Risk averse dynamic optimization Progress in continuous time Linz - PowerPoint PPT Presentation

Risk averse dynamic optimization Progress in continuous time Linz Alois Pichler 1 Ruben Schlotter 1 November 14, 2019 1

Stochastic optimization [Markowitz, 1952] Primal Dual � � maximize E x ⊤ ξ x ⊤ ξ minimize var subject to x ∈ R d , subject to x ∈ R d , � � x ⊤ ξ ≤ q , var E x ⊤ ξ ≥ µ, d d � � x i = 1 x i = 1 i =1 i =1 ( x i ≥ 0) ( x i ≥ 0) A. Pichler risk averse 2

Assessment of risk Proposition (Axioms, cf. [Deprez and Gerber, 1985], [Artzner et al., 1999]) R : Y → R ∪{±∞} Monotonicity: if Y ≤ Y ′ , then R ( Y ) ≤ R ( Y ′ ) , 1 � Y + Y ′ � ≤ R ( Y )+ R ( Y ′ ) , Subadditivity: R 2 Translation equivariance: R ( Y + c ) = R ( Y )+ c for Y ∈ Y and 3 c ∈ R , Positive homogeneity, R ( λ Y ) = λ ·R ( Y ) for λ > 0 . 4 Equivalence principle R ( Y ) := E Y most fair, risk neutral R ( Y ) := esssup Y most unfair, totally risk averse . A. Pichler risk averse 3

Reformulation [Markowitz, 1952] Primal Dual � � � � maximize E x ⊤ ξ minimize E x ⊤ ξ + R − x ⊤ ξ − x ⊤ ξ =: D � � − x ⊤ ξ s.t. −R ≤ q , s.t. E x ⊤ ξ ≥ µ, d d � � x i = 1 x i = 1 i =1 i =1 (and x i ≥ 0) . (and x i ≥ 0) . A. Pichler risk averse 4

Outline Spanning horizons 3 The discrete setting 1 Nested Expressions The general multistage Explicit definition problem Hamilton Jacobi Bellman 4 Dynamic programming Hamilton Jacobi Continuous time Further assessments of risk 2 Generators Applications Risk generator References 5 A. Pichler risk averse 5

Outline Spanning horizons 3 The discrete setting 1 Nested Expressions The general multistage Explicit definition problem Hamilton Jacobi Bellman 4 Dynamic programming Hamilton Jacobi Continuous time Further assessments of risk 2 Generators Applications Risk generator References 5 A. Pichler risk averse 6

Multistage problem Non-Markovian difficulties Multistage optimization � ξ, x ( ξ ) � minimize E c subject to x ∈ X , x ( · ) is adapted (nonanticipative) x nonanticipative iff   x 0 ( ξ 0 )   x 1 ( ξ 0 ,ξ 1 )     .  .  � = � ξ 0 ,...,ξ T .   x   x t ( ξ 0 ,...,ξ t )     .   .  .  x T ( ξ 0 ,ξ 1 ,...,ξ T ) A. Pichler risk averse 7

Problem description Discrete time In a discrete framework, the sequence of decisions is x 0 � ξ 1 � x 1 ··· � ξ T � x T . � � � ξ, x ( ξ ) � : x ( · ) ∈ X , x ( · ) adapted inf E c Problem (Risk aversion) The risk averse stochastic problem is � � minimize R c 0 ( x 0 ) , c 1 ( ξ, x 1 ) ,..., c T ( ξ, x T ) x 0 ∈ X 0 ,..., x t ∈ X t ( x t − 1 ,ξ ) A. Pichler risk averse 8

Problem description Discrete time In a discrete framework, the sequence of decisions is x 0 � ξ 1 � x 1 ··· � ξ T � x T . � � � ξ, x ( ξ ) � : x ( · ) ∈ X , x ( · ) adapted inf E c Problem (Risk aversion) The risk averse stochastic problem is � � minimize R c 0 ( x 0 ) , c 1 ( ξ, x 1 ) ,..., c T ( ξ, x T ) x 0 ∈ X 0 ,..., x t ∈ X t ( x t − 1 ,ξ ) A. Pichler risk averse 8

