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¨

• ckner)

Bielefeld University

• 06. 07. 2019

ISIPTA 2019 Ghent, Belgium

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Contents

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Things that are not on the poster

2

Things that are on the poster

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Contents

1

Things that are not on the poster

2

Things that are on the poster

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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.

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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

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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

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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.

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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.

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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¨

• llmer-Schied (2011) and

upper/lower previsions by Walley (1991).

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Additional properties

We are particularly interested in...

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Additional properties

We are particularly interested in... ...sublinear (pre-)expectations, i.e. E(λX) = λE(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.

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Additional properties

We are particularly interested in... ...sublinear (pre-)expectations, i.e. E(λX) = λE(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

• λX + (1 − λ)Y
• ≤ λE(X) + (1 − λ)E(Y )

for X, Y ∈ M and λ ∈ [0, 1].

Max Nendel (Bielefeld University) Markov chains under nonlinear expectation

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Additional properties

We are particularly interested in... ...sublinear (pre-)expectations, i.e. E(λX) = λE(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

• λX + (1 − λ)Y
• ≤ λE(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(Xn) ց E(X)

• r

E(Xn) ր E(X) for (Xn)n∈N ⊂ M with Xn ց X ∈ M or Xn ր X ∈ M, respectively.

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What do nonlinear expectations look like?

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What do nonlinear expectations look like?

If E is a linear expectation, then E(X) =

• X dP =: EP(X), where P is a

finitely additive probability measure.

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What do nonlinear expectations look like?

If E is a linear expectation, then E(X) =

• X dP =: EP(X), where P is a

finitely additive probability measure. If E is a sublinear expectation, then, E(X) = sup

P∈P

EP(X), where P is nonempty a set of finitely additive probability measures.

Max Nendel (Bielefeld University) Markov chains under nonlinear expectation

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What do nonlinear expectations look like?

If E is a linear expectation, then E(X) =

• X dP =: EP(X), where P is a

finitely additive probability measure. If E is a sublinear expectation, then, E(X) = sup

P∈P

EP(X), where P is nonempty a set of finitely additive probability measures. If E is a sublinear expectation, then, E(X) = sup

P∈P

EP(X) − αP, where P is a nonempty set of finitely additive probability measures and αP ≥ 0 is a penalization for the model P.

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What do nonlinear expectations look like?

If E is a linear expectation, then E(X) =

• X dP =: EP(X), where P is a

finitely additive probability measure. If E is a sublinear expectation, then, E(X) = sup

P∈P

EP(X), where P is nonempty a set of finitely additive probability measures. If E is a sublinear expectation, then, E(X) = sup

P∈P

EP(X) − αP, where P is a nonempty set of finitely additive probability measures and αP ≥ 0 is a penalization for the model P. If E is, additionally, continuous from above, the set P contains only countably additive probability measures.

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Extension of pre-expectations (Denk-Kupper-N. (2018))

We consider two extension procedures for pre-expectations to an expectation:

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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 E(X) := inf

• E(Y )
• Y ∈ M, Y ≥ X
• .

◮ 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

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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 E(X) := inf

• E(Y )
• Y ∈ M, Y ≥ X
• .

◮ 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.

2) Extension if E is continuous from above: For X ∈ L ∞(Ω, F), let E(X) := sup

• inf

n∈N E(Xn)

• (Xn)n∈N ⊂ M, Xn ≥ Xn+1, inf

n∈N Xn ≤ X

• .

◮ Inspired by Choquet’s theorem on capacitability and outer measures, ◮ Again, linked to superhedging (≈ NFLVR, Delbaen-Schachermayer (1994)), ◮ Preserves convexity and sublinearity, ◮ Uniqueness (in a certain sense) and representation in terms of countably

additive measures.

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Stochastic Processes under nonlinear expectations

The extension procedures from the last slide can be used to derive an imprecise version Kolmogorov’s theorem on the existence of stochastic processes. This reduces the existence of Markov process to certain properties of a family of transition operators (so-called regular kernels) (Es,t)0≤s<t. The construction of the transition operators is inspired by Nisio (1976). Main examples are:

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Stochastic Processes under nonlinear expectations

