Exponentially concave functions and multiplicative cyclical monotonicity

Exponentially concave functions and

multiplicative cyclical monotonicity

W. Schachermayer

11th December, RICAM, Linz

Based on the paper S. Pal, T.L. Wong:

Exponential concavity

Fix a convex subsetU⊆Rn.TypicallyUwill equal the unit simplex ∆n={(p1, . . . ,pn)∈[0,1]n:

n X i=1

pi= 1} or a convex subset of ∆n_,e.g.int(∆n_).

Definition:

A functionϕ:U7→_Ris calledexponentially concaveif Φ = exp(ϕ) is a concave function.

An exponentially concave function is concave but not vice versa.

For example, considerU= (0,1).An affine functionϕ(x) =ax+bonUis concave but not exponentially concave. On the other hand, the functionϕ(x) = log(x) is the arch-example of an exponentially concave function onU.

Exponential concavity

Fix a convex subsetU⊆Rn.TypicallyUwill equal the unit simplex ∆n={(p1, . . . ,pn)∈[0,1]n:

n X i=1

pi= 1} or a convex subset of ∆n_,e.g.int(∆n_).

Definition:

A functionϕ:U7→_Ris calledexponentially concaveif Φ = exp(ϕ) is a concave function.

An exponentially concave function is concave but not vice versa.

For example, considerU= (0,1).An affine functionϕ(x) =ax+bonUis concave but not exponentially concave. On the other hand, the functionϕ(x) = log(x) is the arch-example of an exponentially concave function onU.

Exponential concavity

Fix a convex subsetU⊆Rn.TypicallyUwill equal the unit simplex ∆n={(p1, . . . ,pn)∈[0,1]n:

n X i=1

pi= 1} or a convex subset of ∆n_,e.g.int(∆n_).

Definition:

A functionϕ:U7→_Ris calledexponentially concaveif Φ = exp(ϕ) is a concave function.

An exponentially concave function is concave but not vice versa.

For example, considerU= (0,1).An affine functionϕ(x) =ax+bonUis concave but not exponentially concave. On the other hand, the functionϕ(x) = log(x) is the arch-example of an exponentially concave function onU.

Lemma:

LetU be a convex subset ofRn. A concave functionϕ:U7→Ris

exponentially concave iff for everyµ0, µ1∈U the function

f(t) =ϕ(tµ0_{+ (1−t)µ}1_{) satisfies}

f00(t)≤ −(f0(t))2,

for almost allt in [0,1].

Lemma:

LetU be a convex subset ofRn. A concave functionϕ:U7→Ris

exponentially concave iff for everyµ0, µ1∈U the function

f(t) =ϕ(tµ0_{+ (1−t)µ}1_{) satisfies}

f00(t)≤ −(f0(t))2,

for almost allt in [0,1].

Definition:

LetU⊆RnandT :U7→Rn a (possibly multi-valued) map (interpreted

as a transport map).

We callT multiplicatively cyclically monotoneif, for all

µ1_{, . . . , µ}m_{, µ}m+1 _=µ1_{∈U and all valuesT(µ}1_{), . . . ,T(µ}m_{) we have}

hT(µj_{), µ}j+1_−µj_{i>−1 and} m Y j=1 (1 +hT(µj), µj+1−µji)≥1, (1) or, equivalently n X j=1 log(1 +hT(µj), µj+1−µji)≥0. (2)

Recall thatT iscyclically monotoneif n

X

j=1

Theorem [PW14]:

LetU⊆_Rn_{be a convex set andϕ:U 7→}

Ra concave function. Thenϕ

is exponentially concave iff its super-differentialT :=∂ϕonU is multiplicatively cyclically monotone. In this case,ϕis unique up to an additive constant.

Sketch of proof

Assume thatϕis exponentially concave so that Φ(µ) = exp(ϕ(µ)) is a concave function onU. Denote byS(µ) the super-differential of Φ for which we have Φ(µj+1)≤Φ(µj) +hS(µj), µj+1−µji, or Φ(µj+1₎ Φ(µj₎ ≤1 + DS(µj) Φ(µj₎, µ j+1 −µjE.

