markov chains under nonlinear expectation

Markov chains under nonlinear expectation Max Nendel (joint work - PowerPoint PPT Presentation

Markov chains under nonlinear expectation Max Nendel (joint work with Robert Denk, Michael Kupper and Michael R ockner) Bielefeld University 06. 07. 2019 ISIPTA 2019 Ghent, Belgium Max Nendel (Bielefeld University) Markov chains under


  1. Markov chains under nonlinear expectation Max Nendel (joint work with Robert Denk, Michael Kupper and Michael R¨ ockner) Bielefeld University 06. 07. 2019 ISIPTA 2019 Ghent, Belgium Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 1 / 16

  2. Contents Things that are not on the poster 1 Things that are on the poster 2 Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 2 / 16

  3. Contents Things that are not on the poster 1 Things that are on the poster 2 Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 3 / 16

  4. Definition of nonlinear expectations Let (Ω , F ) be a measurable space. We denote by L ∞ (Ω , F ) the space of all bounded measurable random variables X : Ω → R . Throughout, let M ⊂ L ∞ (Ω , F ) be a subspace that contains the constant gambles. Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 4 / 16

  5. Definition of nonlinear expectations Let (Ω , F ) be a measurable space. We denote by L ∞ (Ω , F ) the space of all bounded measurable random variables X : Ω → R . Throughout, let M ⊂ L ∞ (Ω , F ) be a subspace that contains the constant gambles. Definition (Peng (2005)) A (nonlinear) pre-expectation E is a functional E : M → R with the following properties: Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 4 / 16

  6. Definition of nonlinear expectations Let (Ω , F ) be a measurable space. We denote by L ∞ (Ω , F ) the space of all bounded measurable random variables X : Ω → R . Throughout, let M ⊂ L ∞ (Ω , F ) be a subspace that contains the constant gambles. Definition (Peng (2005)) A (nonlinear) pre-expectation E is a functional E : M → R with the following properties: (i) Monotone: E ( X ) ≤ E ( Y ) for all X , Y ∈ M with X ≤ Y . Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 4 / 16

  7. Definition of nonlinear expectations Let (Ω , F ) be a measurable space. We denote by L ∞ (Ω , F ) the space of all bounded measurable random variables X : Ω → R . Throughout, let M ⊂ L ∞ (Ω , F ) be a subspace that contains the constant gambles. Definition (Peng (2005)) A (nonlinear) pre-expectation E is a functional E : M → R with the following properties: (i) Monotone: E ( X ) ≤ E ( Y ) for all X , Y ∈ M with X ≤ Y . (ii) Constant preserving: E ( m ) = m for all m ∈ R . Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 4 / 16

  8. Definition of nonlinear expectations Let (Ω , F ) be a measurable space. We denote by L ∞ (Ω , F ) the space of all bounded measurable random variables X : Ω → R . Throughout, let M ⊂ L ∞ (Ω , F ) be a subspace that contains the constant gambles. Definition (Peng (2005)) A (nonlinear) pre-expectation E is a functional E : M → R with the following properties: (i) Monotone: E ( X ) ≤ E ( Y ) for all X , Y ∈ M with X ≤ Y . (ii) Constant preserving: E ( m ) = m for all m ∈ R . If M = L ∞ (Ω , F ), we say that E is a (nonlinear) expectation . Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 4 / 16

  9. Definition of nonlinear expectations Let (Ω , F ) be a measurable space. We denote by L ∞ (Ω , F ) the space of all bounded measurable random variables X : Ω → R . Throughout, let M ⊂ L ∞ (Ω , F ) be a subspace that contains the constant gambles. Definition (Peng (2005)) A (nonlinear) pre-expectation E is a functional E : M → R with the following properties: (i) Monotone: E ( X ) ≤ E ( Y ) for all X , Y ∈ M with X ≤ Y . (ii) Constant preserving: E ( m ) = m for all m ∈ R . If M = L ∞ (Ω , F ), we say that E is a (nonlinear) expectation . Pre-expectations are closely related to (monetary) risk measures introduced by Artzner et al. (1999), Delbaen (2000, 2002), see also F¨ ollmer-Schied (2011) and upper/lower previsions by Walley (1991). Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 4 / 16

  10. Additional properties We are particularly interested in... Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 5 / 16

