Exponentially concave functions and multiplicative cyclical - PowerPoint PPT Presentation

Exponentially concave functions and multiplicative cyclical monotonicity W. Schachermayer University of Vienna Faculty of Mathematics 11th December, RICAM, Linz Based on the paper S. Pal, T.L. Wong: The Geometry of relative arbitrage. (ArXiv:1402.3720)

Exponential concavity Fix a convex subset U ⊆ R n . Typically U will equal the unit simplex n ∆ n = { ( p 1 , . . . , p n ) ∈ [0 , 1] n : � p i = 1 } i =1 or a convex subset of ∆ n , e.g. int (∆ n ) . Definition: A function ϕ : U �→ R is called exponentially concave if Φ = exp( ϕ ) is a concave function. An exponentially concave function is concave but not vice versa. For example, consider U = (0 , 1) . An affine function ϕ ( x ) = ax + b on U is concave but not exponentially concave. On the other hand, the function ϕ ( x ) = log( x ) is the arch-example of an exponentially concave function on U .

Exponential concavity Fix a convex subset U ⊆ R n . Typically U will equal the unit simplex n ∆ n = { ( p 1 , . . . , p n ) ∈ [0 , 1] n : � p i = 1 } i =1 or a convex subset of ∆ n , e.g. int (∆ n ) . Definition: A function ϕ : U �→ R is called exponentially concave if Φ = exp( ϕ ) is a concave function. An exponentially concave function is concave but not vice versa. For example, consider U = (0 , 1) . An affine function ϕ ( x ) = ax + b on U is concave but not exponentially concave. On the other hand, the function ϕ ( x ) = log( x ) is the arch-example of an exponentially concave function on U .

Exponential concavity Fix a convex subset U ⊆ R n . Typically U will equal the unit simplex n ∆ n = { ( p 1 , . . . , p n ) ∈ [0 , 1] n : � p i = 1 } i =1 or a convex subset of ∆ n , e.g. int (∆ n ) . Definition: A function ϕ : U �→ R is called exponentially concave if Φ = exp( ϕ ) is a concave function. An exponentially concave function is concave but not vice versa. For example, consider U = (0 , 1) . An affine function ϕ ( x ) = ax + b on U is concave but not exponentially concave. On the other hand, the function ϕ ( x ) = log( x ) is the arch-example of an exponentially concave function on U .

Lemma: Let U be a convex subset of R n . A concave function ϕ : U �→ R is exponentially concave iff for every µ 0 , µ 1 ∈ U the function f ( t ) = ϕ ( t µ 0 + (1 − t ) µ 1 ) satisfies f ′′ ( t ) ≤ − ( f ′ ( t )) 2 , for almost all t in [0 , 1] . Indeed, for F ( t ) = exp( f ( t )) we have F ′′ ( t ) = F ( t )[ f ′′ ( t ) + ( f ′ ( t )) 2 ] .

Lemma: Let U be a convex subset of R n . A concave function ϕ : U �→ R is exponentially concave iff for every µ 0 , µ 1 ∈ U the function f ( t ) = ϕ ( t µ 0 + (1 − t ) µ 1 ) satisfies f ′′ ( t ) ≤ − ( f ′ ( t )) 2 , for almost all t in [0 , 1] . Indeed, for F ( t ) = exp( f ( t )) we have F ′′ ( t ) = F ( t )[ f ′′ ( t ) + ( f ′ ( t )) 2 ] .

