On the dynamic representation of some time-inconsistent ...

[Pages:21]On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration

Julio Backhoff Veraguas Vienna University of Technology

Based on joint work with Ludovic Tangpi (Vienna University) Workshop on Variational and Stochastic Analysis

Santiago 15/03/2017

Introduction Main results Wrap up

Plan

Introduction

Let (, F, P) a probability space. L the space of essentially bounded random variables Losses.

A functional : L R is a (monetary-convex) risk measure if

? (Monotonicity) X Y (X ) (Y ); ? (Convexity) (X + (1 - )Y ) (X ) + (1 - )(Y ); ? (Cash-invariance) (X + c) = (X ) + c, for c R.

Interpretation: diversification reduces risk of big losses; subtracting (X ) units of cash from X yields an "acceptable" (risk-less) position.

Examples:

(X ) = E[X ] , esssup(X ) , -1

1 1-

FX-1 (t )dt

,

log E[exp(X )] ,

...

Artzner, Delbaen, Eber, Heath, F?ollmer, Schied, Ru?schendorf, Ruszczyn?ski, Shapiro, Rockafellar, Uryasev, Acerbi, Weber, ....

Introduction

It may be desirable that the risk of X L depends only on its distribution; often is just a mathematical gadget!

? (Law-invariance) (X ) = (Y ) if Law(X ) = Law(Y ). .... Kusuoka, Schachermayer, Touzi, Jouini ....

A large family of particularly tractable law-invariant convex risk measures is given by the Optimized certainty equivalents (OCE):

Definition (Ben-Tal & Teboulle)

Let : R R convex and increasing. The OCE with loss function is (X ) = inf{ E[ (X - r )] + r : r R }.

Examples: Entropic case: ex , Monotone Mean-Variance: [x]2+, Conditional Value-at-Risk: [x]+ ....

Introduction

This is a static picture. For a dynamic version, let

(, FT , {Ft }Tt=0, P) be a filtered probability space.

A functional t : L(FT ) L(Ft ) is a conditional risk measure if for every L(Ft ) [0, 1], c L(Ft ):

? (Monotonicity) X Y (X ) (Y ); ? (Convexity) (X + (1 - )Y ) (X ) + (1 - )(Y ); ? (Cash-invariance) (X + c) = (X ) + c.

For a family {t }Tt=0 of such operators, one defines:

? (Time-consistency) t+h(X ) t+h(Y ) t (X ) t (Y ).

Equivalently,

t (t+h(X )) = t (X ).

.... Detlefsen, Scandolo, Cheridito, Kupper, Acciaio, Penner ....

Introduction

A crucial result of Kupper & Schachermayer:

The only families of time-consistent, law invariant convex risk measures: expected values, essential suprema, and entropic risk measures.

OCE risk measures are most often not time-consistent.

Time inconsistency in stochastic optimization: .... Zhou, Li, Ekeland, Lazrak, B?auerle, Ott, Shapiro, Pflug, Pichler, Chow, Tamar, Mannor, Pavone, Miller, Yang ....

Our Goal: to nevertheless understand the dynamic behaviour of OCEs.

From now on, the setting is: ? continuous time, t [0, T ]; ? Brownian filtration, Ft = (Ws : s t), where W is a B.M.

Introduction

The time-consistent case is well understood in this framework! .... Delbaen, Peng, Rosazza Gianin, Coquet, Hu, M?emin ....

There is a correspondence between time-consistent convex risk measures and certain Backward Stochastic Differential Equations (BSDE)1, in the

sense that Yt := t (X ) solves, along some process Z :

T

T

Yt = X + g (s, Zs )ds - Zs dWs ,

t

t

for suitable generator g .

? This provides a dynamic way of computing (X ) = 0(X ).

? If X is a "Markovian claim" on a diffusion process HJB equation.

We ask: For Markovian claims, and OCE risk measures, is there a HJB? Can this characterize (X )?.

1A.K.A. g -expectations or non-linear expectations.

Setup

We want to compute the risk of a claim written on a diffusion process2:

dYt = b(t, Yt )dt + (t, Yt )dWt .

We make standard Lipschitz and linear growth assumptions existence and uniqueness of strong solution.

We shall consider positions/claims written on Y , such as3

X = f (YT ) +

T 0

g (t,

Yt )

dt ;

we assume f , g bounded and "Lipschitz in space variable." Let be an OCE risk measure with reasonable loss function .

2W is a d-dimensional B.M. and Y is an m-dimensional process. 3one says X is a Markovian/static/additive function of Y .

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