A new functional analytic approach to robust utility maximization in the dominated case Julio Daniel Backhoff Humboldt-Universit¨ at zu Berlin Universit¨ at Wien Joint work with Joaqu´ ın Fontbona of Universidad de Chile I thank the Berlin Mathematical School for full support. Part of this work was undertaken during a visit to the Hausdorff Research Institute for Mathematics at the University of Bonn within the Trimester Program Stochastic Dynamics in Economics and Finance. N OVEMBER , 2014 Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 1 / 20
Outline Introduction 1 Utility maximization in continuous time financial markets The convex duality approach Robust problem under “model compactness” Open questions and motivation Robust problem without model compactness 2 A Modular space formulation Our main result Worst-case measure for “linear uncertainty” in complete case 3 Setting and an abstract result Example Conclusions, open problems 4 Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 2 / 20
Outline Introduction 1 Utility maximization in continuous time financial markets The convex duality approach Robust problem under “model compactness” Open questions and motivation Robust problem without model compactness 2 A Modular space formulation Our main result Worst-case measure for “linear uncertainty” in complete case 3 Setting and an abstract result Example Conclusions, open problems 4 Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 2 / 20
Outline Introduction 1 Utility maximization in continuous time financial markets The convex duality approach Robust problem under “model compactness” Open questions and motivation Robust problem without model compactness 2 A Modular space formulation Our main result Worst-case measure for “linear uncertainty” in complete case 3 Setting and an abstract result Example Conclusions, open problems 4 Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 2 / 20
Outline Introduction 1 Utility maximization in continuous time financial markets The convex duality approach Robust problem under “model compactness” Open questions and motivation Robust problem without model compactness 2 A Modular space formulation Our main result Worst-case measure for “linear uncertainty” in complete case 3 Setting and an abstract result Example Conclusions, open problems 4 Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 2 / 20
Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 3 / 20
Continuous Time Financial Market Filtered probability space (Ω , F , ( F ) t ≤ T , P ) , P reference law. S i � � Market consists of d stocks and a risk-less bond, S = 0 ≤ i ≤ d . S continuous (or loc. bounded) semimartingale. � t The value of portfolio ( X 0 , π ) at time t is X t = X 0 + 0 π u dS u . � � ˜ P ∼ P : S is a ˜ M e ( S ) = P -loc. martingale � = ∅ . Admissible wealths starting from x � t � � X ( x ) = X ≥ 0 : X t = X 0 + H u dS u with X 0 ≤ x 0 Utility Functions on ( 0 , ∞ ) U : ( 0 , ∞ ) → ( −∞ , ∞ ) is strictly increasing, strictly concave and continuously differentiable. It satisfies INADA if U ′ ( 0 +) = ∞ and xU ′ ( x ) U ′ ( ∞ ) = 0. Its asymptotic elasticity is AE ( U ) := l´ ım sup U ( x ) . x →∞ Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 3 / 20
Utility maximization problems Standard utility maximization Agent tries to maximize expected final utility starting from x > 0, under the fixed (subjective) model Q ∼ = P . Value function is E Q [ U ( X T )] . u Q ( x ) := sup X ∈X ( x ) Robust utility maximization Actual probabilistic model (law) possibly unknown (model uncertainty) but there is a set Q of reasonable possible models. Pessimistic agent tries to maximize expected final utility of the worst-case model. Value function is Q ∈Q E Q [ U ( X T )] . u ( x ) := sup ´ ınf X ∈X ( x ) Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 4 / 20