Theorem [PW14]:

LetU⊆_Rn_{be a convex set andϕ:U 7→}

Ra concave function. Thenϕ

is exponentially concave iff its super-differentialT :=∂ϕonU is multiplicatively cyclically monotone. In this case,ϕis unique up to an additive constant.

Sketch of proof

Assume thatϕis exponentially concave so that Φ(µ) = exp(ϕ(µ)) is a concave function onU. Denote byS(µ) the super-differential of Φ for which we have Φ(µj+1)≤Φ(µj) +hS(µj), µj+1−µji, or Φ(µj+1₎ Φ(µj₎ ≤1 + DS(µj) Φ(µj₎, µ j+1 −µjE.

Sketch of proof contd.

Assuming that Φ is differentiable, the super-differentialS(µ) equals

∇Φ(µ) so thatT :=∇ϕ=∇(log(Φ) = ∇Φ_Φ = S Φ.Therefore Φ(µj+1) Φ(µj₎ ≤1 +hT(µ j_{), µ}j+1_−µj_i. Ifµ1_{, µ}2_{, . . . , µ}m_{, µ}m+1_=µ1_{is a roundtrip we have} 1 = Φ(µ m+1₎ Φ(µ1₎ ≤ m Y i=1 (1 +hT(µj), µj+1−µji).

Applications in Finance

Stochastic portfolio Theory

R. Fernholz (1999), Karatzas & Fernholz (2009),. . .

We considernstocks with relative market capitalization at timet

µ(t) = (µ1(t), . . . , µn(t))∈int(∆n).

Aportfolio is a map

π:int(∆n)7→∆n.

Interpretation:

The agent invests her wealthV(t) according toπ(µ(t)) during ]t,t+ 1].

We associate toπtheweights w(µ) = (π1(µ)

µ1

, . . . ,πn(µ) µn

Stochastic portfolio Theory

R. Fernholz (1999), Karatzas & Fernholz (2009),. . .

We considernstocks with relative market capitalization at timet

µ(t) = (µ1(t), . . . , µn(t))∈int(∆n).

Aportfolio is a map

π:int(∆n)7→∆n.

Interpretation:

The agent invests her wealthV(t) according toπ(µ(t)) during ]t,t+ 1].

We associate toπtheweights

w(µ) = (π1(µ)

µ1

, . . . ,πn(µ) µn

Example:

a) themarket portfolio:

π(µ) =µ w(µ) = (1, . . . ,1)

b) Theequal weight portfolio: π(µ) = (1 n, . . . , 1 n) w(µ) = 1 n( 1 µ1 , . . . , 1 µn ) c) Let 0<p<1 and π(µ) = µ p 1 Pn i=1µ p i , . . . , µ p n Pn i=1µ p i ,w(µ) = µ p−1 1 1 p Pn i=1µ p i , . . . , µ p−1 n 1 p Pn i=1µ p i

Example:

a) themarket portfolio:

π(µ) =µ w(µ) = (1, . . . ,1)

b) Theequal weight portfolio:

π(µ) = (1 n, . . . , 1 n) w(µ) = 1 n( 1 µ1 , . . . , 1 µn ) c) Let 0<p<1 and π(µ) = µ p 1 Pn i=1µ p i , . . . , µ p n Pn i=1µ p i ,w(µ) = µ p−1 1 1 p Pn i=1µ p i , . . . , µ p−1 n 1 p Pn i=1µ p i

Example:

a) themarket portfolio:

π(µ) =µ w(µ) = (1, . . . ,1)

b) Theequal weight portfolio:

π(µ) = (1 n, . . . , 1 n) w(µ) = 1 n( 1 µ1 , . . . , 1 µn ) c) Let 0<p<1 and π(µ) = µ p 1 Pn i=1µ p i , . . . , µ p n Pn i=1µ p i ,w(µ) = µ p−1 1 1 p Pn i=1µ p i , . . . , µ p−1 n 1 p Pn i=1µ p i

Question:

Can you beat themarket portfolio?

