MONETARY UTILITY FUNCTIONS AND BSDE

Delbaen Freddy ¨rich ETH Zu Paris, March 30, 2007 http://www.math.ethz.ch/˜delbaen

(1) Origin: 1994 (Soc-Gen) questions asked by J.M. Eber on risk. ADEH look at this problem, an axiomatic approach, first presentations by Dave Heath in 1995. Alternatives proposed (Tailvar). (2) 1996 Bremen, 1997 Trento, Milano, Lausanne, Tokyo many talks on coherent risk measures, warning for the use of VaR, ... . (3) May 31, 2006: “Banque de France” officially warns for using VaR as the basic risk measure. Two reasons: the 10 day period (liquidity problems, complexity of the credit derivatives) and the size of the losses (not only P).

References (sorry for the ones forgotten) Aumann-Anscombe, Aumann, Yaari, Denneberg, GilboaSchmeidler, Huber, Maccheroni-Marinacci-Rustichini, ... Billera-Heath, Artzner-Ostroy, Aubin, ... H.U. Gerber (1985!!), S. Wang, Mary Hardy, ... F¨ ollmer, Schied, Maass, Kusuoka, Tsukahara, JouiniSchachermayer-Touzi, Dana, Carlier, .... Koopmans (1960 !!!), Kreps-Porteus, Epstein, Zin, Schneider, Duffie, ...

Artzner, D., Eber, Heath, Ku, Cheridito, Kupper, ... S. Wang, Rosazza-Gianin, Penner, Biagini, Fritelli, Scandolo, Riedel, S. Peng, Pardoux, Y. Hu, S.Tang, Chen, M. Dai, Zhou, Barrieu, El Karoui, Bion-Nadal, Kl¨ oppel, .... Uryasev-Rockafellar, ... Choquet (1955): capacity theory, envelopes of measures. Relation with convex analysis in infinite dimensional spaces and relation with Backward Stochastic Differential Equations and non-linear expectations.

Definition of Risk Measures Notation: (Ω, (Ft )0≤t≤T , P) a filtered probability space with the usual assumptions. L∞ (Ω, F, P) space of bounded random variables, L1 (Ω, F, P) space of integrable RV. Liabilities are with − sign!! Wealth is with + sign. Bankruptcy means “under zero”. Utility functions are defined on random variables, not on “lotteries”.

Definition. u: L∞ → R is called a monetary utility function if u(ξ + a) = u(ξ) + a for all a ∈ R. Definition. u: L∞ → R is called a (Fatou) monetary concave utility function if (1) u(ξ) ≥ 0 if ξ ≥ 0 (2) u is concave (3) u(ξ + a) = u(ξ) + a for all a ∈ R (4) Fatou property. If supn ξn ∞ < ∞, if ξn → ξ in probability, then u(ξ) ≥ lim sup u(ξn ).

A utility u is characterised by the acceptance set A = {ξ | u(ξ) ≥ 0} , u(ξ) = max{a ∈ R | ξ − a ∈ A}. In case u is a monetary utility function we define ρ(ξ) = −u(ξ) and call it a convex risk measure. It describes the amount of money to be added to become acceptable, i.e. to be in A. ρ(ξ + ρ(ξ)) = 0

and u(ξ − u(ξ)) = 0.

In economic terms: once the needed capital is added, the position becomes acceptable. In utility terms: only the set preferred to 0 is important. All other preferred sets are obtained by translation, nivelloids. Each utility function where the preferred sets are translates (along the diagonal) of each other can be represented by a “monetary” utility function. A lot of economists do not like monetary utility functions. The theory is disjoint from the von Neumann– Morgenstern theory.

P = {Q  P | Q is a probability}. c : P → R+ ∪ +∞ is a convex function, for each k ∈ R+ the set {Q | c(Q) ≤ k} is convex and closed. inf Q∈P c(Q) = 0, most of the time we will use c(P) = 0. Characterisation of such utility functions. For given u (Fatou) there is c as above so that u(ξ) = inf{EQ [ξ] + c(Q) | Q ∈ P}. Conversely such a function c defines a Fatou utility function.

Depending on c we get different examples, some of them easy to calculate some are more difficult. Essentially it becomes a linear programme in infinite dimensions. The proof is essentially the Hahn-Banach theorem together with the Krein-Smulian theorem (needed to get weak∗ closed sets in L∞ ). We also need that on bounded sets of L∞ the topology of convergence in measure is the Mackey topology, a result that goes back to Grothendieck and which is based on the characterisation of relatively weakly compact sets (in L1 ) as the uniformly integrable sets, the so-called Dunford-Pettis theorem.

u is positively homogeneous (coherent) if and only if there is a closed convex set S ⊂ P so that c(Q) = 0 on S and c(Q) = +∞ if Q ∈ / S. u(ξ) = inf{EQ [ξ] | Q ∈ S}. In continuous time this suggests ut (ξ) = essinf{EQ [ξ | Ft ] | Q ∈ S}.

