Market environments, stability and equlibria

Transcription

1 Market environments, stability and equlibria Gordan Žitković Department of Mathematics University of exas at Austin Austin, Aug 03, Summer School in Mathematical Finance

2 he Information Flow two states of the world: Ω = {ω 1, ω 2} one period t {0, 1} A oy Model nothing is known at t = 0, everything is known at t = 1: F 0 = {, Ω}, F 1 = 2 Ω. agents two economic agents characterized by random endowments (stochastic income) j ff E 1 3 =, E 2 = 1 j ff 1 4 utility functions U 1 ( U 2 ( j x1 x 2 j x1 x 2 ff ) = 1 2 log(x1) log(x2) ff ) = 1 7 x1/ x1/3 2

3 he Information Flow two states of the world: Ω = {ω 1, ω 2} one period t {0, 1} A oy Model nothing is known at t = 0, everything is known at t = 1: F 0 = {, Ω}, F 1 = 2 Ω. agents two economic agents characterized by random endowments (stochastic income) j ff E 1 3 =, E 2 = 1 j ff 1 4 utility functions U 1 ( U 2 ( j x1 x 2 j x1 x 2 ff ) = 1 2 log(x1) log(x2) ff ) = 1 7 x1/ x1/3 2

4 A toy example the financial instrument j ff 1 S 0 = p, S 1 =, B 0 0 = B 1 = 1: 10 p Market clearing he demand functions: i (p) = argmax U i (E i +q(s 1 p)) q Equilibrium conditions: 5 1 (p) + 2 (p) = 0

5 A toy example the financial instrument j ff 1 S 0 = p, S 1 =, B 0 0 = B 1 = 1: 10 p Market clearing he demand functions: i (p) = argmax U i (E i +q(s 1 p)) q Equilibrium conditions: 5 1 (p) + 2 (p) = 0

6 What happens when markets are incomplete and trading is dynamic? p 0 S 11 p 1 S 12 S 21 p 2 S 22 S 23 S 31 In the IC&mp case, the equilibrium conditions determine both prices and the geometry (degree of incompleteness) of the market. Another complication : no representative-agent analysis. he First Welfare heorem does not hold anymore. p 3 S 32 S 33 Instead of one price p, we need to determine the whole price process (p 0, (p 1, p 2, p 3)). C IC 1p * * mp * +

7 What happens when markets are incomplete and trading is dynamic? p 0 S 11 p 1 S 12 S 21 p 2 S 22 S 23 S 31 In the IC&mp case, the equilibrium conditions determine both prices and the geometry (degree of incompleteness) of the market. Another complication : no representative-agent analysis. he First Welfare heorem does not hold anymore. p 3 S 32 S 33 Instead of one price p, we need to determine the whole price process (p 0, (p 1, p 2, p 3)). C IC 1p * * mp * +

8 What happens when markets are incomplete and trading is dynamic? p 0 S 11 p 1 S 12 S 21 p 2 S 22 S 23 S 31 In the IC&mp case, the equilibrium conditions determine both prices and the geometry (degree of incompleteness) of the market. Another complication : no representative-agent analysis. he First Welfare heorem does not hold anymore. p 3 S 32 S 33 Instead of one price p, we need to determine the whole price process (p 0, (p 1, p 2, p 3)). C IC 1p * * mp * +

9 What happens when markets are incomplete and trading is dynamic? p 0 S 11 p 1 S 12 S 21 p 2 S 22 S 23 S 31 In the IC&mp case, the equilibrium conditions determine both prices and the geometry (degree of incompleteness) of the market. Another complication : no representative-agent analysis. he First Welfare heorem does not hold anymore. p 3 S 32 S 33 Instead of one price p, we need to determine the whole price process (p 0, (p 1, p 2, p 3)). C IC 1p * * mp * +

10 Financial frameworks Information A filtered probability space (Ω, F, {F t} t [0, ], P), where P is used only to determine the null-sets. Agents A number I (finite or infinite) of economic agents, each of which is characterized by a random endowment E i F, a utility function U : Dom(U) R, a subjective probability measure P i P. Completeness constraints A set S of {F t} t [0, ] -semimartingales (possibly several-dimensional) - the allowed asset-price dynamics.