The extension procedures from the last slide can be used to derive an imprecise version Kolmogorov’s theorem on the existence of stochastic processes. This reduces the existence of Markov process to certain properties of a family of transition operators (so-called regular kernels) (Es,t)0≤s<t. The construction of the transition operators is inspired by Nisio (1976). Main examples are: A Brownian Motion with imprecise drift µ ∈ [µ, µ] (≈ BSDEs, El Karoui- Peng-Quenez (1997), Coquet et al. (2002)) A Brownian Motion with imprecise volatility σ ∈ [σ, σ] (≈ 2BSDEs, Peng (2007, 2008), Denis-Hu-Peng (2011), Soner-Touzi-Zhang (2011a, 2011b)) L´ evy Processes with imprecise L´ evy triplet (Hu-Peng (2009), Neufeld-Nutz (2014), Hollender (2016), K¨ uhn (2018), Denk-Kupper-N. (2017)) Discrete/Continuous-time Markov chains (Hartfiel (1998), De Cooman- Hermans-Quaeghebeur (2009), ˇ Skulj (2009, 2015), Krak-De Bock-Siebes (2017), N. (2018)) Imprecise Ornstein-Uhlenbeck processes and Geometric Brownian Motions (Epstein-Ji (2013), Vorbrink (2014), R¨

• ckner-N. (2019))

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Contents

1

Things that are not on the poster

2

Things that are on the poster

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Q-matrices

Definition

A matrix q = (qij)1≤i,j≤d ∈ Rd×d is called a Q-matrix if it satisfies the following: (i) qii ≤ 0 for all i ∈ {1, . . . , d}, (ii) qij ≥ 0 for all i, j ∈ {1, . . . , d} with i = j, (iii) d

j=1 qij = 0 for all i ∈ {1, . . . , d}.

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Q-matrices

Definition

A matrix q = (qij)1≤i,j≤d ∈ Rd×d is called a Q-matrix if it satisfies the following: (i) qii ≤ 0 for all i ∈ {1, . . . , d}, (ii) qij ≥ 0 for all i, j ∈ {1, . . . , d} with i = j, (iii) d

j=1 qij = 0 for all i ∈ {1, . . . , d}.

We say that a (possibly nonlinear) map Q: Rd → Rd satisfies the positive maximum principle (PMP) if for v = (v1, . . . , vd)T ∈ Rd and i ∈ {1, . . . , d} the following implication holds: vi = max

j=1,...,d vj

= ⇒ (Qv)i ≤ 0

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Q-matrices

Definition

A matrix q = (qij)1≤i,j≤d ∈ Rd×d is called a Q-matrix if it satisfies the following: (i) qii ≤ 0 for all i ∈ {1, . . . , d}, (ii) qij ≥ 0 for all i, j ∈ {1, . . . , d} with i = j, (iii) d

j=1 qij = 0 for all i ∈ {1, . . . , d}.

We say that a (possibly nonlinear) map Q: Rd → Rd satisfies the positive maximum principle (PMP) if for v = (v1, . . . , vd)T ∈ Rd and i ∈ {1, . . . , d} the following implication holds: vi = max

j=1,...,d vj

= ⇒ (Qv)i ≤ 0 − → q ∈ Rd×d is a Q-matrix if and only if it satisfies the PMP and 1 ∈ ker q.

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Examples for Q-matrices

Examples for Q-matrices are:

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Examples for Q-matrices

Examples for Q-matrices are: d = 3 : q = µ   −1 1 −1 1 1 −1   ← → µ∂x

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Examples for Q-matrices

Examples for Q-matrices are: d = 3 : q = µ   −1 1 −1 1 1 −1   ← → µ∂x d = 3 : q = σ2

2

  −2 1 1 1 −2 1 1 1 −2   ← →

σ2 2 ∂xx

Max Nendel (Bielefeld University) Markov chains under nonlinear expectation

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Examples for Q-matrices

Examples for Q-matrices are: d = 3 : q = µ   −1 1 −1 1 1 −1   ← → µ∂x d = 3 : q = σ2

2

  −2 1 1 1 −2 1 1 1 −2   ← →

σ2 2 ∂xx

d = 4 : q = σ2

2

    −2 1 1 1 −2 1 1 −2 1 1 1 −2     ← →

σ2 2 ∂xx

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Overview

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Q-operators: A generalization of Q-matrices

We now want to generalize the concept of a Q-matrix to a nonlinear setup.