Definition: Let U ⊆ R n and T : U �→ R n a (possibly multi-valued) map (interpreted as a transport map). We call T multiplicatively cyclically monotone if, for all µ 1 , . . . , µ m , µ m +1 = µ 1 ∈ U and all values T ( µ 1 ) , . . . , T ( µ m ) we have � T ( µ j ) , µ j +1 − µ j � > − 1 and m � (1 + � T ( µ j ) , µ j +1 − µ j � ) ≥ 1 , (1) j =1 or, equivalently � n log(1 + � T ( µ j ) , µ j +1 − µ j � ) ≥ 0 . (2) j =1 Recall that T is cyclically monotone if n � � T ( µ j ) , µ j +1 − µ j � ≥ 0 . (3) j =1

Theorem [PW14]: Let U ⊆ R n be a convex set and ϕ : U �→ R a concave function. Then ϕ is exponentially concave iff its super-differential T := ∂ϕ on U is multiplicatively cyclically monotone. In this case, ϕ is unique up to an additive constant. Sketch of proof Assume that ϕ is exponentially concave so that Φ( µ ) = exp( ϕ ( µ )) is a concave function on U . Denote by S ( µ ) the super-differential of Φ for which we have Φ( µ j +1 ) ≤ Φ( µ j ) + � S ( µ j ) , µ j +1 − µ j � , or � S ( µ j ) Φ( µ j ) , µ j +1 − µ j � Φ( µ j +1 ) ≤ 1 + . Φ( µ j )

Theorem [PW14]: Let U ⊆ R n be a convex set and ϕ : U �→ R a concave function. Then ϕ is exponentially concave iff its super-differential T := ∂ϕ on U is multiplicatively cyclically monotone. In this case, ϕ is unique up to an additive constant. Sketch of proof Assume that ϕ is exponentially concave so that Φ( µ ) = exp( ϕ ( µ )) is a concave function on U . Denote by S ( µ ) the super-differential of Φ for which we have Φ( µ j +1 ) ≤ Φ( µ j ) + � S ( µ j ) , µ j +1 − µ j � , or � S ( µ j ) Φ( µ j ) , µ j +1 − µ j � Φ( µ j +1 ) ≤ 1 + . Φ( µ j )

Sketch of proof contd. Assuming that Φ is differentiable, the super-differential S ( µ ) equals ∇ Φ( µ ) so that T := ∇ ϕ = ∇ (log(Φ) = ∇ Φ Φ = S Φ . Therefore Φ( µ j +1 ) ≤ 1 + � T ( µ j ) , µ j +1 − µ j � . Φ( µ j ) If µ 1 , µ 2 , . . . , µ m , µ m +1 = µ 1 is a roundtrip we have � m 1 = Φ( µ m +1 ) (1 + � T ( µ j ) , µ j +1 − µ j � ) . ≤ Φ( µ 1 ) i =1 Hence T is multiplicatively cyclically monotone.

Applications in Finance Relative Capital distribution 1929-1999 of NY stock exchange:

Stochastic portfolio Theory R. Fernholz (1999), Karatzas & Fernholz (2009), . . . We consider n stocks with relative market capitalization at time t µ ( t ) = ( µ 1 ( t ) , . . . , µ n ( t )) ∈ int (∆ n ) . A portfolio is a map π : int (∆ n ) �→ ∆ n . Interpretation: The agent invests her wealth V ( t ) according to π ( µ ( t )) during ] t , t + 1] . We associate to π the weights w ( µ ) = ( π 1 ( µ ) , . . . , π n ( µ ) ) ∈ R n + . µ 1 µ n

Stochastic portfolio Theory R. Fernholz (1999), Karatzas & Fernholz (2009), . . . We consider n stocks with relative market capitalization at time t µ ( t ) = ( µ 1 ( t ) , . . . , µ n ( t )) ∈ int (∆ n ) . A portfolio is a map π : int (∆ n ) �→ ∆ n . Interpretation: The agent invests her wealth V ( t ) according to π ( µ ( t )) during ] t , t + 1] . We associate to π the weights w ( µ ) = ( π 1 ( µ ) , . . . , π n ( µ ) ) ∈ R n + . µ 1 µ n