11 Financial frameworks Information A filtered probability space (Ω, F, {F t} t [0, ], P), where P is used only to determine the null-sets. Agents A number I (finite or infinite) of economic agents, each of which is characterized by a random endowment E i F, a utility function U : Dom(U) R, a subjective probability measure P i P. Completeness constraints A set S of {F t} t [0, ] -semimartingales (possibly several-dimensional) - the allowed asset-price dynamics.

12 Financial frameworks Information A filtered probability space (Ω, F, {F t} t [0, ], P), where P is used only to determine the null-sets. Agents A number I (finite or infinite) of economic agents, each of which is characterized by a random endowment E i F, a utility function U : Dom(U) R, a subjective probability measure P i P. Completeness constraints A set S of {F t} t [0, ] -semimartingales (possibly several-dimensional) - the allowed asset-price dynamics.

13 he equilibrium problem Problem Does there exist S S such that X ˆπ t(s) i = 0, for all t [0, ], a.s, i I where Z ˆπ i (S) = argmax E Pi [U i (E i + π u ds u)] π 0 denotes the optimal trading strategy for the agent i when the market dynamics is given by S. Problem If such an S exists, is it unique? Problem If such an S exists, can we characterize it analytically or numerically?

14 he equilibrium problem Problem Does there exist S S such that X ˆπ t(s) i = 0, for all t [0, ], a.s, i I where Z ˆπ i (S) = argmax E Pi [U i (E i + π u ds u)] π 0 denotes the optimal trading strategy for the agent i when the market dynamics is given by S. Problem If such an S exists, is it unique? Problem If such an S exists, can we characterize it analytically or numerically?

15 he equilibrium problem Problem Does there exist S S such that X ˆπ t(s) i = 0, for all t [0, ], a.s, i I where Z ˆπ i (S) = argmax E Pi [U i (E i + π u ds u)] π 0 denotes the optimal trading strategy for the agent i when the market dynamics is given by S. Problem If such an S exists, is it unique? Problem If such an S exists, can we characterize it analytically or numerically?

16 Examples of Completeness Constraints Complete markets. S contains all {F t} t [0, ] -semimartingales. (If an equilibrium exists, a complete one will exist). Constraints on the number of assets. S is the set of all d-dimensional {F t} t [0, ] -semimartingales. If d < n, where n is the spanning number of the filtration, no complete markets are allowed. Information-constrained markets. Let {G t} t [0, ] be a sub-filtration of {F t} t [0, ], and let S be the class of all {G t} t [0, ] -semimartingales. Partial-equilibrium models. Let {S 0 t } t [0, ] be a d-dimensional semimartingale. S is the collection of all m-dimensional {F t} t [0, ] -semimartingales such that its first d < m components coincide with S 0. Marketed-Set Constrained markets Let V be a subspace of L (F ), satisfying an appropriate set of regularity conditions. Let S be the collection of all finite dimensional semimartingales {S t} t [0, ] such that Z {x + π t ds t : x R, π A} L (F ) = V, 0

17 Examples of Completeness Constraints Complete markets. S contains all {F t} t [0, ] -semimartingales. (If an equilibrium exists, a complete one will exist). Constraints on the number of assets. S is the set of all d-dimensional {F t} t [0, ] -semimartingales. If d < n, where n is the spanning number of the filtration, no complete markets are allowed. Information-constrained markets. Let {G t} t [0, ] be a sub-filtration of {F t} t [0, ], and let S be the class of all {G t} t [0, ] -semimartingales. Partial-equilibrium models. Let {S 0 t } t [0, ] be a d-dimensional semimartingale. S is the collection of all m-dimensional {F t} t [0, ] -semimartingales such that its first d < m components coincide with S 0. Marketed-Set Constrained markets Let V be a subspace of L (F ), satisfying an appropriate set of regularity conditions. Let S be the collection of all finite dimensional semimartingales {S t} t [0, ] such that Z {x + π t ds t : x R, π A} L (F ) = V, 0

