Introduction g -Deviation Measures Characterizations & Representation Results Portfolio Optimisation Generalized Dynamic Deviation Measures in Risk Analysis Martijn Pistorius Mitja Stadje ISM-UUlm Workshop: Risk and Statistics October 8th, 2019 1 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction Motivation g -Deviation Measures Conditional Deviation Measures Characterizations & Representation Results Dynamic Deviation Measures Portfolio Optimisation Motivation One traditional way of thinking about risk is in terms of the extent that random realisations deviate from the mean. In this context an axiomatic framework for static (generalized) deviation measures was introduced and developed in Rockafellar, Uryasev, Zabarankin (2006a). They were inspired by the axiomatic approach by Artzner et al. (1999) for coherent risk measures which can be seen as (generalized) expectations , see also F¨ ollmer and Schied (2002). Various aspects of portfolio optimisation and financial decision making under general deviation measures have been explored in the literature, in particular regarding CAPM, asset betas, one- and two-fund theorems and equilibrium theory; 2 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction Motivation g -Deviation Measures Conditional Deviation Measures Characterizations & Representation Results Dynamic Deviation Measures Portfolio Optimisation Motivation We present an axiomatic approach to deviation measures in dynamic continuous-time settings. We give characterisations of dynamic deviation measures in terms of additively m -stable sets and certain (backward) SDEs. We give applications for portfolio optimisation in continuous-time. 3 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction Motivation g -Deviation Measures Conditional Deviation Measures Characterizations & Representation Results Dynamic Deviation Measures Portfolio Optimisation Conditional Deviation Measures (Rockafellar et al. (2006a)). For any given t ∈ [0 , T ], D t : L 2 ( F T ) → L 2 + ( F t ) is called an F t -conditional deviation measure if it is normalised ( D t (0) = 0) and the following properties are satisfied: (D1) Translation Invariance : D t ( X + m ) = D t ( X ) for any m ∈ L ∞ ( F t ); (D2) Positive Homogeneity : D t ( λ X ) = λ D t ( X ) for any X ∈ L 2 ( F T ) and λ ∈ L ∞ + ( F t ); (D3) Subadditivity : D t ( X + Y ) ≤ D t ( X ) + D t ( Y ) for any X , Y ∈ L 2 ( F T ); (D4) Positivity : D t ( X ) ≥ 0 for any X ∈ L 2 ( F T ) , and D t ( X ) = 0 if and only if X is F t -measurable. (D5) Lower Semi-Continuity: If X n converges to X in L 2 ( F T ) then D t ( X ) ≤ lim inf n D t ( X n ) . 4 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction Motivation g -Deviation Measures Conditional Deviation Measures Characterizations & Representation Results Dynamic Deviation Measures Portfolio Optimisation Conditional Risk Measures (Artzner et al. (1999), F¨ ollmer and Schied (2002)). Rockafellar et. al (2006a) were inspired by the axiomatic approach of Artzner et al. (1999) and F¨ ollmer and Schied (2002) for coherent and convex risk measures (generalized expectations). ρ t is a coherent risk measure if the following properties are satisfied: (R1) Translation Invariance : ρ t ( X + m ) = ρ t ( X ) + m for any m ∈ L ∞ ( F t ); (R2) Positive Homogeneity : ρ t ( λ X ) = λρ t ( X ) for any X ∈ L 2 ( F T ) and λ ∈ L ∞ + ( F t ); (R3) Subadditivity : ρ t ( X + Y ) ≤ ρ t ( X ) + ρ t ( Y ) for any X , Y ∈ L 2 ( F T ); (R4) Monotonicity: If X ≤ Y then ρ ( X ) ≥ ρ ( Y ) . 5 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction Motivation g -Deviation Measures Conditional Deviation Measures Characterizations & Representation Results Dynamic Deviation Measures Portfolio Optimisation Dynamic Deviation Measures In a dynamic setting we need the following two axioms: (D6) Consistency: For all s , t ∈ [0 , T ] with s ≤ t and X ∈ L 2 ( F T ) D s ( X ) = D s ( E [ X |F t ]) + E [ D t ( X ) |F s ] . 6 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction Motivation g -Deviation Measures Conditional Deviation Measures Characterizations & Representation Results Dynamic Deviation Measures Portfolio Optimisation Dynamic Deviation Measures Definition A family ( D t ) t ∈ [0 , T ] is called a dynamic deviation measure if D t , t ∈ [0 , T ], are F t -conditional deviation measures satisfying (D6). 