Definition (Just for the sublinear case)

A (possibly nonlinear) map Q: Rd → Rd is called a Q-operator if the following conditions are satisfied: (i)

• Q(ei)
• i ≤ 0 for all i ∈ {1, . . . , d},

(ii)

• Q(−ei)
• j ≤ 0 for all i, j ∈ {1, . . . , d} with i = j,

(iii) Q(m1) = 0 for all m ∈ R, where 1 := (1, . . . , 1)T ∈ Rd.

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Q-operators: A generalization of Q-matrices

We now want to generalize the concept of a Q-matrix to a nonlinear setup.

Definition (Just for the sublinear case)

A (possibly nonlinear) map Q: Rd → Rd is called a Q-operator if the following conditions are satisfied: (i)

• Q(ei)
• i ≤ 0 for all i ∈ {1, . . . , d},

(ii)

• Q(−ei)
• j ≤ 0 for all i, j ∈ {1, . . . , d} with i = j,

(iii) Q(m1) = 0 for all m ∈ R, where 1 := (1, . . . , 1)T ∈ Rd. − → One immediately sees that a linear Q-operator is a Q-matrix and vice versa.

Max Nendel (Bielefeld University) Markov chains under nonlinear expectation

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Q-operators: A generalization of Q-matrices

We now want to generalize the concept of a Q-matrix to a nonlinear setup.

Definition (Just for the sublinear case)

A (possibly nonlinear) map Q: Rd → Rd is called a Q-operator if the following conditions are satisfied: (i)

• Q(ei)
• i ≤ 0 for all i ∈ {1, . . . , d},

(ii)

• Q(−ei)
• j ≤ 0 for all i, j ∈ {1, . . . , d} with i = j,

(iii) Q(m1) = 0 for all m ∈ R, where 1 := (1, . . . , 1)T ∈ Rd. − → One immediately sees that a linear Q-operator is a Q-matrix and vice versa. Example: For d = 3 and 0 < σ ≤ σ we consider the mapping Qv := sup

σ∈[σ,σ]

σ2 2   −2 1 1 1 −2 1 1 1 −2   v ≈ sup

σ∈[σ,σ]

σ2 2 ∂xxv

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Theorem (Main result, the sublinear case)

Let Q: Rd → Rd be a mapping. Then the following statements are equivalent: (i) Q is a sublinear Q-operator. (ii) Q is sublinear, satisfies the PMP and Q(m1) = 0 for all m ∈ R. (iii) There exists a set P ⊂ Rd×d of Q-matrices such that Qu0 = sup

q∈P

qu0 for all u0 ∈ Rd. (iv) There exists a sublinear Markov semigroup

• S(t)
• t≥0 such that

u(t) := S(t)u0 defines the unique solution u ∈ C 1 [0, ∞); Rd to the initial value problem u′(t) = Qu(t) for all t ≥ 0, u(0) = u0. (ODE) (v) There exists a sublinear Markov chain

• Ω, F, E, (Xt)t≥0
• such that

Qu0 = lim

hց0

E

• u0(Xh)
• − u0

h for all u0 ∈ Rd.

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Some remarks and conclusions

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Some remarks and conclusions

The previous theorem gives an axiomatization of the generators of sublinear Markov chains.

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Some remarks and conclusions

The previous theorem gives an axiomatization of the generators of sublinear Markov chains. The transition operators are given by S(t)u0 = E(u0(Xt)) and have an “explicit” primal and dual representation.

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Some remarks and conclusions

The previous theorem gives an axiomatization of the generators of sublinear Markov chains. The transition operators are given by S(t)u0 = E(u0(Xt)) and have an “explicit” primal and dual representation. Solutions to (ODE) remain bounded. Therefore, a Picard iteration can be used for numerical computations and the convergence rate (depending on the size of the initial value u0) can be explicitly computed. Other numerical methods such as Runge-Kutta methods can also be applied.

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Some remarks and conclusions

The previous theorem gives an axiomatization of the generators of sublinear Markov chains. The transition operators are given by S(t)u0 = E(u0(Xt)) and have an “explicit” primal and dual representation. Solutions to (ODE) remain bounded. Therefore, a Picard iteration can be used for numerical computations and the convergence rate (depending on the size of the initial value u0) can be explicitly computed. Other numerical methods such as Runge-Kutta methods can also be applied. By solving (ODE) (for example with Euler’s method), we can compute “prices” for European contingent claims of the form u(t) = E(u0(Xt)) under model uncertainty. More precisely, E(u0(Xt)) ≈

• I + t

nQ n u0.

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Thank you very much for your attention and see you at the poster! :-)

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