Mathemat i Kun Do Eko No Mie

8/13/2019 Mathemat i Kun Do Eko No Mie

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Mathematics and Economics

Frank Riedel

Institute for Mathematical EconomicsBielefeld University

Mathematics Colloquium Bielefeld, January 2011
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Three Leading Questions

Mathematics and Economics: Big Successes in History

Leon Walras,Elements deconomie politique pure 1874

Francis Edgeworth,Mathematical Psychics, 1881

John von Neumann, Oskar Morgenstern,

Theory of Games and Economic Behavior, 1944

Paul Samuelson,Foundations of Economic Analysis, 1947

Kenneth Arrow, Gerard Debreu,

Competitive Equilibrium 1954

John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory

Fischer Black, Myron Scholes, Robert Merton, 1973,

Mathematical Finance
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Competitive Equilibrium 1954

John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory


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Competitive Equilibrium 1954John Nash 1950, Reinhard Selten, 1965,Noncoperative Game Theory



Th L di Q i
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Th L di Q ti


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Fischer Black, Myron Scholes, Robert Merton, 1973,Mathematical Finance

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1 Rationality ?Isnt it simply wrong to impose heroic foresight and

intellectual abilities to describe humans?

2 Egoism ?Humans show altruism, envy, passions etc.

3 Probability ?Doesnt the crisis show that mathematics is useless, even

dangerous in markets?

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g Q







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g Q

Three Leading Questions: Details

Rationality ? Egoism ?These assumptions are frequently justified

Aufklarung! . . . answers Kants Was soll ich tun?

design of institutions: good regulation must be robust against

rational, egoistic agents (Basel II was not, e.g.)Doubts remain . . .; Poincare to Walras:

Par exemple, en mechanique, on neglige souvent lefrottement et on regarde les corps comme infiniment polis.Vous, vous regardez les hommes comme infiniment egoisteset infiniment clairvoyants. La premiere hypothese peut etreadmise dans une premiere approximation, mais la deuxiemenecessiterait peut-etre quelques reserves

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g







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Three Leading Doubts: Details ctd.

Probability ?

Option Pricing based on probability theory assumptions isextremelysuccessful

some blame mathematicians for financial crisis nonsense, butdoes probability theory apply to single events like

Greece is going bankrupt in 2012SF Giants win the World SeriesmedinForm in Bielefeld wird profitabel

Ellsberg experiments



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Probability ?







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Probability ?





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Probability ?




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Probability ?





Egoism
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Egoism

based on Dufwenberg, Heidhues, Kirchsteiger, R., Sobel, Review ofEconomic Studies 2010

Empirical Evidence

Humans react on their environment

relative concerns, in particular with peers, are important

especially in situations with few players

not in anonymous situations

FehrSchmidt Otherregarding preferences matter in games, but not inmarkets

Egoism
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Egoism


Empirical Evidence






Egoism
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Egoism


Empirical Evidence






Egoism
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Egoism


Empirical Evidence






Egoism
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Egoism


Empirical Evidence






Egoism
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Egoism


Empirical Evidence






Egoism

E i
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Egoism

General Equilibrium is the general theory of free, competitive markets withrational,selfinterestedagents

The Big Theorems

ExistenceFirst Welfare Theorem: Equilibrium Allocations are efficient

. . . in the core, even

Second Welfare Theorem: efficient allocations can be implemented

via free markets and lumpsum transfersCoreEquivalence: in large economies, the outcome of rationalcooperation (core) is close to market outcomes

Egoism

E i
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Egoism


The Big Theorems





Egoism

E i
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Egoism


The Big Theorems





Egoism

E i
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Egoism


The Big Theorems





Egoism

Eg is
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Egoism


The Big Theorems





Egoism

Egoism
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Egoism


The Big Theorems





Egoism

Egoism
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Egoism


The Big Theorems





Egoism

Simple yet Famous Models
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Simple, yet Famous Models

OtherRegarding Utility functions used to explain experimental data

FehrSchmidt (BoltonOckenfels) introduce fairness and envy:Ui=mi

iI1 kmax{(mkmi), 0}

iI1 kmax{(mimk), 0}

CharnessRabin: mi+ iI1

imin{m1, . . . , mI} + (1 i)

Ij=1mj

Edgeworth already has looked at mi +mj

Shaked: such ad hoc models are not science (and Poincare would

agree)