7 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction Motivation g -Deviation Measures Conditional Deviation Measures Characterizations & Representation Results Dynamic Deviation Measures Portfolio Optimisation Assume from now on that we are in a continuous-time setting with two independent stochastic processes: (i) A standard d -dimensional Brownian motion W = ( W 1 , . . . , W d ) ⊺ . (ii) A real-valued Poisson random measure p on [0 , T ] × R k \ { 0 } . We denote by N ( ds , dx ) the associated random (counting) measure with L´ evy measure ν ( dx ) and set ˜ N ( ds , dx ) := N ( ds , dx ) − ν ( dx ) ds . By the martingale representation theorem for any X ∈ L 2 ( F T ) there exist unique predictable square integrable H X and ˜ H X satisfying � T � T � ˜ s ( x ) ˜ H X H X X = E [ X ] + s dW s + N p ( ds , dx ) . 0 0 R k \{ 0 } Henceforth, we will refer to ( H X , ˜ H X ) as the representing pair. 8 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction g -Deviation Measures Definitions Characterizations & Representation Results Results for g -Deviation Measures Portfolio Optimisation g -Deviation Measures Definition We call a P ⊗ B ( R d ) ⊗ U -measurable function R d L 2 ( ν ( d x )) g : [0 , T ] × Ω × × → R + ˜ g ( t , ω, h , ˜ ( t , ω, h , h ) �− → h ) a driver function if for d P × d t a.e. ( ω, t ) ∈ Ω × [0 , T ]: h ) ∈ R d × L 2 ( ν ( d x )) g ( t , h , ˜ (i) (Positivity) For any ( h , ˜ h ) ≥ 0 with equality if and only if ( h , ˜ h ) = 0. (ii) (Lower semi-continuity) If h n → h , ˜ h n → ˜ h L 2 ( ν ( d x ))-a.e. then g ( t , h , ˜ h ) ≤ lim inf n g ( t , h n , ˜ h n ) . 9 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction g -Deviation Measures Definitions Characterizations & Representation Results Results for g -Deviation Measures Portfolio Optimisation Definition We call a driver function g convex if g ( t , h , ˜ h ) is convex in ( h , ˜ h ), d P × d t a.e.; positively homogeneous if g ( t , h , ˜ h ) is positively homogeneous in ( h , ˜ h ), i.e., for λ > 0, g ( t , λ h , λ ˜ h ) = λ g ( t , h , ˜ h ) , d P × d t a.e. and of linear growth if for some K > 0 we have d P × d t a.e. � h ) | 2 ≤ K 2 + K 2 | h | 2 + K 2 | g ( t , h , ˜ h ( x ) 2 ν ( d x ) . ˜ R k \{ 0 } 10 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction g -Deviation Measures Definitions Characterizations & Representation Results Results for g -Deviation Measures Portfolio Optimisation g -Deviation Measures Suppose that g is a convex and positively homogeneous driver function of linear growth. Let ( Y , Z , ˜ Z ) be the unique solution of the SDE with terminal condition 0 given in terms of the representing pair ( H X , ˜ H X ) of X by � t , ˜ Z t ( x ) ˜ ˜ d Y t = − g ( t , H X H X N ( d t × d x ) , t ∈ [0 , T ) , t ) d t + Z t d W t + R k \{ 0 } Y T = 0 . The g-deviation measure D g = ( D g t ) t ∈ [0 , T ] is equal to the collection D t : L 2 ( F T ) → L 2 + ( F t ), t ∈ [0 , T ], given by D g X ∈ L 2 ( F T ) . t ( X ) = Y t , 11 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction g -Deviation Measures Definitions Characterizations & Representation Results Results for g -Deviation Measures Portfolio Optimisation Remark: g -expectations We remark that there is very rich literature on g -expectations. g -expecations can be seen as generalized expectations and as special examples of coherent risk measures satisfying the tower property. A g -expectation ρ t ( X ) := E g t ( X ) := Y t for a terminal payoff X is defined as a the first component Y of a unique triple ( Y , Z , ˜ Z ) satisfying � d Y t = − g ( t , Z t , ˜ Z t ( x ) ˜ ˜ Z t ) d t + Z t d W t + N ( d t × d x ) , t ∈ [0 , T ) , R k \{ 0 } Y T = X . 12 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction g -Deviation Measures Definitions Characterizations & Representation Results Results for g -Deviation Measures Portfolio Optimisation A Large Class of Dynamic Deviation Measures Proposition Let g be a convex and positively homogeneous driver function of linear growth. (i) D g is a dynamic deviation measure. In particular, D g satisfies (D6). (ii) For given X ∈ L 2 ( F T ) , we have �� T � � D g s , ˜ g ( s , H X H X � t ( X ) = E s ) d s � F t , t ∈ [0 , T ] . � t 13 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
Introduction g -Deviation Measures Definitions Characterizations & Representation Results Results for g -Deviation Measures Portfolio Optimisation Properties of g -Deviation Measures Proposition Let g and ˜ g be driver functions of linear growth. (i) D g is convex if and only if g is convex. (ii) D g is positively homogeneous if and only if g is positively homogeneous. (iii) D g ≥ D ˜ g if and only if g ≥ ˜ g d P × d t a.e. 14 / 25 Mitja Stadje (Ulm University) Dynamic Deviation Measures
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