Given the portfolio mapπ:int(∆n_)7→∆n _{and a sequence (µ(t))}m t=1, we

obtain for therelative wealth(in terms of the market portfolio) V(t+ 1) V(t) = n X i=1 πi(µ(t)) µi(t+ 1) µi(t) = n X i=1 wi(µ(t)) µi(t+ 1). =hw(µ(t)), µ(t+ 1)i = 1 +hw(µ(t)), µ(t+ 1)−µ(t)i

Question:

Can you beat themarket portfolio?

Given the portfolio mapπ:int(∆n_)7→∆n _{and a sequence (µ(t))}m t=1, we

obtain for therelative wealth(in terms of the market portfolio)

V(t+ 1) V(t) = n X i=1 πi(µ(t)) µi(t+ 1) µi(t) = n X i=1 wi(µ(t)) µi(t+ 1). =hw(µ(t)), µ(t+ 1)i = 1 +hw(µ(t)), µ(t+ 1)−µ(t)i

Question:

Can you beat themarket portfolio?

Given the portfolio mapπ:int(∆n_)7→∆n _{and a sequence (µ(t))}m t=1, we

obtain for therelative wealth(in terms of the market portfolio)

V(t+ 1) V(t) = n X i=1 πi(µ(t)) µi(t+ 1) µi(t) = n X i=1 wi(µ(t)) µi(t+ 1). =hw(µ(t)), µ(t+ 1)i = 1 +hw(µ(t)), µ(t+ 1)−µ(t)i

Suppose that the market makes a“round trip” µ1_{, µ}2_{, . . . , µ}m_{, µ}m+1_=µ1_. Then V(m+ 1) V(1) = m Y t=1 V(t+ 1) V(t) = m Y t=1 [1 +hw(µ(t)), µ(t+ 1)−µ(t)i] (4)

is precisely the term appearing in the definition ofmultiplicative cyclical monotonocity. Taking logarithms yields

logh m Y t=1 V(t+ 1) V(t) i = m X t=1 logh1 +hw(µ(t)), µ(t+ 1)−µ(t)ii (5)

Note thatw ismultiplicatively cyclically monotoneiff (4) is always≥1 or, equivalently, (5) is always≥0.

Suppose that the market makes a“round trip” µ1_{, µ}2_{, . . . , µ}m_{, µ}m+1_=µ1_. Then V(m+ 1) V(1) = m Y t=1 V(t+ 1) V(t) = m Y t=1 [1 +hw(µ(t)), µ(t+ 1)−µ(t)i] (4)

is precisely the term appearing in the definition ofmultiplicative cyclical monotonocity. Taking logarithms yields

logh m Y t=1 V(t+ 1) V(t) i = m X t=1 logh1 +hw(µ(t)), µ(t+ 1)−µ(t)ii (5)

Note thatw ismultiplicatively cyclically monotoneiff (4) is always≥1 or, equivalently, (5) is always≥0.

Suppose that the market makes a“round trip” µ1_{, µ}2_{, . . . , µ}m_{, µ}m+1_=µ1_. Then V(m+ 1) V(1) = m Y t=1 V(t+ 1) V(t) = m Y t=1 [1 +hw(µ(t)), µ(t+ 1)−µ(t)i] (4)

is precisely the term appearing in the definition ofmultiplicative cyclical monotonocity. Taking logarithms yields

logh m Y t=1 V(t+ 1) V(t) i = m X t=1 logh1 +hw(µ(t)), µ(t+ 1)−µ(t)ii (5)

Note thatw ismultiplicatively cyclically monotoneiff (4) is always≥1 or, equivalently, (5) is always≥0.

Suppose that the market makes a“round trip” µ1_{, µ}2_{, . . . , µ}m_{, µ}m+1_=µ1_. Then V(m+ 1) V(1) = m Y t=1 V(t+ 1) V(t) = m Y t=1 [1 +hw(µ(t)), µ(t+ 1)−µ(t)i] (4)

is precisely the term appearing in the definition ofmultiplicative cyclical monotonocity. Taking logarithms yields

logh m Y t=1 V(t+ 1) V(t) i = m X t=1 logh1 +hw(µ(t)), µ(t+ 1)−µ(t)ii (5)

Note thatw ismultiplicatively cyclically monotoneiff (4) is always≥1 or, equivalently, (5) is always≥0.