Egoism

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iI1 kmax{(mkmi), 0}

iI1 kmax{(mimk), 0}


imin{m1, . . . , mI} + (1 i)

Ij=1mj



agree)

Egoism

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iI1 kmax{(mkmi), 0}

iI1 kmax{(mimk), 0}


imin{m1, . . . , mI} + (1 i)

Ij=1mj



agree)

Egoism

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iI1 kmax{(mkmi), 0}

iI1 kmax{(mimk), 0}


imin{m1, . . . , mI} + (1 i)

Ij=1mj



agree)

Egoism

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iI1 kmax{(mkmi), 0}

iI1 kmax{(mimk), 0}


imin{m1, . . . , mI} + (1 i)

Ij=1mj



agree)

Egoism

Mathematical Formulation
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in anonymous situations, an agent cannot debate prices or influence

what others consumeown consumption x RL+, others consumption y R

K, pricesp RL+, income w>0

utilityu(x, y), strictly concave and smooth in x

when is the solution d(y, p, w) of

maximize u(x, y) subject to p x=w

independent ofy ?

Definition

We say that agent ibehaves asif selfishif her demand functiondi

p, w, xi

does not depend on others consumption plans xi.

Egoism

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independent ofy ?

Definition


p, w, xi


Egoism

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independent ofy ?

Definition


p, w, xi


Egoism



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independent ofy ?

Definition


p, w, xi


Egoism



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independent ofy ?

Definition


p, w, xi


Egoism

AsIf Selfish Demand Examples


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p

Clearly, standard egoistic utility functions vi(xi) =vi(xi1, . . . , viL)lead to as-if selfish behavior

Additive social preferences: let Ui(xi, xj) =vi(xi) +vj(xj). Then

marginal utilities are independent ofxj,Product Preferences:Ui(xi) =vi(xi)vj(xj) =vi(xi1, . . . , viL)vj(xj1, . . . , vjL)

marginal utilities do depend on others consumption bundlesbut marginal rates of substitution do not!

asif selfish behavior

Egoism



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Egoism



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Egoism



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Egoism



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Egoism



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Egoism

AsIf Selfish Preferences
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Theorem

Agent i behaves as if selfish if and only if her preferences can berepresented by a separable utility function

Vi(mi(xi), xi)

where mi :Xi R is theinternal utility function, continuous, strictly

monotone, strictly quasiconcave, and Vi :D R R(I1)L+ R is an

aggregator, increasing in own utility mi.

Technical AssumptionPreferences are smooth enough such that demand is continuouslydifferentiable. Needed for the only if.

Egoism

AsIf Selfish Preferences
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Theorem

Agent i behaves as if selfish if and only if her preferences can berepresented by a separable utility function

Vi(mi(xi), xi)

where mi :Xi R is theinternal utility function, continuous, strictly

monotone, strictly quasiconcave, and Vi :D R R(I1)L+ R is an

aggregator, increasing in own utility mi.

Technical AssumptionPreferences are smooth enough such that demand is continuouslydifferentiable. Needed for the only if.

Egoism

General Consequences
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Free Markets are a good institution in the sense that they maximizematerialefficiency (in terms ofmi(xi))

but not necessarily good as a social institution, i.e. in terms of realutilityui(xi, xi)

Egoism

General Consequences
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Free Markets are a good institution in the sense that they maximizematerialefficiency (in terms ofmi(xi))

but not necessarily good as a social institution, i.e. in terms of realutilityui(xi, xi)

Egoism

Inefficiency
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ExampleTake two agents, two commodities with the same internal utility andUi=m1+m2. Take as endowment an internally efficient allocation closeto the edge of the box. Unique Walrasian equilibrium, but not efficient, as

the rich agent would like to give endowment to the poor. Markets cannotmake gifts!

Remark

Public goods are a way to make gifts. Heidhues/R. have an example in

which the rich agent uses a public good to transfer utility to the pooragent (but still inefficient allocation).

Egoism

Inefficiency
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ExampleTake two agents, two commodities with the same internal utility andUi=m1+m2. Take as endowment an internally efficient allocation closeto the edge of the box. Unique Walrasian equilibrium, but not efficient, as

the rich agent would like to give endowment to the poor. Markets cannotmake gifts!

Remark

Public goods are a way to make gifts. Heidhues/R. have an example in

which the rich agent uses a public good to transfer utility to the pooragent (but still inefficient allocation).

