Background The problem Results References And finally . . . Using stochastic control in robust portfolio selection Martin Schweizer Department of Mathematics ETH Z¨ urich Stochastic Processes: Theory and Applications in honor of Wolfgang Runggaldier Bressanone, 20.07.2007 based on joint work with Giuliana Bordigoni (Milano) and Anis Matoussi (Le Mans) Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Results References And finally . . . Background Well-known problem from economics: maximize expected utility from investment and consumption lots of references . . . large majority assumes that underlying model is known Here: what happens if model is not exactly known? � � Abstract formulation: consider E Q U 0 , T ( π, c ) : U 0 , T ( π, c ) stands for utility over (0 , T ] from investment π and consumption c Q stands for probabilistic model maximize with respect to π, c minimize with respect to Q , to capture uncertainty about Q Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Formulation Results More details References And finally . . . Formulation Approach 1: specify set Q of possible models Q � � consider sup Q ∈Q E Q inf U 0 , T ( π, c ) (or inf sup) ( π, c ) Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Formulation Results More details References And finally . . . Formulation Approach 1: specify set Q of possible models Q � � consider sup Q ∈Q E Q inf U 0 , T ( π, c ) (or inf sup) ( π, c ) Approach 2: allow all Q as possible models penalize by some functional R ( Q ) � �� � � � consider sup inf U 0 , T ( π, c ) + β R ( Q ) (or inf E Q Q ∈M 1 ( P ) ( π, c ) sup) Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Formulation Results More details References And finally . . . Formulation Approach 1: specify set Q of possible models Q � � consider sup Q ∈Q E Q inf U 0 , T ( π, c ) (or inf sup) ( π, c ) Approach 2: allow all Q as possible models penalize by some functional R ( Q ) � �� � � � consider sup inf U 0 , T ( π, c ) + β R ( Q ) (or inf E Q Q ∈M 1 ( P ) ( π, c ) sup) Here: some results about partial problem � � � � inf E Q U 0 , T + β R ( Q ) ; Q ∈M 1 ( P ) R ( Q ) is entropic penalty term � � U 0 , T represents abstract utility think of ( π, c ) as fixed Martin Schweizer Using stochastic control in robust portfolio selection
Background The problem Formulation Results More details References And finally . . . More details More precisely: � t � � discount factor S δ t := exp − 0 δ s ds � T S δ α S δ ¯ utility term U δ t , T := α t U s ds + ¯ U T s T S δ S δ t t � T S δ t log Z Q t ds + S δ t log Z Q penalty term R δ t , T ( Q ) := t δ s s s T T S δ Z Q S δ Z Q t Goal: minimize cost functional U δ 0 , T + β R δ � � Q �→ Γ( Q ) := E Q 0 , T ( Q ) U 0 � � Remark: For δ ≡ 0, we get E Q + β H ( Q | P ); so have 0 , T to minimize relative entropy with respect to suitable P U ; solution is Q ∗ = P U . But δ �≡ 0 is not so easy (for us . . . ). Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? Preliminary results Assumptions: (A1) 0 ≤ δ ≤ � δ � ∞ < ∞ for some constant � δ � ∞ � T 0 | U s | ds has all exponential moments (subspace M exp of (A2) Orlicz space L exp ) (A3) | ¯ U T | has all exponential moments Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? Preliminary results Assumptions: (A1) 0 ≤ δ ≤ � δ � ∞ < ∞ for some constant � δ � ∞ � T 0 | U s | ds has all exponential moments (subspace M exp of (A2) Orlicz space L exp ) (A3) | ¯ U T | has all exponential moments First step: for constants depending only on α, α ′ , β, δ, T , U , ¯ U T (but not on Q ), we have estimate � � � � c 1 + H ( Q | P ) ≤ 1 + Γ( Q ) ≤ C 1 + H ( Q | P ) (1) RHS is easy; LHS exploits exponential moment conditions Consequences: � H ( Q | P ) < ∞ � � � can work with Q ∈ Q f := Q ∈ M 1 ( P ) can later use entropy bounds Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? Existence Theorem 1: Under (A1) – (A3) , there exists unique Q ∗ ∈ Q f minimizing Q �→ Γ( Q ) over all Q ∈ Q f . Moreover, Q ∗ ≈ P . Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? Existence Theorem 1: Under (A1) – (A3) , there exists unique Q ∗ ∈ Q f minimizing Q �→ Γ( Q ) over all Q ∈ Q f . Moreover, Q ∗ ≈ P . Sketch of proof: Uniqueness from strict convexity For minimizing sequence of density processes Z n , Koml´ os yields convex combinations ¯ → ¯ Z n Z ∞ T − T P -a.s. Use entropy estimate Q ∞ with density ¯ (1) (to get UI) to show that ¯ Z ∞ T is in Q f Γ( Q ) as functional on density processes Z Q is convex and “morally lower semicontinuous” with respect to Q ∞ must attain infimum P -a.s. convergence of Z Q T ; so ¯ � U δ � Proving lsc for (linear, but unbounded term) E Q is 0 , T tricky; uses full strength of exponential moment conditions Equivalence adapts Frittelli (2000), using FOC for optimality Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? Dynamic formulation Dynamic version: value of problem started at time τ is � � E Q ′ [ U δ τ, T |F τ ] + β E Q ′ [ R δ τ, T ( Q ′ ) |F τ ] V τ := P - ess inf Q ′ ∈ Q f , Q ′ ≈ P Structure of value process V ? − → stochastic control Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? Dynamic formulation Dynamic version: value of problem started at time τ is � � E Q ′ [ U δ τ, T |F τ ] + β E Q ′ [ R δ τ, T ( Q ′ ) |F τ ] V τ := P - ess inf Q ′ ∈ Q f , Q ′ ≈ P Structure of value process V ? − → stochastic control Properties of � � J Q E Q ′ [ U δ 0 , T |F τ ] + β E Q ′ [ R δ 0 , T ( Q ′ ) |F τ ] τ := P - ess inf : Q ′ = Q on F τ Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? Dynamic formulation Dynamic version: value of problem started at time τ is � � E Q ′ [ U δ τ, T |F τ ] + β E Q ′ [ R δ τ, T ( Q ′ ) |F τ ] V τ := P - ess inf Q ′ ∈ Q f , Q ′ ≈ P Structure of value process V ? − → stochastic control Properties of � � J Q E Q ′ [ U δ 0 , T |F τ ] + β E Q ′ [ R δ 0 , T ( Q ′ ) |F τ ] τ := P - ess inf : Q ′ = Q on F τ Martingale optimality principle: J Q is a Q -submartingale for each Q Q ∗ is optimal iff J Q ∗ is a Q ∗ -martingale Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? Dynamic formulation Dynamic version: value of problem started at time τ is � � E Q ′ [ U δ τ, T |F τ ] + β E Q ′ [ R δ τ, T ( Q ′ ) |F τ ] V τ := P - ess inf Q ′ ∈ Q f , Q ′ ≈ P Structure of value process V ? − → stochastic control Properties of � � J Q E Q ′ [ U δ 0 , T |F τ ] + β E Q ′ [ R δ 0 , T ( Q ′ ) |F τ ] τ := P - ess inf : Q ′ = Q on F τ Martingale optimality principle: J Q is a Q -submartingale for each Q Q ∗ is optimal iff J Q ∗ is a Q ∗ -martingale J Q = S δ V + α s ds + β S δ log Z Q S δ δ s S δ s log Z Q � � s U s ds + β Martin Schweizer Using stochastic control in robust portfolio selection
Background Preliminary results The problem Existence Results Dynamic formulation References BSDE And finally . . . What next? BSDE for V Theorem 2: Under (A1) – (A3) and if F is continuous , ( V , M V ) is unique solution in D exp × M 0 , loc ( P ) of 0 backward stochastic differential equation (BSDE) ( δ t Y t − α U t ) dt + 1 dY t = 2 β d � M � t + dM t , (2) α ¯ Y T = ¯ U T . � β M V � is true P -martingale, and M V lies in − 1 Moreover, E martingale space M p 0 ( P ) for every p ∈ [1 , ∞ ). Martin Schweizer Using stochastic control in robust portfolio selection
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