Risk Example In the simplest case, � �� T � � ξ, x 0 ( ξ ) � ,..., c T � ξ, x T ( ξ ) � ξ, x t ( ξ ) � . R c 0 = E c t t =0 Problem � � �� � � ξ, x 0 ( ξ ) � ,..., c T � ξ, x T ( ξ ) R : x ∈ X , x ( · ) adapted inf c 0 . A. Pichler risk averse 9

Outline Spanning horizons 3 The discrete setting 1 Nested Expressions The general multistage Explicit definition problem Hamilton Jacobi Bellman 4 Dynamic programming Hamilton Jacobi Continuous time Further assessments of risk 2 Generators Applications Risk generator References 5 A. Pichler risk averse 10

Towards dynamic programming The Bellman principle � c 0 ( x 0 ) , c 1 ( ξ, x 1 ) ,..., c T ( ξ, x T ) � , min R x 0 ∈ X 0 , x t ∈ X t ( x t − 1 ,ξ ) , s . t . t = 1 ,..., T . Definition (Time consistent) The transition functionals are recursive, if Figure: Lattice R t , u ( Y t ,..., Y u ) approximation � Y t ,..., Y v − 1 , R v , u ( Y v ,..., Y u ) � . = R t , v A. Pichler risk averse 11

Towards dynamic programming Examples Conditional risk functionals Semideviation β ✁ F t � � SD ( Y | F t ) := E [ Y | F t ]+ β · E ( Y − E [ Y | F t ]) + | F t , Average Value-at-Risk α ✁ F t � � 1 AV@R α ( Y | F t ) := essinf q ✁ F t q + ( Y − q ) + | F t 1 − α E , Entropic Value-at-Risk α ✁ F t 1 1 EV@R α ( Y | F t ) := essinf t log 1 − α exp( E [ Y | F t ]) . 0 < t ✁ F t A. Pichler risk averse 12

Towards dynamic programming Examples Conditional risk functionals Semideviation β ✁ F t � � SD ( Y | F t ) := E [ Y | F t ]+ β · E ( Y − E [ Y | F t ]) + | F t , Average Value-at-Risk α ✁ F t � � 1 AV@R α ( Y | F t ) := essinf q ✁ F t q + ( Y − q ) + | F t 1 − α E , Entropic Value-at-Risk α ✁ F t 1 1 EV@R α ( Y | F t ) := essinf t log 1 − α exp( E [ Y | F t ]) . 0 < t ✁ F t A. Pichler risk averse 12

Towards dynamic programming Examples Conditional risk functionals Semideviation β ✁ F t � � SD ( Y | F t ) := E [ Y | F t ]+ β · E ( Y − E [ Y | F t ]) + | F t , Average Value-at-Risk α ✁ F t � � 1 AV@R α ( Y | F t ) := essinf q ✁ F t q + ( Y − q ) + | F t 1 − α E , α ✁ F t Entropic Value-at-Risk 1 1 EV@R α ( Y | F t ) := essinf t log 1 − α exp( E [ Y | F t ]) . 0 < t ✁ F t A. Pichler risk averse 12

Dynamic programming equations Proposition (Bellman equations, recursive transitions [2018]) V T ( ξ, x T − 1 ) := x T ∈X Z ( x T − 1 ,ξ ) c T ( ξ, x T ) , essinf V t ( ξ, x t − 1 ) := x t ∈X t ( ξ, x t − 1 ) R t : t +1 ( c t ( ξ, x t ) , V t +1 ( ξ, x t )) . essinf V 0 solves the problem � � minimize R c 0 ( x 0 ) , c 1 ( ξ, x 1 ) ,..., c T ( ξ, x T ) , x 0 ∈ X 0 , x t ∈ X t ( x t − 1 ,ξ ) , subject to t = 1 ,..., T . A. Pichler risk averse 13