18 Examples of Completeness Constraints Complete markets. S contains all {F t} t [0, ] -semimartingales. (If an equilibrium exists, a complete one will exist). Constraints on the number of assets. S is the set of all d-dimensional {F t} t [0, ] -semimartingales. If d < n, where n is the spanning number of the filtration, no complete markets are allowed. Information-constrained markets. Let {G t} t [0, ] be a sub-filtration of {F t} t [0, ], and let S be the class of all {G t} t [0, ] -semimartingales. Partial-equilibrium models. Let {S 0 t } t [0, ] be a d-dimensional semimartingale. S is the collection of all m-dimensional {F t} t [0, ] -semimartingales such that its first d < m components coincide with S 0. Marketed-Set Constrained markets Let V be a subspace of L (F ), satisfying an appropriate set of regularity conditions. Let S be the collection of all finite dimensional semimartingales {S t} t [0, ] such that Z {x + π t ds t : x R, π A} L (F ) = V, 0

19 Examples of Completeness Constraints Complete markets. S contains all {F t} t [0, ] -semimartingales. (If an equilibrium exists, a complete one will exist). Constraints on the number of assets. S is the set of all d-dimensional {F t} t [0, ] -semimartingales. If d < n, where n is the spanning number of the filtration, no complete markets are allowed. Information-constrained markets. Let {G t} t [0, ] be a sub-filtration of {F t} t [0, ], and let S be the class of all {G t} t [0, ] -semimartingales. Partial-equilibrium models. Let {S 0 t } t [0, ] be a d-dimensional semimartingale. S is the collection of all m-dimensional {F t} t [0, ] -semimartingales such that its first d < m components coincide with S 0. Marketed-Set Constrained markets Let V be a subspace of L (F ), satisfying an appropriate set of regularity conditions. Let S be the collection of all finite dimensional semimartingales {S t} t [0, ] such that Z {x + π t ds t : x R, π A} L (F ) = V, 0

20 Examples of Completeness Constraints Complete markets. S contains all {F t} t [0, ] -semimartingales. (If an equilibrium exists, a complete one will exist). Constraints on the number of assets. S is the set of all d-dimensional {F t} t [0, ] -semimartingales. If d < n, where n is the spanning number of the filtration, no complete markets are allowed. Information-constrained markets. Let {G t} t [0, ] be a sub-filtration of {F t} t [0, ], and let S be the class of all {G t} t [0, ] -semimartingales. Partial-equilibrium models. Let {S 0 t } t [0, ] be a d-dimensional semimartingale. S is the collection of all m-dimensional {F t} t [0, ] -semimartingales such that its first d < m components coincide with S 0. Marketed-Set Constrained markets Let V be a subspace of L (F ), satisfying an appropriate set of regularity conditions. Let S be the collection of all finite dimensional semimartingales {S t} t [0, ] such that Z {x + π t ds t : x R, π A} L (F ) = V, 0

21 Examples of Completenes Constraints Markets with fast-and-slow information. Let {F t} t [0, ] be generated by two orthogonal martingales M 1 and M 2, and let S be the collection of all processes of the form S t = D t + M 1 t, where D is any predictable process of finite variation. For example, M 1 = B (Brownian motion), M 2 = N t t (compensated Poisson process) so that a typical element of S is given by Z S t = λ(u, B u, N u) du + db u. he information in B is fast, and that in N is slow. Another interesting situation: M 1 = B, M 2 = W, where B and W independent Brownian motions.

22 Analysis wo paths to existence Representative agents. Uses the fact that equilibrium allocations are Pareto optimal; works (essentially) only for complete markets. Literature in continuous time: Complete markets: Bank, Dana, Duffie, Huang, Karatzas, Lakner, Lehoczky, Riedel, Shreve, Ž., etc. Incomplete markets: Basak and Cuoco 98 (incompleteness from restrictions in stock-market participation, logarithmic utility) Excess-demand approach. Introduced by Walras (1874): 1. Establish good topological/convexity properties of the excess demand ˆπ(S), and then 2. use a suitable fixed-point-type theorem to show existence (Brouwer, KKM, degree-based, etc.) Literature in continuous time: none, really!