  11. Additional properties We are particularly interested in... ... sublinear (pre-)expectations, i.e. E ( λ X ) = λ E ( X ) and E ( X + Y ) ≤ E ( X ) + E ( Y ) for all X , Y ∈ M and λ > 0. In this case, ρ ( X ) := E ( − X ) defines a coherent risk measure. Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 5 / 16

  12. Additional properties We are particularly interested in... ... sublinear (pre-)expectations, i.e. E ( λ X ) = λ E ( X ) and E ( X + Y ) ≤ E ( X ) + E ( Y ) for all X , Y ∈ M and λ > 0. In this case, ρ ( X ) := E ( − X ) defines a coherent risk measure. ... convex (pre-)expectations, i.e. � � E λ X + (1 − λ ) Y ≤ λ E ( X ) + (1 − λ ) E ( Y ) for X , Y ∈ M and λ ∈ [0 , 1]. Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 5 / 16

  13. Additional properties We are particularly interested in... ... sublinear (pre-)expectations, i.e. E ( λ X ) = λ E ( X ) and E ( X + Y ) ≤ E ( X ) + E ( Y ) for all X , Y ∈ M and λ > 0. In this case, ρ ( X ) := E ( − X ) defines a coherent risk measure. ... convex (pre-)expectations, i.e. � � E λ X + (1 − λ ) Y ≤ λ E ( X ) + (1 − λ ) E ( Y ) for X , Y ∈ M and λ ∈ [0 , 1]. ...(pre-)expectations that are continuous from above or below , i.e. E ( X n ) ց E ( X ) or E ( X n ) ր E ( X ) for ( X n ) n ∈ N ⊂ M with X n ց X ∈ M or X n ր X ∈ M , respectively. Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 5 / 16

  14. What do nonlinear expectations look like? Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 6 / 16

  15. What do nonlinear expectations look like? � If E is a linear expectation, then E ( X ) = X d P =: E P ( X ), where P is a finitely additive probability measure. Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 6 / 16

  16. What do nonlinear expectations look like? � If E is a linear expectation, then E ( X ) = X d P =: E P ( X ), where P is a finitely additive probability measure. If E is a sublinear expectation, then, E ( X ) = sup E P ( X ) , P ∈P where P is nonempty a set of finitely additive probability measures. Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 6 / 16

  17. What do nonlinear expectations look like? � If E is a linear expectation, then E ( X ) = X d P =: E P ( X ), where P is a finitely additive probability measure. If E is a sublinear expectation, then, E ( X ) = sup E P ( X ) , P ∈P where P is nonempty a set of finitely additive probability measures. If E is a sublinear expectation, then, E ( X ) = sup E P ( X ) − α P , P ∈P where P is a nonempty set of finitely additive probability measures and α P ≥ 0 is a penalization for the model P . Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 6 / 16

  18. What do nonlinear expectations look like? � If E is a linear expectation, then E ( X ) = X d P =: E P ( X ), where P is a finitely additive probability measure. If E is a sublinear expectation, then, E ( X ) = sup E P ( X ) , P ∈P where P is nonempty a set of finitely additive probability measures. If E is a sublinear expectation, then, E ( X ) = sup E P ( X ) − α P , P ∈P where P is a nonempty set of finitely additive probability measures and α P ≥ 0 is a penalization for the model P . If E is, additionally, continuous from above, the set P contains only countably additive probability measures. Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 6 / 16

  19. Extension of pre-expectations (Denk-Kupper-N. (2018)) We consider two extension procedures for pre-expectations to an expectation: Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 7 / 16

  20. Extension of pre-expectations (Denk-Kupper-N. (2018)) We consider two extension procedures for pre-expectations to an expectation: 1) Extension without continuity assumptions: For X ∈ L ∞ (Ω , F ), let ˆ � Y ∈ M , Y ≥ X � � � E ( X ) := inf E ( Y ) . ◮ Inspired by Kantorovich’s extension of positive linear functionals, ◮ Closely linked to the idea of superhedging ( ≈ NFL), ◮ Preserves convexity and sublinearity, ◮ The maximal extension and representation in terms of finitely additive measures. Max Nendel (Bielefeld University) Markov chains under nonlinear expectation 06. 07. 2019 7 / 16

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