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 nonlinear expectation 06. 07. 2019 1 / 16

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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