Egoism

Is Charity Enough to Restore Efficiency?
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in example: charity would lead to efficiency

not true for more than 2 agents!Prisoners Dilemma

Egoism

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Egoism



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Egoism

Second Welfare Theorem
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Some (unplausible) preferences have to be ruled out:

Example

Hateful Society: Ui=mi 2mjfor two agents i=j. No consumption isefficient.

Social Monotonicity

For z RL+ \ {0} and any allocation x, there exists a redistribution (zi)with

zi =z such that

Ui(x+z)> Ui(x)

Egoism



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Some (unplausible) preferences have to be ruled out:

Example

Hateful Society: Ui=mi 2mjfor two agents i=j. No consumption isefficient.

Social Monotonicity

For z RL+ \ {0} and any allocation x, there exists a redistribution (zi)with

zi =z such that

Ui(x+z)> Ui(x)


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Egoism



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Theorem

Under social monotonicity, the set of Pareto optima is included in the setof internal Pareto optima.

Corollary

Second Welfare Theorem. The price system does not create inequality.

Uncertainty and Probability

Uncertainty Theory as new Probability Theory


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As P is not exactly known, work with a whole class of probability

measures P, (Huber, 1982, Robust Statistics)

Knightian Decision Making

GilboaSchmeidler: U(X) = minPPiEPu(x)

FollmerSchied, Maccheroni, Marinacci, Rustichinigeneralize tovariational preferences

U(X) = minP

EPu(X) +c(P)

for a cost function cdo not trust your model! be aware of sensitivities! do not believe inyour EXCEL sheet!




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U(X) = minP

EPu(X) +c(P)



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U(X) = minP

EPu(X) +c(P)



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U(X) = minP

EPu(X) +c(P)


Uncertainty and Probability Optimal Stopping

IMW Research on Optimal Stopping and KnightianUncertainty
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U y

Dynamic Coherent Risk Measures, Stochastic Processes and TheirApplications 2004

Optimal Stopping with Multiple Priors, Econometrica, 2009

Optimal Stopping under Ambiguity in Continuous Time (with XueCheng), IMW Working Paper 2010

The Best Choice Problem under Ambiguity (with TatjanaChudjakow), IMW Working Paper 2009

Chudjakow, Vorbrink, Exercise Strategies for American Exotic Optionsunder Ambiguity, IMW Working Paper 2009

Vorbrink, Financial Markets with Volatility Uncertainty, IMW WorkingPaper 2010

JanHenrik Steg, Irreversible Investment in Oligopoly, Finance andStochastics 2011


Optimal Stopping Problems: Classical Version
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Let

,F, P, (Ft)t=0,1,2,...

be a filtered probability space.

Given a sequence X0, X1, . . . , XTof random variables

adapted to the filtration (Ft)

choose a stopping time T

that maximizes EX.

classic: Snell, Chow/Robbins/Siegmund: Great Expectations


Optimal Stopping Problems: Solution, Discrete FiniteTime
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based on R., Econometrica 2009

Solution

Define the Snell envelope Uvia backward induction:

UT =XT

Ut= max {Xt,E [Ut+1|Ft]} (t


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Solution


UT =XT



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Solution


UT =XT



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Two classics

The Parking Problem


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The Parking Problem

You drive along a road towards a theatre

You want to park as close as possible to the theatre

Parking spaces are freeiid with probability p>0

When is the right time to stop? take the first free after 68%1/pdistance

Secretary Problem = When to Marry?

You see sequentially N applicants

maximize the probability to get the best one

rejected applicants do not come backapplicants come in random (uniform) order

optimal rule: take the first candidate (better than all previous) afterseeing 1/eof all applicants

probability of getting the best one approx 1 e Uncertainty and Probability Optimal Stopping

Two classics

The Parking Problem
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The Parking Problem










probability of getting the best one approx 1 e
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Two classics

The Parking Problem


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The Parking Problem











Two classics

The Parking Problem
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The Parking Problem











Two classics

The Parking Problem
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The Parking Problem











Two classics

The Parking Problem
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The Parking Problem











Two classics

The Parking Problem


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g











Two classics

The Parking Problem
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g











Two classics

The Parking Problem
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Two classics

The Parking Problem
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Two classics

