[PPT] - Minimal Sufficient Conditions for a Primal Optimizer in Nonsmooth PowerPoint Presentation

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Minimal Sufficient Conditions for a Primal Optimizer in Nonsmooth Utility Maximization Harry Zheng Imperial College (Joint work with Nicholas Westray, Humboldt Univ. Berlin) Workshop on Stochastic Analysis and Finance Hong Kong City University 2 July 2009

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Classical Utility Maximization

dS(t) = αS(t)dt + σS(t)dW(t), S(0) = s > 0 where α > 0 excess return, σ > 0 stock volatility, W(t) Brownian motion.

satisfies SDE dX(t) = H(t)dS(t)

continuously differentiable, U ′(0) = ∞ and U ′(∞) = 0.

max

(1)

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Martingale Approach

B(x) = {B : B ≥ 0, FT − measurable, E[H(T)B] ≤ x} where H(t) = exp

t θ(s)dW(s) − 1 2 t θ(s)2ds

max

(2)

XH∗(T) = B∗, a.s.

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Dual Problem

˜ U(y) = sup

(U(x) − xy).

we have the relation max

U(yH(T))] + xy

min

U(yH(T))] + xy

E[U(B∗)] = E[ ˜ U(y∗H(T))] + xy∗ then B∗ solves primal problem and y∗ solves dual problem.

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Construction of Optimal Solutions

U′(y). Duality relation implies E[U(B∗)] = E[ ˜ U(yH(T))] + yE[H(T)B∗].

E[H(T)I(yH(T))] = x. Then y∗ is optimal dual solution and B∗ optimal primal solution.

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Conditions for Martingale Approach Success of martingale approach above is based on following conditions:

definition of function I and for existence of y∗.

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Wealth Process T finite time horizon, a market consisting of one bond, equal to 1, and d stocks, S1, . . . , Sd modelled by an Rd-valued semimartingale on a filtered probability space (Ω, F, (Ft)0≤t≤T, P), satisfying the usual conditions. For a predictable S-integrable process, the wealth processes is defined by Xx,H = x + H ∙ S, (3) where x > 0 initial endowment. The set for optimization is following, X(x) :=

for some Xx,H as defined in (3)

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Smooth Utility Maximization

and U ′(∞) = 0.

u(x) := sup

E

Y(y) :=

w(x) := inf

U(Y )] + xy

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Kramkov and Schachermayer (1999) Result Theorem 1. Let M(S) = ∅ (set of equivalent local martingale measures for S) and AE(U) = lim sup

xU ′(x) U(x) < 1 (asymptotic elasticity condition). Let x > 0 be such that u(x) < ∞, then (i) There exists a unique solution y∗ > 0 and Y∗ ∈ Y(y∗) to dual problem w(x) = E ˜ U(Y∗)] + xy∗

(ii) There exists a unique solution X∗ to primal problem u(x) = E[U(X∗)]. (iii) E[X∗Y∗] = xy∗ and X∗ = − ˜ U′(Y∗). (i)-(iii) imply u(x) = w(x). In classical case, Y+(y) = {yH(T)}, which implies Y∗ = y∗H(T) and y∗ being determined from (iii).

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Nonsmooth Utility Functions Assumption 1 (Inada Condition). inf

∂U(x) = 0, sup

∂U(x) = ∞. Assumption 2 (Concave Increasing Condition). U(0) = 0, U(∞) = ∞ and U is concave and increasing on R+. Assumption 3 (Asymptotic Elasticity Condition). AE( ˜ U) := lim sup

sup

|q|y ˜ U(y) < ∞. Assumption 4 (No Arbitrage Condition). M(S) = ∅.

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Deelstra, Pham, Touzi (2001) Result DPT prove following theorem using quadratic inf-convolution method. Theorem 2. Let Assumptions 1-4 hold. Let x > 0 be such that u(x) < ∞, then (i) There exists some y∗ > 0 and Y∗ ∈ Y(y∗) to dual problem w(x) = E ˜ U (Y∗) + xy∗

(ii) There exists some X∗ ∈ X(x) to primal problem u(x) = E[U(X∗)]. (iii) E[X∗Y∗] = xy∗ and X∗ ∈ −∂ ˜ U (Y∗).