23 Analysis wo paths to existence Representative agents. Uses the fact that equilibrium allocations are Pareto optimal; works (essentially) only for complete markets. Literature in continuous time: Complete markets: Bank, Dana, Duffie, Huang, Karatzas, Lakner, Lehoczky, Riedel, Shreve, Ž., etc. Incomplete markets: Basak and Cuoco 98 (incompleteness from restrictions in stock-market participation, logarithmic utility) Excess-demand approach. Introduced by Walras (1874): 1. Establish good topological/convexity properties of the excess demand ˆπ(S), and then 2. use a suitable fixed-point-type theorem to show existence (Brouwer, KKM, degree-based, etc.) Literature in continuous time: none, really!

24 Analysis wo paths to existence Representative agents. Uses the fact that equilibrium allocations are Pareto optimal; works (essentially) only for complete markets. Literature in continuous time: Complete markets: Bank, Dana, Duffie, Huang, Karatzas, Lakner, Lehoczky, Riedel, Shreve, Ž., etc. Incomplete markets: Basak and Cuoco 98 (incompleteness from restrictions in stock-market participation, logarithmic utility) Excess-demand approach. Introduced by Walras (1874): 1. Establish good topological/convexity properties of the excess demand ˆπ(S), and then 2. use a suitable fixed-point-type theorem to show existence (Brouwer, KKM, degree-based, etc.) Literature in continuous time: none, really!

25 Analysis wo paths to existence Representative agents. Uses the fact that equilibrium allocations are Pareto optimal; works (essentially) only for complete markets. Literature in continuous time: Complete markets: Bank, Dana, Duffie, Huang, Karatzas, Lakner, Lehoczky, Riedel, Shreve, Ž., etc. Incomplete markets: Basak and Cuoco 98 (incompleteness from restrictions in stock-market participation, logarithmic utility) Excess-demand approach. Introduced by Walras (1874): 1. Establish good topological/convexity properties of the excess demand ˆπ(S), and then 2. use a suitable fixed-point-type theorem to show existence (Brouwer, KKM, degree-based, etc.) Literature in continuous time: none, really!

26 A convex-analytic (sub)approach A first step towards a solution Work with random variables instead of processes; for example in the fast-and-slow model with ds λ u = λ u du + db u, we perform the following transformations π X λ,π = Z 0 π u ds λ u, λ Z λ = E( λ M), and consider a more tractable version i of the demand function so that i (Z λ ) = X λ,ˆπi (S λ ), i : E M L 0 + L 0 + L +. he problem now becomes simple to state: Can we solve the equation (Z) = 0, a.s. on E M?

27 A convex-analytic (sub)approach A first step towards a solution Work with random variables instead of processes; for example in the fast-and-slow model with ds λ u = λ u du + db u, we perform the following transformations π X λ,π = Z 0 π u ds λ u, λ Z λ = E( λ M), and consider a more tractable version i of the demand function so that i (Z λ ) = X λ,ˆπi (S λ ), i : E M L 0 + L 0 + L +. he problem now becomes simple to state: Can we solve the equation (Z) = 0, a.s. on E M?

28 A convex-analytic (sub)approach A first step towards a solution Work with random variables instead of processes; for example in the fast-and-slow model with ds λ u = λ u du + db u, we perform the following transformations π X λ,π = Z 0 π u ds λ u, λ Z λ = E( λ M), and consider a more tractable version i of the demand function so that i (Z λ ) = X λ,ˆπi (S λ ), i : E M L 0 + L 0 + L +. he problem now becomes simple to state: Can we solve the equation (Z) = 0, a.s. on E M?

