Convex duality and intertemporal consumption choice Peter Bank and - - PowerPoint PPT Presentation

Convex duality and intertemporal consumption choice

Peter Bank

and joint work with

Helena Kauppila, Ph.D. 2009 at Columbia University AnStAp10 Conference in Honor of W. Schachermayer Vienna, July 12–16, 2010

Utility maximization

How to optimally invest in a financial market?

◮ financial market: tradable assets S with equivalent local

martingale measures M = ∅, S0 ≡ 1

◮ trade: Xt = x +

t

0 Hs dSs ≥ 0 ❀ X ∈ X (x) ◮ primal problem: Maximize EU(XT) with U strictly increasing,

concave, U′(0) = ∞, U′(∞) = 0

◮ u(x) supX∈X (x) EU(XT) < ∞

Utility maximization

How to optimally invest in a financial market?

◮ financial market: tradable assets S with equivalent local

martingale measures M = ∅, S0 ≡ 1

◮ trade: Xt = x +

t

0 Hs dSs ≥ 0 ❀ X ∈ X (x) ◮ primal problem: Maximize EU(XT) with U strictly increasing,

concave, U′(0) = ∞, U′(∞) = 0

◮ u(x) supX∈X (x) EU(XT) < ∞

Kramkov-Schachermayer (1999, 2003):

◮ AE(U) = limx→∞ U′(x) U(x)/x < 1 ◮ bipolar theorem generalizing financiability constraint

X ≤ XT ∈ X (x) ⇔ supQ∈M EQX ≤ x

◮ convex duality ◮ . . .

Convex duality à la Kramkov-Schachermayer

◮ Legendre-Fenchel transform: V (y) = supx≥0{U(x) − xy} ◮ dual processes: Y s.t. Y0 = y, XY supermartingale for all

X ∈ X (1) ❀ Y ∈ Y (y)

◮ dual problem: Minimize EV (YT) over Y ∈ Y (y) ◮ v(y) = infY ∈Y (y) EV (YT)

Convex duality à la Kramkov-Schachermayer

◮ Legendre-Fenchel transform: V (y) = supx≥0{U(x) − xy} ◮ dual processes: Y s.t. Y0 = y, XY supermartingale for all

X ∈ X (1) ❀ Y ∈ Y (y)

◮ dual problem: Minimize EV (YT) over Y ∈ Y (y) ◮ v(y) = infY ∈Y (y) EV (YT)

Theorem (Kramkov-Schachermayer)

If v(y) < ∞ for all y > 0:

• 1. u and v are C 1, strictly convex, and conjugate to each other:

v(y) = sup

x≥0

{u(x) − xy}, u(x) = inf

y≥0{v(y) + xy}

• 2. For y = u′(x), the optimizers for primal and dual problem

exists and are related by U′(X ∗

T(x)) = Y ∗ T(y),

X ∗

T(x) = −V ′(Y ∗ T(y)) .

Convex duality and intertemporal consumption

Our aim:

In analogy to Kramkov-Schachermayer, develop a theory of convex duality for utility functionals of the form EU(C) = E T F(t, Ct) dt where

◮ F = F(t, c) is an agent’s felicity function: strictly concave and

increasing in c, Inada conditions, AE(F)(t, c) < 1

◮ C = (Ct)0≤t≤T is the agent’s cumulative consumption: ≥ 0,

increasing, adapted, left-continuous

Why should one consider such a utility functional?

EU(C) = E T F(t, Ct) dt

◮ F = F(t, c) is an agent’s felicity function: C 1,2, strictly

concave and increasing in c, Inada conditions, AE(F)(t, c) < 1

◮ C = (Ct)0≤t≤T is the agent’s cumulative consumption: ≥ 0,

increasing, adapted, left-continuous

Different answers from different people, e.g., from a . . .

Why should one consider such a utility functional?

EU(C) = E T F(t, Ct) dt

◮ F = F(t, c) is an agent’s felicity function: C 1,2, strictly

concave and increasing in c, