On the dual of the solvency cone

Transcription

1 On the dual of the solvency cone Andreas Löhne Friedrich-Schiller-Universität Jena Joint work with: Birgit Rudloff (WU Wien) Wien, April, 0

2 Simplest solvency cone example Exchange between: Currency : Nepalese Rupee Currency : Euro π = 0 π = 0 ( ) Rupee -portfolios: Euro ( ) 0 ( ) 0 ( ) 0 0 {( 0 ( ) K = cone 0 0 ), ( )} 0 price systems: ( ) ( ) 0 0 ( ) ( ) K+ = cone /0 /0 {( ), 0 ( )} 0

3 Solve a problem stated in Bouchard, B., Touzi, N. (000): Explicit solution to the multivariate super-replication problem under transaction costs, Ann. Appl. Probab. provide explicitly a generating family for the polar [or dual] cone [of K d for d > ]

4 Basic facts about transportation problem s > 0 s > 0 d = s > 0 d = s > 0 A = s > 0 Variables: x = x x x x x Neg. cost: c = c c c c c Supply s = s s s s s max c T x s.t. Ax = s, x 0

5 Dual transportation problem min s T y s.t. A T y c s > 0 s > 0 d = s > 0 d = s > 0 A = s > 0 y y + c y y + c y y + c y y + c y y + c c = 0 (primal problem = feasibility problem) y y y y y y y y y y s T y = 0

6 Modified transportation problem s > 0 s > 0 d = s > 0 d = s > 0 A = π π π π π s > 0 Variables: x = x x x x x Neg. cost: c = Supply s = s s s s s max c T x s.t. Ax = s, x 0

7 Modified transportation problem (dual) min s T y s.t. A T y c s > 0 s > 0 d = s > 0 d = s > 0 A = π π π π π s > 0 π y y π y y π y y π y y π y y s T y = 0

8 Definition (solvency cone) Let d {,,...}, V = {,..., d} and let Π = (π ij ) be a (d d)-matrix such that Sometimes, () and () is replaced by i V : π ii =, () i, j V : 0 < π ij, () i, j, k V : π ij π ik π kj, () i, j, k V : π ij < π ik π kj. () i, j V, k V \ {i, j} : π ij < π ik π kj. () The polyhedral convex cone K d := cone { π ij e i e j ij V V } is called solvency cone induced by Π.

9 The dual cone K + d := { y R d x K d : x T y 0 }... (positive) dual cone of K d Proposition. One has K + d = { y R d i, j V : π ij y i y j }. Proof: obvious Recall: K d := cone { π ij e i e j ij V V } Proposition. One has R d + \ {0} int K d and K + d \ {0} int Rd +. Proof: Follows from () to (), a separation argument is used. Proposition. One has K d R d + = {0}. Proof: Elementary.

10 Feasible tree solution V = {,..., d} (P, N)... bi-partition of V, i.e., = P V, N = V \ P G(P, N)... bi-partite digraph with arc set E = P N y R d is called generated by a tree T if T is a spanning tree of G(P, N) such that ij E(T ) P N : π ij y i = y j > 0. () y R d is called feasible with respect to (P, N) if ij P N : π ij y i y j > 0. (7) y is called feasible tree solution w.r.t (P, N) if both properties hold.

11 Feasible tree solution V = {,,,,,, 7}, P = {,,, }, N = {,, 7} y = T y = π y = π y = π π y = π π 7 π π 7 π y = π π 7 π y 7 = π π 7 π Tree solution: π ij y i = y j for ij E(T )

12 Feasible tree solution V = {,,,,,, 7}, P = {,,, }, N = {,, 7} y = T y = π y = π y = π π y = π π 7 π π 7 π y = π π 7 π y 7 = π π 7 π Feasibility: e.g. π 7 y y 7

13 Characterization of K + d Theorem. For y R d, the following statements are equivalent. (i) y is an extremal direction of K + d ; (ii) y is a feasible tree solution w.r.t. some bipartition (P, N) of V.

14 Questions: Existence of extremal directions/feasible tree solutions Construction of extremal directions/feasible tree solutions Structure of extremal directions/feasible tree solutions

15 Degree vectors P T N deg T (P ) = deg T (N) = 7

16 Degree vectors of spanning trees P N 7 c N P is called P -configuration if c i = d i P b N N is called N-configuration if N = {,,...} i N b i = d

17 Degree vectors of spanning trees P T N 7 c N P is called P -configuration if c i = d i P b N N is called N-configuration if N = {,,...} i N b i = d

18 Degree vectors of spanning trees P T N 7 c N P is called P -configuration if c i = d i P b N N is called N-configuration if N = {,,...} i N b i = d

19 Degree vectors of spanning trees P T N 7 c N P is called P -configuration if c i = d i P b N N is called N-configuration if N = {,,...} i N b i = d

20 Existence of feasible tree solutions Theorem. For every bi-partition (P, N) of V and every P -configuration c N P there exists a feasible tree solution y R d generated by a spanning tree T of the bi-partite graph G(P, N) with deg T (P ) = c. An analogous statement holds if an N-configuration is given.