29 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

30 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

31 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

32 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

33 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

34 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

35 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

36 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

37 Stability of utility maximization in incomplete markets (Note: fix an agent and drop the index i.) heorem (Larsen and Ž. (2006), to appear in SPA) Suppose that E 0. Let {λ n} n N Λ be a sequence such that Z λn is a martingale for each n, the collection {V + (Z λn ) : n N} is uniformly integrable, and Z λn Zλ in probability. hen, for x n x > 0 we have u λn (x n) u λ (x), and ˆX λn,xn ˆX λ,x in probability. Here, V is the convex conjugate of the utility function U, i.e., x,λ V (y) = sup x>0 (U(x) xy), and ˆX is the optimal terminal wealth in the market S λ with initial wealth x. Remarks: he uniform-integrability condition is practically necessary. Completes the Hadamard-style analysis of the utility maximization problem - repercussions for estimation. Further generalized to the general semimartingale case - under a different perturbation family - including general E L (Kardaras and Ž. (2007)). herefore, (under suitable conditions) is (L 0, L 0 )-continuous.

38 he KKM-theorem Some fixed-point theory heorem (Knaster, Kuratowski and Mazurkiewicz, 1929) Let S be the unit simplex in R m, and let V = {e 1,..., e m} be the set of its vertices. A mapping F : V 2 Rm is said to be a KKM-map if conv(e i, i J) i JF (e i), J {1,..., m}. If F (e i) is a closed subset of R m for all i {1,..., m}, then i {1,...,n} F (e i).

39 Convex compactness he KKM-heorem can easily be extended to infinite-dimensional vector-spaces as long as mild topological properties are imposed and local convexity is required (Kakutani, Fan, Browder, etc.) How about L 0 - the prime example of a non-locally-convex space? Yes, if one can fake compactness there: Convex-compactness (Nikišin, Buhvalov, Lozanovskii, Delbaen, Schahermayer, etc.) A subset B of a topological vector space is said to be convex-compact if any family (F α) α A of closed and convex subsets of B has the finite-intersection property, i.e. D fin A \! F α 0 \ F α. α D α A

40 Convex compactness he KKM-heorem can easily be extended to infinite-dimensional vector-spaces as long as mild topological properties are imposed and local convexity is required (Kakutani, Fan, Browder, etc.) How about L 0 - the prime example of a non-locally-convex space? Yes, if one can fake compactness there: Convex-compactness (Nikišin, Buhvalov, Lozanovskii, Delbaen, Schahermayer, etc.) A subset B of a topological vector space is said to be convex-compact if any family (F α) α A of closed and convex subsets of B has the finite-intersection property, i.e. D fin A \! F α 0 \ F α. α D α A

41 Convex compactness he KKM-heorem can easily be extended to infinite-dimensional vector-spaces as long as mild topological properties are imposed and local convexity is required (Kakutani, Fan, Browder, etc.) How about L 0 - the prime example of a non-locally-convex space? Yes, if one can fake compactness there: Convex-compactness (Nikišin, Buhvalov, Lozanovskii, Delbaen, Schahermayer, etc.) A subset B of a topological vector space is said to be convex-compact if any family (F α) α A of closed and convex subsets of B has the finite-intersection property, i.e. D fin A \! F α 0 \ F α. α D α A

42 Convex compactness he KKM-heorem can easily be extended to infinite-dimensional vector-spaces as long as mild topological properties are imposed and local convexity is required (Kakutani, Fan, Browder, etc.) How about L 0 - the prime example of a non-locally-convex space? Yes, if one can fake compactness there: Convex-compactness (Nikišin, Buhvalov, Lozanovskii, Delbaen, Schahermayer, etc.) A subset B of a topological vector space is said to be convex-compact if any family (F α) α A of closed and convex subsets of B has the finite-intersection property, i.e. D fin A \! F α 0 \ F α. α D α A

43 A characterization Proposition. A closed and convex subset C of a topological vector space X is convex-compact if and only if for any net (x α) α A in C there exists a subnet (y β ) β B of convex combinations of (x α) α A such that y β y for some y C. (A net (y β ) β B is said to be a subnet of convex combinations of (x α) α A if there exists a mapping D : B Fin(A) such that y β conv{x α : α D(β)} for each β B, and for each α A there exists β B such that α α for each α S β β D(β ).) Examples. Any convex and compact subset of a VS is convex-compact. A closed and convex subset of a unit ball in a dual X of a normed vector space X is convex-compact under any compatible topology (essentially Mazur), Any convex, closed and bounded-in-probability subset of L 0 + is convex-compact (essentially Komlós).