The Parking Problem
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Optimal Stopping with Multiple Priors: Discrete Time
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We choose the following modeling approachLet X0, X1, . . . , XTbe a (finite) sequence of random variables

adapted to a filtration (Ft)

on a measurable space (,F),

let Pbe a set of probability measures


that maximizesinfPP

EPX


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that maximizesinfPP

EPX


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that maximizesinfPP

EPX


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that maximizesinfPP

EPX


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that maximizesinfPP

EPX
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Assumptions


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(Xt) bounded by a Puniformly integrable random variable

there exists a reference measure P0: all PPare equivalent to P0

(wlog,Tutsch, PhD 07)

agent knows all null sets,Epstein/Marinacci 07

Pweakly compact in L1

,F, P0

inf is always min,Follmer/Schied 04, Chateauneuf, Maccheroni, Marinacci, Tallon 05


Extending the General Theory to Multiple Priors
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Aims

Work as close as possible along the classical lines

Time ConsistencyMultiple Prior Martingale Theory

Backward Induction


Time Consistency

With general P, one runs easily into inconsistencies in dynamictti (S i /W kk )
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settings (Sarin/Wakker)

Time consistency law of iterated expectations:

minQP

EQ

ess infPP

EP [ X |Ft]

= min

PPEPX

Literature on time consistency in decision theory /risk measure theoryEpstein/Schneider, R. , Artzner et al., Detlefsen/Scandolo, Peng,Chen/Epsteintime consistency is equivalent to stability under pasting:

let P,QPand let (pt), (qt) be the density processesfix a stopping time

define a new measure Rvia setting

rt=

pt if tpqt/q else

then RPas well

Uncertainty and Probability Multiple Prior Martingale Theory

Multiple Prior Martingales

Definition
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Definition

An adapted, bounded process (St) is called a multiple priorsupermartingale iff

St ess infPP

EP [ St+1 |Ft]

holds true for all t 0.

multiple prior martingale: =multiple prior submartingale:

Remark

Nonlinear notion of martingales.Different from Pmartingale (martingale for all PPsimultaneously)



Definition
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Definition


St ess infPP

EP [ St+1 |Ft]



Remark




Definition
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Definition


St ess infPP

EP [ St+1 |Ft]



Remark




Definition
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St ess infPP

EP [ St+1 |Ft]



Remark



Characterization of Multiple Prior Martingales
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Theorem(St) is a multiple prior submartingale iff(St) is a Psubmartingale.

(St) is a multiple prior supermartingale iff there exists a PP suchthat(St) is a Psupermartingale.

(Mt) is a multiple prior martingale iff(Mt) is a Psubmartingale andfor some PPa Psupermartingale.

Remark

For multiple prior supermartingales: holds always true. needs

timeconsistency.


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Remark


timeconsistency.


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Remark


timeconsistency.


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Remark


timeconsistency.


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(Mt) is a multiple prior martingale iff(Mt) is aP

submartingale andfor some PPa Psupermartingale.

Remark


timeconsistency.


Doob Decomposition
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Theorem

Let(St) be a multiple prior supermartingale.Then there exists a multiple prior martingale M and a predictable,nondecreasing process A with A0 = 0such that S=M A. Such a

decomposition is unique.

Remark

Standard proof goes through.


Doob Decomposition
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Theorem

Let(St) be a multiple prior supermartingale.Then there exists a multiple prior martingale M and a predictable,nondecreasing process A with A0 = 0such that S=M A. Such a

decomposition is unique.

Remark

Standard proof goes through.


Optional Sampling Theorem
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Theorem

Let(St)0tTbe a multiple prior supermartingale. Let T bestopping times. Then

ess infPP EP

[S|F] S.

Remark

Not true without time consistency.


Optional Sampling Theorem
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Theorem

Let(St)0tTbe a multiple prior supermartingale. Let T bestopping times. Then

ess infPP EP

[S|F] S.

Remark

Not true without time consistency.

Uncertainty and Probability Optimal Stopping Rules

Optimal Stopping under Ambiguity
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With the concepts developed, one can proceed as in the classical case!Solution

Define the multiple priorSnell envelope Uvia backward induction:

UT =XT

Ut= max

Xt, ess inf

PPEP [Ut+1|Ft]

(t


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UT =XT

Ut= max

Xt, ess inf

PPEP [Ut+1|Ft]

(t


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UT =XT

Ut= max

Xt, ess inf

PPEP [Ut+1|Ft]

(t


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With the concepts developed, one can proceed as in the classical case!