Theorem:

((i)⇒(ii): Fernholz (1999), (ii)⇒(i): Pal, Wong (2014))

Fix a portfolio mapπ:int(∆n)7→∆nand letw(µ) = (π1(µ)_µ1 , . . . ,πn(µ)

µn ) be the weight function. TFAE

(i) There is an exponentially concave function ϕ:int(∆n)7→R such thatw(µ) is in the super-gradient ofϕ.

Theorem:

((i)⇒(ii): Fernholz (1999), (ii)⇒(i): Pal, Wong (2014))

Fix a portfolio mapπ:int(∆n)7→∆nand letw(µ) = (π1(µ)_µ1 , . . . ,πn(µ)

µn ) be the weight function. TFAE

(i) There is an exponentially concave function

ϕ:int(∆n)7→R

such thatw(µ) is in the super-gradient ofϕ.

Theorem:

((i)⇒(ii): Fernholz (1999), (ii)⇒(i): Pal, Wong (2014))

Fix a portfolio mapπ:int(∆n)7→∆nand letw(µ) = (π1(µ)_µ1 , . . . ,πn(µ)

µn ) be the weight function. TFAE

(i) There is an exponentially concave function

ϕ:int(∆n)7→R

such thatw(µ) is in the super-gradient ofϕ.

Mass Transport

LetPbe a Probability measure onint(∆n) (e.g. normalized Lebesgue)

and letQ be a probability measure onRn+.We interpretQ as a

probability measure on theweights w ∈Rn+which arenot yet normalized.

Note that forπ:int(∆n)7→∆nand

w(µ) =π1(µ) µ1 , . . . ,πn(µ) µn we have hµ,w(µ)i= 1 (6)

Hence: If we associate (via the desired mass transport)µ∈(∆n_, P) with w(µ)∈_Rn

Mass Transport

LetPbe a Probability measure onint(∆n) (e.g. normalized Lebesgue)

and letQ be a probability measure onRn+.We interpretQ as a

probability measure on theweights w ∈Rn+which arenot yet normalized.

Note that forπ:int(∆n)7→∆nand

w(µ) =π1(µ) µ1 , . . . ,πn(µ) µn we have hµ,w(µ)i= 1 (6)

Hence: If we associate (via the desired mass transport)µ∈(∆n_,

P) with w(µ)∈_Rn

Consider the cost function

c(µ,w) = log(hµ,wi) µ∈int(∆n),w ∈_Rn

+.

For the probabilitiesPonint(∆n) andQ onRn+we consider the optimal

transport problem

E[c(µ,w(µ))]7→min!

Where we optimize over all (non-normalized) functions

Normalization Lemma:

Leth:Rn+7→R+ be a function and

Sh:_Rn₊7→Rn+ Sh(y) =h(y)y

Givenw :int(∆n)7→Rn+ we definew

h_=Sh_◦w.

Thenw is an optimal transport for the pair (_P,Q) iffwh _{is an optimal} transport for the pair (P,S#h(Q)).

Proof.

EP[log(hµ,w

h_{(µ)i)] =}

EP[log(h(w(µ))hµ,w(µ)i]

Normalization Lemma:

Leth:Rn+7→R+ be a function and

Sh:_Rn₊7→Rn+ Sh(y) =h(y)y

Givenw :int(∆n)7→Rn+ we definew

h_=Sh_◦w.

Thenw is an optimal transport for the pair (_P,Q) iffwh _{is an optimal} transport for the pair (P,S#h(Q)).

Proof.

EP[log(hµ,w

h_{(µ)i)] =}

EP[log(h(w(µ))hµ,w(µ)i]

Theorem [PW14]:

Let_P(a.c. with respect to Lebesgue) onint(∆n_{) and Q on}

Rn+be as

above such that

EP[log(hµ,w(µ)i)]7→min!

has a finite value. Then there is an optimal transportw_b :int(∆n_)7→

Rn+.

Assuming (w.l.g.) thathµ,w_b(µ)i= 1,for allµ, the functionw is in the supergradient of anexponentiallyconcave functionϕand thereforew is