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Deelstra-Pham-Touzi Conjecture (2001) For conjugate functions U and ˜ U, U(x) ≤ ˜ U(y) + xy for all y ≥ 0, U(x) = ˜ U(y) + xy if and only if x ∈ −∂ ˜ U(y). Suppose (y∗, Y∗) are optimal for dual problem and ˆ X satisfies ˆ X ∈ −∂ ˜ U(Y∗) and E[ ˆ XY∗] = xy∗. Then E[U( ˆ X)] = E[ ˜ U(Y∗) + ˆ XY∗] = E[ ˜ U(Y∗)] + xy∗ ≥ sup

E[U(X)]. It is clear that ˆ X is optimal if and only if it is an element of X(x).

X ∈ −∂ ˜ U (Y∗) and E[ ˆ XY∗] = xy∗, then ˆ X ∈ X(x). Note that if U strictly concave or if a market is complete, then conjecture is trivially positive. To make conjecture nontrivial, U must be not strictly concave and market not complete.

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Utility Function U(x) :=    2√x x ∈ [0, 1] x + 1 x ∈ (1, 5) 2√x − 4 + 4 x ∈ [5, ∞) . U is C1 and satisfies Assumptions 1 and 2, but not strictly concave. Dual function is given by ˜ U(y) =

y ∈ (0, 1)

y ∈ [1, ∞) . ˜ U satisfies Assumption 3, but ˜ U is not continuously differentiable at y = 1. Subdifferential of ˜ U is given by ∂ ˜ U(y) =      −4 − 1

y ∈ (0, 1) [−5, −1] y = 1 − 1

y ∈ (1, ∞) . (4)

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Probability Space Next we describe our market. Let (Ω, F, (Gt)t≥0, P) be a filtered probability space on which a process W and a random variable ξ are defined. (i) W is a standard Brownian motion and (Gt)t≥0 is augmented filtration generated by W. (ii) ξ is independent of entire path of W and valued in {0, 1} with probabilities P(ξ = 0) = 1 3, P(ξ = 1) = 2 3. To construct the filtration for our example we take (Gt)t≥0 and define Ft := σ(Gt, σ(ξ)).

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Asset Process We introduce stopping time τ, defined by τ := inf

2 / ∈ (− log 4, log 4)

and set terminal time horizon for utility maximization T := τ. A calculation based on first exit time of Brownian motion with drift shows that P

2 = log 4

5, P

2 = − log 4

5. Asset process S is defined by St := 1{ξ=0} + exp

2(t ∧ τ)

We observe that S has continuous paths, is uniformly bounded, nonnegative and at time T is valued in set {1/4, 1, 4}.

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Equivalent Martingale Measures We may construct a large family of martingale measures simply by changing distribution of ξ. This is equivalent to determining pair (q0, q1) where Q(ξ = 0) = q0, Q(ξ = 1) = q1. For consistency we set p0 = 1

processes Zq, defined for t ≥ 0 by Zq

p0 1{ξ=0} + S−1

q1 p1 1{ξ=1}. (5) Using independence of W and ξ one can verify that Zq is a P-martingale for nonnegative q such that q0 + q1 = 1. Moreover, for any such q, ZqS is a P-martingale (it is constant). This implies that if we define dQq dP = Zq

then Qq ∈ M and therefore market is incomplete.

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Primal Solution with Constant Investment Strategies We focus on u(1) and consider X const(1) =

3, 4 3

Define maximization problem uconst(1) := sup

E

uconst(1) = sup

8 15U(1 + 3H) + 2 15U

4H

3U(1)

Since U is C1, one finds maximum solution H∗ = 85

strategy with optimal terminal wealth X∗ := 1 + (H∗ ∙ S)T = 1{ξ=0} + 319 64 1{ST=4,ξ=1} + 1 2561{ST=1

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Dual Solution with Constant Investment Strategies U ′(X∗) = 1{ξ=0} + 1{ST=4,ξ=1} + 161{ST=1

(6) We may equivalently write this as U′(X∗) = 3 q∗ p0 1{ξ=0} + S−1

q∗

p1 1{ξ=1}

where q∗

We compare this with (5) and write Q∗ for risk neutral measure having Radon- Nikodym derivative Zq∗

Y∗ := U′(X∗) = 3dQ∗ dP ∈ Y(3). Using (4), −∂ ˜ U(Y∗) = [1, 5] Y∗ = 1

Y∗ = 16 . (7)