Dynamic programming equations Proposition (Bellman equations, recursive transitions [2018]) V T ( ξ, x T − 1 ) := x T ∈X Z ( x T − 1 ,ξ ) c T ( ξ, x T ) , essinf � � V t ( ξ, x t − 1 ) := x t ∈X t ( ξ, x t − 1 ) R t : t +1 essinf c t ( ξ, x t ) , V t +1 ( ξ, x t ) . V 0 solves the problem � � R minimize c 0 ( x 0 ) , c 1 ( ξ, x 1 ) ,..., c T ( ξ, x T ) , x 0 ∈ X 0 , x t ∈ X t ( x t − 1 ,ξ ) , subject to t = 1 ,..., T . A. Pichler risk averse 13

Outline Spanning horizons 3 The discrete setting 1 Nested Expressions The general multistage Explicit definition problem Hamilton Jacobi Bellman 4 Dynamic programming Hamilton Jacobi Continuous time Further assessments of risk 2 Generators Applications Risk generator References 5 A. Pichler risk averse 14

Decisions under uncertainty The Wiener setting The motion is generated by d X t = b d t + σ d W t Definition (Generator) For a smooth function φ , � � � � 1 φ ( t +∆ t , X t +∆ t ) � G φ ( t ,ξ ) := lim ∆ t E � X t = ξ . � − φ ( t ,ξ ) ∆ t → 0 A. Pichler risk averse 15

Ito’s formula Definition Recall the generator, � � � � 1 φ ( t +∆ t , X t +∆ t ) � G φ ( t ,ξ ) := lim � X t = ξ ∆ t E . � − φ ( t ,ξ ) ∆ t → 0 Lemma (Ito) For d X t = b d t + σ d W t it holds that For φ = 1 , G φ = 0 ; 1 for φ ( ξ ) = ξ , then G φ = b, the drift; 2 for φ ( ξ ) = ξ 2 , then G φ = 2 b ξ + σ 2 , the volatility; 3 for general φ ( ξ ) , 4 2 σ 2 ∂ 2 G = ∂ ∂ t + b ∂ ∂ξ + 1 ∂ξ 2 . A. Pichler risk averse 16

Ito’s formula Definition Recall the generator, � � � � 1 φ ( t +∆ t , X t +∆ t ) � G φ ( t ,ξ ) := lim � X t = ξ ∆ t E . � − φ ( t ,ξ ) ∆ t → 0 Lemma (Ito) For d X t = b d t + σ d W t it holds that For φ = 1 , G φ = 0 ; 1 for φ ( ξ ) = ξ , then G φ = b , the drift; 2 for φ ( ξ ) = ξ 2 , then G φ = 2 b ξ + σ 2 , the volatility; 3 for general φ ( ξ ) , 4 2 σ 2 ∂ 2 G = ∂ ∂ t + b ∂ ∂ξ + 1 ∂ξ 2 . A. Pichler risk averse 16

Ito’s formula Definition Recall the generator, � � � � 1 φ ( t +∆ t , X t +∆ t ) � G φ ( t ,ξ ) := lim � X t = ξ ∆ t E � . − φ ( t ,ξ ) ∆ t → 0 Lemma (Ito) For d X t = b d t + σ d W t it holds that For φ = 1 , G φ = 0 ; 1 for φ ( ξ ) = ξ , then G φ = b, the drift; 2 for φ ( ξ ) = ξ 2 , then G φ = 2 b ξ + σ 2 , the volatility; 3 for general φ ( ξ ) , 4 2 σ 2 ∂ 2 ∂ξ + 1 G = ∂ ∂ t + b ∂ ∂ξ 2 . A. Pichler risk averse 16

Ito’s formula Definition Recall the generator, � � � � 1 φ ( t +∆ t , X t +∆ t ) � G φ ( t ,ξ ) := lim � X t = ξ . ∆ t E � − φ ( t ,ξ ) ∆ t → 0 Lemma (Ito) For d X t = b d t + σ d W t it holds that For φ = 1 , G φ = 0 ; 1 for φ ( ξ ) = ξ , then G φ = b, the drift; 2 for φ ( ξ ) = ξ 2 , then G φ = 2 b ξ + σ 2 , the volatility; 3 for general φ ( ξ ) , 4 2 σ 2 ∂ 2 ∂ t + b ∂ ∂ ∂ξ + 1 G = ∂ξ 2 . A. Pichler risk averse 16