Example: a) the market portfolio : π ( µ ) = µ w ( µ ) = (1 , . . . , 1) b) The equal weight portfolio : π ( µ ) = (1 n , . . . , 1 w ( µ ) = 1 n ( 1 , . . . , 1 n ) ) µ 1 µ n c) Let 0 < p < 1 and � � � � µ p − 1 µ p µ p µ p − 1 1 n 1 n � n � n π ( µ ) = , . . . , , w ( µ ) = � n , . . . , � n i =1 µ p i =1 µ p 1 i =1 µ p 1 i =1 µ p i i i i p p

Example: a) the market portfolio : π ( µ ) = µ w ( µ ) = (1 , . . . , 1) b) The equal weight portfolio : π ( µ ) = (1 n , . . . , 1 w ( µ ) = 1 n ( 1 , . . . , 1 n ) ) µ 1 µ n c) Let 0 < p < 1 and � � � � µ p − 1 µ p µ p µ p − 1 1 n 1 n � n � n π ( µ ) = , . . . , , w ( µ ) = � n , . . . , � n i =1 µ p i =1 µ p 1 i =1 µ p 1 i =1 µ p i i i i p p

Example: a) the market portfolio : π ( µ ) = µ w ( µ ) = (1 , . . . , 1) b) The equal weight portfolio : π ( µ ) = (1 n , . . . , 1 w ( µ ) = 1 n ( 1 , . . . , 1 n ) ) µ 1 µ n c) Let 0 < p < 1 and � � � � µ p − 1 µ p µ p µ p − 1 1 n 1 n � n � n π ( µ ) = , . . . , , w ( µ ) = � n , . . . , � n i =1 µ p i =1 µ p 1 i =1 µ p 1 i =1 µ p i i i i p p

Question: Can you beat the market portfolio ? Given the portfolio map π : int (∆ n ) �→ ∆ n and a sequence ( µ ( t )) m t =1 , we obtain for the relative wealth (in terms of the market portfolio) n � V ( t + 1) µ i ( t + 1) = π i ( µ ( t )) V ( t ) µ i ( t ) i =1 � n = w i ( µ ( t )) µ i ( t + 1) . i =1 = � w ( µ ( t )) , µ ( t + 1) � = 1 + � w ( µ ( t )) , µ ( t + 1) − µ ( t ) �

Question: Can you beat the market portfolio ? Given the portfolio map π : int (∆ n ) �→ ∆ n and a sequence ( µ ( t )) m t =1 , we obtain for the relative wealth (in terms of the market portfolio) n � V ( t + 1) µ i ( t + 1) = π i ( µ ( t )) V ( t ) µ i ( t ) i =1 � n = w i ( µ ( t )) µ i ( t + 1) . i =1 = � w ( µ ( t )) , µ ( t + 1) � = 1 + � w ( µ ( t )) , µ ( t + 1) − µ ( t ) �

Question: Can you beat the market portfolio ? Given the portfolio map π : int (∆ n ) �→ ∆ n and a sequence ( µ ( t )) m t =1 , we obtain for the relative wealth (in terms of the market portfolio) n � V ( t + 1) µ i ( t + 1) = π i ( µ ( t )) V ( t ) µ i ( t ) i =1 � n = w i ( µ ( t )) µ i ( t + 1) . i =1 = � w ( µ ( t )) , µ ( t + 1) � = 1 + � w ( µ ( t )) , µ ( t + 1) − µ ( t ) �

Suppose that the market makes a “round trip” µ 1 , µ 2 , . . . , µ m , µ m +1 = µ 1 . Then � m � m V ( m + 1) V ( t + 1) = = [1 + � w ( µ ( t )) , µ ( t + 1) − µ ( t ) � ] (4) V (1) V ( t ) t =1 t =1 is precisely the term appearing in the definition of multiplicative cyclical monotonocity . Taking logarithms yields � m � � � � � m V ( t + 1) log = log 1 + � w ( µ ( t )) , µ ( t + 1) − µ ( t ) � (5) V ( t ) t =1 t =1 Note that w is multiplicatively cyclically monotone iff (4) is always ≥ 1 or, equivalently, (5) is always ≥ 0 .