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Optimal Solutions with General Investment Strategies A calculation shows that E[X∗Y∗] = 3. Set y∗ := 3 conjugacy relation implies E[ ˜ U(Y∗)] + 3 = E

U

dQ∗ dP

dQ∗ dP

(8) Combining this with inequality E[U(X∗)] ≤ u(1) ≤ inf

U

dP

(9) we have shown X∗ is in fact optimal terminal wealth when non constant strategies are allowed. It follows that optimal solutions are H∗ = 85 64, X∗ =   

ST = 4 1 ST = 1

ST = 1

, (y∗, Y∗) =

dP

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Subgradient Valued Random Variable Define random variables X− : = −D+ ˜ U(Y∗) = 1{ξ=0} + 1{ST=4,ξ=1} + 1 2561{ST=1

X+ : = −D− ˜ U(Y∗) = 51{ξ=0} + 51{ST=4,ξ=1} + 1 2561{ST=1

We can show −∂ ˜ U(Y∗) = [X−, X+], E[X−Y∗] ≤ xy∗, E[X+Y∗] ≥ xy∗. Set ˆ X := λX− + (1 − λ)X+ for λ ∈ [0, 1] and choose λ such that E[ ˆ XY∗] = xy∗(= 3). A calculation shows that λ = 161

ˆ X := 359 1041{ξ=0} + 359 1041{ST=4,ξ=1} + 1 2561{ST=1

(10) We have constructed a ˆ X which satisfies ˆ X ∈ −∂ ˜ U(Y∗) and E[ ˆ XY∗] = xy∗.

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Non-Feasibility To show ˆ X / ∈ X(1), we apply famous paper by Delbaen and Schachermayer (1994), which says ˆ X ∈ X(1) if and only if sup

EQ[ ˆ X] ≤ 1. In particular we need only to find ˆ q such that E[ ˆ XZ ˆ

X > 0, E[ ˆ XZq

XZq

104q0. We can choose any q0 ∈ (0, 1) such that E[ ˆ XZ ˆ

X is not an element of X(x). We have completed the construction of a counterexample to Conjecture.

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A Generalization to Conjecture Any random variable ˆ X is primal optimal solution if it satisfies all of (I) ˆ X ∈ X(x), (II) ˆ X ∈ −∂ ˜ U(Y∗), (III) E[ ˆ XY∗] = xy∗. From present viewpoint Conjecture simply asks whether (II) and (III) auto- matically imply (I)? It therefore seems natural to investigate whether there are any other dependence relations between (I), (II) and (III).

We can show that if we choose ˆ X = 1{ξ=0} + 51{ST=4,ξ=1} + 01{ST=1

then ˆ X satisfies (I), (III), but not (II). If we choose ˆ X := 1{ξ=0} + 1{ST=4,ξ=1} + 1 2561{ST=1

then ˆ X satisfies (I), (II), but not (III).

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Minimum Sufficient Condition for Primal Optimal Solution Suppose that Assumptions 1, 2, 3 and 4 hold. Let y∗ > 0, Y∗ ∈ Y(y∗) be solution to dual problem and x > 0 such that w(x) < ∞. Then a random variable ˆ X is optimal for primal problem if and only if (I) ˆ X ∈ X(x), (II) ˆ X ∈ −∂ ˜ U(Y∗), (III) E[ ˆ XY∗] = xy∗. Note that when U is strictly concave it is shown in Kramkov and Schacher- mayer (1999) that (II) implies both (I) and (III). On the other hand when U is not strictly concave our discussion shows that (I), (II) and (III) are minimum sufficient conditions for optimality.

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Constrained Nonsmooth Utility Maximization Westray and Zheng (2009) have extended results of Deelstra-Pham-Touzi (2001) to constrained case. Let K be a closed polyhedral convex cone in

H := {H : H predictable and S-integrable, Ht ∈ K P-a.s. for all t} . Theorem 3. Suppose that Assumptions 1, 2, 3 and 4 hold. Let w(x) < ∞. Then (i) There exists some y∗ ≥ 0 and an optimal Y∗ ∈ Y(y∗) such that, w(x) = E ˜ U (Y∗) + xy∗

(ii) There exists some X∗ ∈ X(x) satisfying E[X∗Y∗] = xy∗ and X∗ ∈ −∂ ˜ U (Y∗) and u(x) = E[U(X∗)]. (iii) u(x) = w(x).

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