Solution


UT =XT

Ut= max

Xt, ess inf

PPEP [Ut+1|Ft]

(t


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Question: what is the relation between the Snell envelopes U for fixedPPand the multiple prior Snell envelope U?

Theorem

U= ess infPP

UP .

Corollary

Under our assumptions, there exists a measure P P such thatU=UP

. The optimal stopping rule corresponds to the optimal stopping

rule under P.
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Monotonicity and Stochastic Dominance

P


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Suppose that (Yt) are iid under P P

andfor all PP

P[Yt x] P[Yt x] (x R)

and suppose that the payoffXt=g(t, Yt) for a function g that isisotone in y,

then P is for all optimal stopping problems(Xt) the worstcasemeasure,

i.e. the robust optimal stopping rule is the optimal stopping ruleunder P.



P
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andfor all PP







P
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andfor all PP







P
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andfor all PP







P
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Suppose that (Yt) are iid under P andfor all PP






Easy Examples
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Parking problem: choose the smallest pfor open lots

House sale: presume the least favorable distribution of bids in the

sense of firstorder stochastic dominanceAmerican Put: just presume the most positive possible drift

Uncertainty and Probability Continuous Time: gexpectations

Diffusion Models

based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P , (Ft)) with the usual conditions
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0

Typical Example: Ambiguous Drift t() [, ]

P= {P :Wis Brownian motion with drift t() [, ]}

(for timeconsistency: stochastic drift important!)

worst case: either + or, depending on the stateLet EtX= minPPE

P[X|Ft]

we have the representation

EtX = Z

tdt+Z

tdW

t

for some predictable process Z

Knightian expectations solve a backward stochastic differentialequation


Diffusion Models

based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P , (Ft)) with the usual conditions
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0





P[X|Ft]


EtX = Z

tdt+Z

tdW

t




Diffusion Models

based on Cheng, R., IMW Working Paper 429Framework now: Brownian motion Won a filtered probability space(,F, P0, (Ft)) with the usual conditions
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P[X|Ft]


EtX = Ztdt+ZtdWt




Diffusion Models

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P[X|Ft]


EtX = Ztdt+ZtdWt




Diffusion Models

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P[X|Ft]


EtX = Ztdt+ZtdWt




Diffusion Models

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P[X|Ft]


EtX = Ztdt+ZtdWt




Diffusion Models

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P[X|Ft]


EtX = Ztdt+ZtdWt




Timeconsistent multiple priors are gexpectations

gexpectations

F
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Conditional gexpectation of an FTmeasurable random variable Xat time t is Et(X) :=Yt

where (Y, Z) solves the backward stochastic differential equation

YT =X, ,dYt=g(t, Yt, Zt) ZtdWt

the probability theory for gexpectations has been mainly developedbyShige Peng

Theorem (Delbaen, Peng, Rosazza Giannin)

All timeconsistent multiple prior expectations are gexpectations.



gexpectations

C di i l i f F bl d i bl X
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gexpectations

C di i l i f F bl d i bl X
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gexpectations

C diti l t ti f F bl d i bl X
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gexpectations

Co ditio al g e ectatio of a F eas able a do a iable X
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gexpectations

Conditional g expectation of an F measurable random variable X
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Uncertainty and Probability Optimal Stopping under gexpectations: Theory

Optimal Stopping under gexpectations: Theory

Our Problem recall
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Our Problem - recall

Let (Xt) be continuous, adapted, nonnegative process withsuptT |Xt| L

2 (P0).Let g=g(, t, z) be a standard concave driver (in particular,

Lipschitzcontinuous).Find a stopping time T that maximizes

E0(X) .


Optimal Stopping under gexpectations: General Structure

LetVt= ess sup

t

Et(X) .

be the value function of our problem
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be the value function of our problem.

Theorem

1 (Vt) is the smallest rightcontinuous gsupermartingale dominating(Xt);

2 = inf{t 0 :Vt=Xt}is an optimal stopping time;3 the value function stopped at, (Vt) is a gmartingale.

Proof.

Our proof uses the properties ofgexpectations like regularity,timeconsistency, Fatou, etc. to mimic directly the classical proof (as, e.g.,inPeskir,Shiryaev) with one additional topping: rightcontinous versions ofgsupermartingales



LetVt= ess sup

t

Et(X) .