44 A characterization Proposition. A closed and convex subset C of a topological vector space X is convex-compact if and only if for any net (x α) α A in C there exists a subnet (y β ) β B of convex combinations of (x α) α A such that y β y for some y C. (A net (y β ) β B is said to be a subnet of convex combinations of (x α) α A if there exists a mapping D : B Fin(A) such that y β conv{x α : α D(β)} for each β B, and for each α A there exists β B such that α α for each α S β β D(β ).) Examples. Any convex and compact subset of a VS is convex-compact. A closed and convex subset of a unit ball in a dual X of a normed vector space X is convex-compact under any compatible topology (essentially Mazur), Any convex, closed and bounded-in-probability subset of L 0 + is convex-compact (essentially Komlós).

45 A characterization Proposition. A closed and convex subset C of a topological vector space X is convex-compact if and only if for any net (x α) α A in C there exists a subnet (y β ) β B of convex combinations of (x α) α A such that y β y for some y C. (A net (y β ) β B is said to be a subnet of convex combinations of (x α) α A if there exists a mapping D : B Fin(A) such that y β conv{x α : α D(β)} for each β B, and for each α A there exists β B such that α α for each α S β β D(β ).) Examples. Any convex and compact subset of a VS is convex-compact. A closed and convex subset of a unit ball in a dual X of a normed vector space X is convex-compact under any compatible topology (essentially Mazur), Any convex, closed and bounded-in-probability subset of L 0 + is convex-compact (essentially Komlós).

46 A characterization Proposition. A closed and convex subset C of a topological vector space X is convex-compact if and only if for any net (x α) α A in C there exists a subnet (y β ) β B of convex combinations of (x α) α A such that y β y for some y C. (A net (y β ) β B is said to be a subnet of convex combinations of (x α) α A if there exists a mapping D : B Fin(A) such that y β conv{x α : α D(β)} for each β B, and for each α A there exists β B such that α α for each α S β β D(β ).) Examples. Any convex and compact subset of a VS is convex-compact. A closed and convex subset of a unit ball in a dual X of a normed vector space X is convex-compact under any compatible topology (essentially Mazur), Any convex, closed and bounded-in-probability subset of L 0 + is convex-compact (essentially Komlós).

47 Attainment of minima heorem. Let A be a convex-compact subset of X, and let f : A R be a convex lower-semicontinuous function. hen f attains its minimum on A. A minimax-type theorem heorem. Let A, B be a convex-compact subsets of VS X and Y, respectively. Let f : A B R be a function with the following properties: x f(x, y) is usc and (quasi)-concave for each y B, y f(x, y) is lsc and (quasi)-convex for each x A. hen max x min y f(x, y) = min y max f(x, y). x

48 Attainment of minima heorem. Let A be a convex-compact subset of X, and let f : A R be a convex lower-semicontinuous function. hen f attains its minimum on A. A minimax-type theorem heorem. Let A, B be a convex-compact subsets of VS X and Y, respectively. Let f : A B R be a function with the following properties: x f(x, y) is usc and (quasi)-concave for each y B, y f(x, y) is lsc and (quasi)-convex for each x A. hen max x min y f(x, y) = min y max f(x, y). x

49 Generalized KKM theorem heorem. Let A be convex-compact subset of a VS X. Let {F (x)} x A be a family of closed and convex subsets of A such that conv(x 1,..., x n) n i=1f (x i), n N, x 1,..., x n A. hen x AF (x). he state of affairs Using the generalized KKM theorem, we can show existence of equilibria in many cases of some interest (it works for an infinity of agents, too). he requirement of (quasi)-convexity it places on the excess-demand function is a serious one. We are trying to sort the situation out (work in progress with Malamud, Anthropelos)... Kardaras ( 08) uses convex-compactness to give a general abstract framework for existence of numéraire portfolios.