21 Towards a proof of Theorem 7

22 Towards a proof of Theorem 7

23 Towards a proof of Theorem k = k arg max{y j /π j j N} 7

24 Towards a proof of Theorem 7 Is there an N-configuration b N N and a feasible tree solution y generated by T such that b = deg T (N) and c = deg T (P )?

25 Towards a proof of Theorem 7 7

26 Towards a proof of Theorem 7 7 k arg min{y i π ij i P }

27 Remaining question: Given a P -configuration c N P. Is there an N-configuration b N N and a feasible tree solution y generated by T such that b = deg T (N) and c = deg T (P )?

28 Towards a proof of Theorem T (H)... set of all spanning trees of a graph H Lemma. Let H = H(P, N) be a bi-partite graph. Then {deg T (P ) T T (H)} = {deg T (N) T T (H)}. P H N 7 Sang-Il Oum Postnikov 009 (about generalized permutohedra)

29 Toward a proof of Theorem For a feasible tree solution y, define subgraph H(y) of G = G(P, N) V (H(y)) := V (G), E(H(y)) := { ij P N π ij y i = y j } P(y) := {deg T (P ) T T (H(y))} N (y) := {deg T (N) T T (H(y))} Lemma. Let x, y be two feasible tree solutions such that x αy for all α > 0. Then P(x) P(y) = and N (x) N (y) =.

30 Illustration of Lemma and Lemma x H(x) y H(y) x y x y x y x y 7 x 7 7 y 7 x y π ij x i = x j, π ij x i > x j P(x) = N (x) = π ij y i = y j, π ij y i > y j P(y) = N (y) =,,

31 Consequences of Theorem and Corollary. Assume that also () holds. Let x, y be two feasible tree solutions with respect to bi-partitions (P x, N x ) and (P y, N y ) of V, respectively. Then (P x, N x ) (P y, N y ) implies x αy for all α > 0. Moreover, K d + has at least d extremal directions. Corollary. K + d has at most ( ) d d d p= p )( p extremal directions. Example. The upper bound in Corollary cannot be improved. Let the non-diagonal entries be pairwise different prime numbers such that ( min { πij ij V V, i j }) > max { πij ij V V, i j }

32 Example. d = 0, π i i =, π = 9, π =... π 0,9 = 7 9 > 7 = () K 0 + has exactly ( p= p )( p ) = extremal directions. P = {,, 7, 8, 9, 0, }, N = {,...,,,..., 0}. ( ) ( d p = 8 ) = 8 P -configurations for this bi-partition (p := P ). c = (,,,,,, ) T N P Algorithm (Matlab, about minutes): y = ( , , , 77 89, , , 89, 7 9 7, 7, , 7, 7 7, , 7 7, 7 7, 89 89, 89 89, , 7 09, ) T b = (,,,,,,,,,,,, ) T N N

33 Special case π ii := and π ij := a j /b i (i j), 0 < b i a i for all i V, 0 < b k < a k for at least one k V () to () Recursion formula ( ) a b Y = b a Direct description Y d = b a Y d. Y d. b d a d... a d a d b d... b d b d a d. K + d = cone { y R d (P, N) bi-part. of V, i P : y i = b i, j N : y j = a j } Consequence K + d has at most d extremal directions.

34 Special case π ii := and π ij := a j /b i (i j), 0 < b i < a i for all i V, } () to () The same as in special case, but now K + d has exactly d extremal directions.

35 Special case π ii := and π ij := a j /b i (i j), 0 < b i a i for all i V, 0 < b k < a k for at least one k V b k = a k for some k V () to () Recursion formula (w.l.o.g. a = b = ) Y = ( a b ) Y d = Y d Y d. a d... a d b d... b d Direct description K d + = cone { y R d } Q V \ {k}, i Q : y i = b i, j V \ Q : y j = a j. Consequence K + d has at most d extremal directions.

36 References Jewell, W. S. (9): Optimal flow through networks with gains. Operations Research, 0, 7-99 Kabanov, Y. M. (999): Hedging and liquidation under transaction costs in currency markets. Finance and Stochastics, 7-8 Postnikov, A (009): Permutohedra, associahedra, and beyond, Int. Math. Res. Not., 0-0 L., Rudloff, B. (0): On the dual of the solvency cone, Discrete Applied Mathematics 8, 7-8