl` ag process with D 0 = 1 . It can be verified by Itˆ o’s formula that y E ( − λ · M ) E ( L ) ∈ Y ( y ) . Then, from the fact that V is strictly decreasing, one can deduce that � y E ( − λ · M ) E ( � Y = � L ) , which is a local martingale, namely, � D ≡ 1 . �

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: alternative method We would like to provide an alternative method to prove the same theorem [Larson-ˇ Zitkovi´ c, 2007]. Based on the same idea, we could generalize this theorem to the case of bounded random endowment in the next section. The idea is as follows: we first stop the process � X by a sequence of stopping times { τ k } k ∈ N , such that before each τ k , � X is bounded away from 0. Precisely, define a localizing sequence τ k := inf { t : � X t < 1 /k } ∧ T. Since � X is continuous, � X τ k ≥ 1 /k and P (lim k τ k = T ) = 1 . Then, we shall construct a process � Y , such that � Y T = � h � y and prove that the stopped process � Y ·∧ τ k is a martingale by means of that � X � Y is a uniformly integrable martingale. Finally, we prove that � Y ∈ D ( � y ) .

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: alternative method From the result in [Kramkov-Schachermayer, 1999], one can find a sequence { Q n } ∞ n =1 from M e ( S ) such that yd Q n → � � d P − h � y , a.s.. We denote by Y n the associated density process, which is a martingale, i.e., � � yd Q n � Y n t := � . � d P F t Then, we construct a process � Y in terms of { Y n } ∞ n =1 , such that Y T = � � h � y . To this end, we need the following lemma.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: alternative method Lemma ([Czichowsky-Schachermayer, 2014]) Let { Y n } ∞ n =1 be a sequence of non-negative optional strong supermartingales Y n = { Y n t } 0 ≤ t ≤ T starting at Y n 0 = y . Then there is a sequence { � Y n } ∞ n =1 of convex combinations Y n ∈ conv ( Y n , Y n +1 , · · · ) � and a non-negative optional strong supermartingale Y = { � � Y t } 0 ≤ t ≤ 1 such that for every [0 , T ] -valued stoppting time τ , we have convergence in probability, i.e., � → � Y n τ − Y τ . WLOG, we may choose a subsequence such that for each k , � → � Y n τ k − Y τ k , a.s..