50 he direct (sub)approach Let us consider the fast-and-slow model with Let {F t} t [0, ] be generated by a Brownian motion B and a one-jump-poisson process N with intensity µ > 0. We let S be the collection of all processes of the form S t = Z t 0 λ(u, B u, N u) du + db t, where λ : [0, ] R {0, 1} R ranges through bounded measurable functions. here is a finite number I of agents, each agent has the exponential utility U i (x) = exp( γ ix), the random endowments are of the form E i = g i (B, N ). heorem. Under the assumption that g i C 2+δ (R), i I, δ (0, 1), there exists 0 > 0 such that an equilibrium market, unique in the class C 2+δ,1+δ/2 ([0, ] R), exists whenever 0. heorem* he restriction < 0 is superfluous.

51 he direct (sub)approach Let us consider the fast-and-slow model with Let {F t} t [0, ] be generated by a Brownian motion B and a one-jump-poisson process N with intensity µ > 0. We let S be the collection of all processes of the form S t = Z t 0 λ(u, B u, N u) du + db t, where λ : [0, ] R {0, 1} R ranges through bounded measurable functions. here is a finite number I of agents, each agent has the exponential utility U i (x) = exp( γ ix), the random endowments are of the form E i = g i (B, N ). heorem. Under the assumption that g i C 2+δ (R), i I, δ (0, 1), there exists 0 > 0 such that an equilibrium market, unique in the class C 2+δ,1+δ/2 ([0, ] R), exists whenever 0. heorem* he restriction < 0 is superfluous.

52 he direct (sub)approach Let us consider the fast-and-slow model with Let {F t} t [0, ] be generated by a Brownian motion B and a one-jump-poisson process N with intensity µ > 0. We let S be the collection of all processes of the form S t = Z t 0 λ(u, B u, N u) du + db t, where λ : [0, ] R {0, 1} R ranges through bounded measurable functions. here is a finite number I of agents, each agent has the exponential utility U i (x) = exp( γ ix), the random endowments are of the form E i = g i (B, N ). heorem. Under the assumption that g i C 2+δ (R), i I, δ (0, 1), there exists 0 > 0 such that an equilibrium market, unique in the class C 2+δ,1+δ/2 ([0, ] R), exists whenever 0. heorem* he restriction < 0 is superfluous.

53 he direct (sub)approach Let us consider the fast-and-slow model with Let {F t} t [0, ] be generated by a Brownian motion B and a one-jump-poisson process N with intensity µ > 0. We let S be the collection of all processes of the form S t = Z t 0 λ(u, B u, N u) du + db t, where λ : [0, ] R {0, 1} R ranges through bounded measurable functions. here is a finite number I of agents, each agent has the exponential utility U i (x) = exp( γ ix), the random endowments are of the form E i = g i (B, N ). heorem. Under the assumption that g i C 2+δ (R), i I, δ (0, 1), there exists 0 > 0 such that an equilibrium market, unique in the class C 2+δ,1+δ/2 ([0, ] R), exists whenever 0. heorem* he restriction < 0 is superfluous.

54 Express the optimal portfolio in the form Sketch of the proof π i t = 1 γ i λ(t, B, N t) u i b(t, B t, N t), where solves the semi-linear system of two interacting PDEs ( 0 = u i t ui bb λu i b + 1 2γ i λ 2 µ γ (exp( γui n) 1) u i (,, ) = g i. where u i n(t, b, 0) = u i (t, b, 1) u i (t, b, 0), u i n(t, b, 1) = 0. Write the market-clearing condition in the form where 1 γ = P I i=1 1 γ i. 0 = IX ˆπ t(λ) i = 1 γ λ i=1 F (λ) = IX u i b(λ), i=1 IX γu i b(λ) = λ. i=1