We suppose P ∈ S and set S e = {Q ∈ S | Q ∼ P} We get good properties if S is m–stable (rectangular) 
Q ∈ S is identified with Zt = E dQ dP | Ft Definition. If Z, Z  ∈ S e if τ is a stopping time then L ∈ S where Lt = Zt if t < τ and Lt =

Zt
Zτ Z
τ

if t ≥ τ .

We take the case of d−dimensional Brownian Motion B, with the usual filtration. 0 ∈ C(t, ω) ⊂ Rd closed convex, C is predictable.   S e = E(q · B)∞ | E[E(q · B)∞ ] = 1; q ∈ C; 0

m–stable sets are always of this form.

∞

 |qu |2 du < ∞ .

ut (ξ) = essinf{EQ [ξ | Ft ] | Q ∈ S} m-stable is equivalent to time consistency: If t < s and us (ξ) ≤ us (η) then ut (ξ) ≤ ut (η). It is for finite horizon, also equivalent to a form of Bellman’s principle. And it is also equivalent to: for each Q ∈ S the process ut (ξ) is a Q−submartingale. This has a good economic interpretation.

Relation with BSDE Wiener space with d-dimensional Brownian Motion. Finite horizon. Consider a function g: Rd → R+ convex, non-negative, g(0) = 0. Suppose that g has at most quadratic growth g(x) ≤ k(1 + |x|2 ).

For ξ ∈ L∞ consider (remark Z ∈ Rd ) dYt = g(Zt ) − Zt dBt YT = ξ This equation has for bounded ξ a unique bounded solution Y . (Kobylanski) Define ut (ξ) = Yt . This is a time consistent utility function, not homogeneous but it is concave and monetary.

Dual representation 





T

 ∗

g (qu ) du | Ft

ut (ξ) = essinf EQ ξ + t

dQ = E(q · B)T | dP

g ∗ is the Legendre transform of g (superquadratic). This leads to the “weakly”-compact case for concave monetary utility functions.



The above expression almost describes the class of all monetary Fatou time consistent utility functions (in the Brownian case). (Forthcoming work of FD, Peng and Rosazza-Gianin). Moreover one can prove that there is Q0 ∈ P so that Q0 ∼ P and so that for all t:    T

ut (ξ) = EQ0 ξ +

g ∗ (qu ) du | Ft .

t

The uniqueness of such Q0 is related but not equivalent to the strict convexity of the function g ∗ .

What happens if g is not subquadratic? Are equivalent: (1) g is “subquadratic” g ∗ (x) ∗ (2) g is “superquadratic”, lim inf x→∞ |x|2 > 0


 T ∗ (3) the sets {Q | c(Q) = EQ 0 g (qu ) du ≤ k} are weakly compact in L1 . ∞ there is Q(∼ P) with u(ξ) = (4) for every ξ ∈ L 


T ∗ EQ ξ + 0 g (qu ) du (5) the BSDE has a (unique) bounded solution for every ξ ∈ L∞ (6) u is strictly monotone, i.e. ξ, η ∈ L∞ , ξ ≤ η, P[ξ < η] > 0 implies u(ξ) < u(η).

Example: dimension = 1. Put g(z) = |z|β with β > 2, (g(z)/(1 + z 2 ) is unbounded). Then there is ξ ∈ L∞ so that there is a bounded solution of dYt = g(Zt ) dt − Zt dBt ,

YT = ξ.

But there is also a set A, P[A] > 0 so that for every η = ξ + h where h ≥ 0, P[h > 0] > 0 and h = 0 outside A, there is no bounded solution with YT = η. In fact we get ut (ξ) = ut (η) for all t < T.

It can be shown that if g is superquadratic then: (1) if for ξ there is a bounded solution starting at Y0 , then for every x < Y0 there are infinitely many bounded solutions that start at x, (2) also there exists a ξ such that there are infinitely many bounded solutions starting at u0 (ξ), (3) every bounded solution Y satisfies Yt ≤ ut (ξ), (4) if ξ is minimal, i.e. η ≤ ξ and P[η < ξ] > 0 imply u0 (η) < u0 (ξ), then for ξ there is a bounded solution Y .

The related quasi-linear PDE has a solution. 1 ∂t u + ∂xx u − g(∂x u) = 0, 2

u(T, x) = j(x),

for bounded (smooth) j has a bounded solution. 1 ∂t u + ∂xx u − |∂x u|β = 0, 2

u(T, x) = j(x)

This shows a difference between the Markovian setting and the general setting for BSDE.