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Theorem



Proof.




LetVt= ess sup

t

Et(X) .

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Theorem



Proof.




LetVt= ess sup

t

Et(X) .

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p

Theorem



Proof.




LetVt= ess sup

t

Et(X) .

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p

Theorem



Proof.



WorstCase Priors

Drift ambiguity

V is a gsupermartingale

from the DoobMeyerPeng decomposition
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y g p

dVt=g(t, Zt)dt ZtdWt+dAt

for some increasing process A

= |Zt|dt ZtdWt+dAt

Girsanov: = ZtdWt +dAtwith kernel sgn(Zt)

Theorem (Duality for ambiguity)

There exists a probability measure P

P

such thatVt= ess suptEt(X) = ess suptE [X|Ft]. In particular:

max

minPP

EP [X|Ft] = minPP

max

EP [X|Ft]


WorstCase Priors

Drift ambiguity


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y g p



= |Zt|dt ZtdWt+dAt




P


max

minPP

EP [X|Ft] = minPP

max

EP [X|Ft]


WorstCase Priors

Drift ambiguity


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= |Zt|dt ZtdWt+dAt




P


max

minPP

EP [X|Ft] = minPP

max

EP [X|Ft]


WorstCase Priors

Drift ambiguity


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= |Zt|dt ZtdWt+dAt




P


max

minPP

EP [X|Ft] = minPP

max

EP [X|Ft]


WorstCase Priors

Drift ambiguity


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= |Zt|dt ZtdWt+dAt




P


max

minPP

EP [X|Ft] = minPP

max

EP [X|Ft]


WorstCase Priors

Drift ambiguity


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= |Zt|dt ZtdWt+dAtGirsanov: = ZtdW

t +dAtwith kernel sgn(Zt)



P


max

minPP

EP [X|Ft] = minPP

max

EP [X|Ft]

Uncertainty and Probability PDE Approach: Modified HamiltonJacobiBellman Equation

Markov Models

the state variable Ssolves a forward SDE, e.g.

dSt=(St)dt+ (St)dWt, S0 = 1 .

Let
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Let

L =(x)

x + 2(x)

2

x2

be the infinitesimal generator ofS.

By Itos formula, v(t, St) is a martingale if

vt(t, x) + Lv(t, x) = 0 (1)

similarly,v(t, St) is a gmartingale if

vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))= 0 (2)


Markov Models



Let
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Let

L =(x)

x + 2(x)

2

x2



vt(t, x) + Lv(t, x) = 0 (1)




Markov Models



Let
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Let

L =(x)

x + 2(x)

2

x2



vt(t, x) + Lv(t, x) = 0 (1)


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PDE Approach: A Modified HJB Equation

Theorem (Verification Theorem)

Let v be a viscosity solution of the gHJB equation


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y g q

max {f(x) v(t, x), vt(t, x) + Lv(t, x) +g(t, vx(t, x)(x))} = 0 . (3)

Then Vt=v(t, St).

nonlinearity only in the firstorder term

numeric analysis feasible

ambiguity introduces an additional nonlinear drift term




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Then Vt=v(t, St).







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Then Vt=v(t, St).







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Then Vt=v(t, St).




Uncertainty and Probability Examples

More general problems

With monotonicity and stochastic dominance, worstcase prior easy
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y p yto identify

In general, the worstcase prior is pathdependenteven in iid settings

Barrier OptionsShout Options

Secretary Problem



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to identify



Secretary Problem



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to identify



Secretary Problem



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to identify



Secretary Problem



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to identify



Secretary Problem

Uncertainty and Probability Secretary Problem

Ambiguous secretaries (with Tatjana Chudjakow)

based on Chudjakow, R., IMW Working Paper 413

Call applicant j a candidate if she is better than all predecessors
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Call applicant j a candidateif she is better than all predecessors

We are interested in Xj=Prob[jbest|jcandidate]

Here, the payoffXjis ambiguous assume that this conditional

probability is minimal

If you compare this probability with the probability that latercandidates are best, you presume the maximalprobability for them!




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Call applicant j a candidateif she is better than all predecessors

We are interested in Xj=Prob[jbest|jcandidate]

Here, the payoffXjis ambiguous assume that this conditional

probability is minimal

If you compare this probability with the probability that latercandidates are best, you presume the maximalprobability for them!




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Call applicant j a candidateif she is better than all prede

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