Utility maximization problems Standard utility maximization Agent tries to maximize expected final utility starting from x > 0, under the fixed (subjective) model Q ∼ = P . Value function is E Q [ U ( X T )] . u Q ( x ) := sup X ∈X ( x ) Robust utility maximization Actual probabilistic model (law) possibly unknown (model uncertainty) but there is a set Q of reasonable possible models. Pessimistic agent tries to maximize expected final utility of the worst-case model. Value function is Q ∈Q E Q [ U ( X T )] . u ( x ) := sup ´ ınf X ∈X ( x ) Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 4 / 20
Duality in financial market models V ( y ) := sup [ U ( x ) − xy ] , y > 0 conjugate of U . x > 0 “Supermartingale densities ” w.r.t. (subjective) model Q Y Q ( y ) := { Y ≥ 0 , YX is a Q − supermartingale ∀ X ∈ X ( 1 ) , Y 0 = y } . Generalizes set of densities wrt. Q of e.g. risk-neutral measures. For all x > 0 , X ∈ X ( x ) , Q , � � E Q [ U ( X T )] ≤ ´ Y ∈Y Q ( y ) E Q [ V ( Y T )] + xy ınf ´ ınf y > 0 Y ∈Y Q ( y ) E Q [ V ( Y T )] candidate conjugate of u Q ( x ) , = ⇒ v Q ( y ) := ´ ınf = ⇒ v ( y ) := ´ Q ∈Q v Q ( y ) candidate conjugate of u ( x ) . ınf Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 5 / 20
Duality in financial market models V ( y ) := sup [ U ( x ) − xy ] , y > 0 conjugate of U . x > 0 “Supermartingale densities ” w.r.t. (subjective) model Q Y Q ( y ) := { Y ≥ 0 , YX is a Q − supermartingale ∀ X ∈ X ( 1 ) , Y 0 = y } . Generalizes set of densities wrt. Q of e.g. risk-neutral measures. For all x > 0 , X ∈ X ( x ) , Q , � � E Q [ U ( X T )] ≤ ´ Y ∈Y Q ( y ) E Q [ V ( Y T )] + xy ınf ´ ınf y > 0 Y ∈Y Q ( y ) E Q [ V ( Y T )] candidate conjugate of u Q ( x ) , = ⇒ v Q ( y ) := ´ ınf = ⇒ v ( y ) := ´ Q ∈Q v Q ( y ) candidate conjugate of u ( x ) . ınf Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 5 / 20
Duality in financial market models V ( y ) := sup [ U ( x ) − xy ] , y > 0 conjugate of U . x > 0 “Supermartingale densities ” w.r.t. (subjective) model Q Y Q ( y ) := { Y ≥ 0 , YX is a Q − supermartingale ∀ X ∈ X ( 1 ) , Y 0 = y } . Generalizes set of densities wrt. Q of e.g. risk-neutral measures. For all x > 0 , X ∈ X ( x ) , Q , � � E Q [ U ( X T )] ≤ ´ Y ∈Y Q ( y ) E Q [ V ( Y T )] + xy ınf ´ ınf y > 0 Y ∈Y Q ( y ) E Q [ V ( Y T )] candidate conjugate of u Q ( x ) , = ⇒ v Q ( y ) := ´ ınf = ⇒ v ( y ) := ´ Q ∈Q v Q ( y ) candidate conjugate of u ( x ) . ınf Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 5 / 20
Robust case under model compactness � � �� Y T d Q Dual involves v ( y ) = ´ ınf ´ ınf d P V . Y ∈Y P ( y ) E d Q / d P Q ∈Q Q ∈Q E Q [ U ( X T )] = ´ Primal requires Minimax: sup ´ ınf Q ∈Q u Q ( x ) . ınf X ∈X ( x ) Conditions on Q are needed. [SchiedWu05] consider: Q convex, 1 P ( A ) = 0 ⇐ ⇒ Q ( A ) = 0 ∀ Q ∈ Q , and 2 � � d Q d Q closed in L 0 ( P ) (equiv. σ ( L 1 , L ∞ ) − compact ). d P := d P : Q ∈ Q 3 Theorem ([SchiedWu05] (see also Gundel ∼ 03)) Then minimax equality holds and u , v are conjugate. Under additional assumptions (e.g. AE ( U ) < 1 ), everything is attained: X T = ( U ′ ) − 1 ( ˆ ˆ Y T / ˆ u ( x ) = u ˆ Q ( x ) , Z T ) � Z = d ˆ � y ∈ ∂ u ( x ) , ˆ ˆ d P , ˆ Q where ˆ Y ∈ Y (ˆ y ) and the pair Y attains the double infimum in the dual problem for such ( x , ˆ y ) . Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 6 / 20
Robust case under model compactness � � �� Y T d Q Dual involves v ( y ) = ´ ınf ´ ınf d P V . Y ∈Y P ( y ) E d Q / d P Q ∈Q Q ∈Q E Q [ U ( X T )] = ´ Primal requires Minimax: sup ´ ınf Q ∈Q u Q ( x ) . ınf X ∈X ( x ) Conditions on Q are needed. [SchiedWu05] consider: Q convex, 1 P ( A ) = 0 ⇐ ⇒ Q ( A ) = 0 ∀ Q ∈ Q , and 2 � � d Q d Q closed in L 0 ( P ) (equiv. σ ( L 1 , L ∞ ) − compact ). d P := d P : Q ∈ Q 3 Theorem ([SchiedWu05] (see also Gundel ∼ 03)) Then minimax equality holds and u , v are conjugate. Under additional assumptions (e.g. AE ( U ) < 1 ), everything is attained: X T = ( U ′ ) − 1 ( ˆ ˆ Y T / ˆ u ( x ) = u ˆ Q ( x ) , Z T ) � Z = d ˆ � y ∈ ∂ u ( x ) , ˆ ˆ d P , ˆ Q where ˆ Y ∈ Y (ˆ y ) and the pair Y attains the double infimum in the dual problem for such ( x , ˆ y ) . Julio Daniel Backhoff () Workshop London Nov. 2014 29/11/2014 6 / 20
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