Introduction to sublinear expectation

1. Sublinear Expectation and Coherent Measures of Risk
Yusuke Kikuchi
University of California, Berkeley
July 2nd, 2018
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 1 / 14

2. Contents
1 Overview
2 Definition of Sublinear Expectation
3 Coherent measure of risk
4 Representation of a sublinear expectaion
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 2 / 14

3. Overview
Robust expectation
←→ Sublinear expectation
Coherent risk measure
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4. Setup
Let Ω, which is the scenario set, be a set. Let H, which is the set of
random variables, be a linear space of real-valued functions on Ω, which
satisfies
∀c ∈ R c ∈ H, X ∈ H ⇒ |X| ∈ H
We usually impose some additional conditions on H.
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5. Definition of sublinear expectation
Definition
E: H → R is a sublinear expectation on (Ω, H) if
1 (Monotonicity) X ≥ Y ⇒ E[X] ≥ E[Y ].
2 (Constant preserving) E[c] = c for all c ∈ R.
3 (Sub-additivity) E[X + Y ] ≤ E[X] + E[Y ].
4 (Positive homogeneity) E[λX] = λE[X] for all λ ≥ 0.
The triple (Ω, H, E) is called a sublinear expectation space.
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 5 / 14

6. Robust expectation
Suppose we have a coin and we know
p = P(H) ∈ [1/2, 1/3].
The coin is fliped once and the result is denoted by X. Assume we lose
ϕ(X) depending on the result. Then the robust expectation of loss is
E[ϕ(X)] := sup
p∈[1/2,1/3]
Ep[ϕ(X)] = sup
p∈[1/2,1/3]
{pϕ(H) + (1 − p)ϕ(T)}.
Ω = {H, T}, H = {ϕ | ϕ : Ω → R}.
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 6 / 14

7. Robust expectation
Generally, given a mesurable space (Ω, F) and a family of probablity
measures {Pθ : θ ∈ Θ}, then
E[X] := sup
θ∈Θ
Eθ[X]
is a sublinear expectation on (Ω, L0(Ω, F)).
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 7 / 14

8. An invest manager
An invest manager wants to measure the risk of traders’ positions. Let Ω
be the set of all scenarios and H be the set all of possible positions.
Assume the manager has an acceptance set A ⊂ H. Then one way to
measure the risk is:
ρA(X) := ρ(X) := inf{m | m + X ∈ A}, X ∈ H.
If A is coherent, then E[X] := ρ(−X) is a sublinear expectation on (Ω, H).
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 8 / 14

9. Coherent set
Definition
A is coherent if
1 (Monotonicity) X ≤ Y and X ∈ A ⇒ Y ∈ A.
2 0 ∈ A but not −1 ∈ A.
3 (Positive homogeneity) λX ∈ A for λ ≥ 0.
4 (Convexity) X, Y ∈ A ⇒ αX + (1 − α)Y ∈ A for 0 ≤ α ≤ 1.
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 9 / 14

10. Coherent risk measure
Proposition
If A is coherent, then ρA is also coherent i.e.
1 (Monotonicity) X ≥ Y ⇒ ρ(X) ≤ ρ(Y ).
2 (Constatnt preserving) ρ(c) = −c for all c ∈ R.
3 (Sub-additivity) ρ(X + y) ≤ ρ(X) + ρ(Y ).
4 (Positive homegeneity) ρ(λX) = λρ(X) for λ ≥ 0.
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 10 / 14

11. Correspondence results
For a risk measure ρ : H → R, define
Aρ = {X ∈ H | ρ(X) ≤ 0}.
Proposition
Aρ is coherent if and only if ρA is coherent.
Theorem
Given (Ω, H), there exists a one-to-one correspondence between sublinear
expectations and coherent risk measures. The correspondence is given by
E[X] = ρ(−X).
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12. Statement of the result
Theorem
Let E be a sublinear expectation on (Ω, H). Then, there exists a family of
linear expectations {Eθ | θ ∈ Θ} such that
E[X] = sup
θ∈Θ
Eθ[X].
Remark
Eθ is not necessarily σ-additive but finitely additive.
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13. Proof
Let
Q = {E | linear expectation, E[X] ≤ E[X] for all X ∈ H}.
We show Q is not empty. Fix X ∈ H and define L = {aX | a ∈ R},
I : L → R by
I(aX) = aE[X].
I is linear and I ≤ E on L. By Hahn-Banach theorem, there exists an
expansion E of I to H such that
E ≤ E, E|L = E|L .
It is easy to see E is a linear expectation. Therefore Q is not empty and
E[X] = sup
E∈Q
E[X].
Yusuke Kikuchi (UC Berkeley) Sublinear Expectation and Coherent Measures of Risk July 2nd, 2018 13 / 14

14. Reference
S. Peng(2010), Nonlinear Expectations and Stochastic Calculus under
Uncertainty, arXiv:1002.4546.
P. Artzner, F. Delbaen, J.-M. Eber and D. Heath(1999), Coherent
Measures of Risk, Mathematical Finance, Vol. 9, No.3, 203-228.
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