55 Express the optimal portfolio in the form Sketch of the proof π i t = 1 γ i λ(t, B, N t) u i b(t, B t, N t), where solves the semi-linear system of two interacting PDEs ( 0 = u i t ui bb λu i b + 1 2γ i λ 2 µ γ (exp( γui n) 1) u i (,, ) = g i. where u i n(t, b, 0) = u i (t, b, 1) u i (t, b, 0), u i n(t, b, 1) = 0. Write the market-clearing condition in the form where 1 γ = P I i=1 1 γ i. 0 = IX ˆπ t(λ) i = 1 γ λ i=1 F (λ) = IX u i b(λ), i=1 IX γu i b(λ) = λ. i=1

56 Sketch of the proof Show that the mapping λ u i b(λ) is Lipschitz with a small Lipschitz coefficient in a well-chosen function space. he right one turns out to be the weighted Hölder space C (β);1+δ ([0, ] R). Apply the Banach fixed-point theorem to the function F.

57 Sketch of the proof Show that the mapping λ u i b(λ) is Lipschitz with a small Lipschitz coefficient in a well-chosen function space. he right one turns out to be the weighted Hölder space C (β);1+δ ([0, ] R). Apply the Banach fixed-point theorem to the function F.

58 What next Some research directions: (Aumann models) let the number of agents, and study the limiting behavior (mean-field-type ideas) (Alternative sources of incompleteness) jumps, transactions costs, default, etc. (Numerical methods) forward-backward SDEs, iterative approaches (Partial equilibria) with application to pricing in incomplete markets (Statistical issues) calibration, etc. (Dynamics) issues related to uniqueness (Simplification) the most pressing issue!

59 What next Some research directions: (Aumann models) let the number of agents, and study the limiting behavior (mean-field-type ideas) (Alternative sources of incompleteness) jumps, transactions costs, default, etc. (Numerical methods) forward-backward SDEs, iterative approaches (Partial equilibria) with application to pricing in incomplete markets (Statistical issues) calibration, etc. (Dynamics) issues related to uniqueness (Simplification) the most pressing issue!

60 What next Some research directions: (Aumann models) let the number of agents, and study the limiting behavior (mean-field-type ideas) (Alternative sources of incompleteness) jumps, transactions costs, default, etc. (Numerical methods) forward-backward SDEs, iterative approaches (Partial equilibria) with application to pricing in incomplete markets (Statistical issues) calibration, etc. (Dynamics) issues related to uniqueness (Simplification) the most pressing issue!

61 What next Some research directions: (Aumann models) let the number of agents, and study the limiting behavior (mean-field-type ideas) (Alternative sources of incompleteness) jumps, transactions costs, default, etc. (Numerical methods) forward-backward SDEs, iterative approaches (Partial equilibria) with application to pricing in incomplete markets (Statistical issues) calibration, etc. (Dynamics) issues related to uniqueness (Simplification) the most pressing issue!

62 What next Some research directions: (Aumann models) let the number of agents, and study the limiting behavior (mean-field-type ideas) (Alternative sources of incompleteness) jumps, transactions costs, default, etc. (Numerical methods) forward-backward SDEs, iterative approaches (Partial equilibria) with application to pricing in incomplete markets (Statistical issues) calibration, etc. (Dynamics) issues related to uniqueness (Simplification) the most pressing issue!

63 What next Some research directions: (Aumann models) let the number of agents, and study the limiting behavior (mean-field-type ideas) (Alternative sources of incompleteness) jumps, transactions costs, default, etc. (Numerical methods) forward-backward SDEs, iterative approaches (Partial equilibria) with application to pricing in incomplete markets (Statistical issues) calibration, etc. (Dynamics) issues related to uniqueness (Simplification) the most pressing issue!

64 What next Some research directions: (Aumann models) let the number of agents, and study the limiting behavior (mean-field-type ideas) (Alternative sources of incompleteness) jumps, transactions costs, default, etc. (Numerical methods) forward-backward SDEs, iterative approaches (Partial equilibria) with application to pricing in incomplete markets (Statistical issues) calibration, etc. (Dynamics) issues related to uniqueness (Simplification) the most pressing issue!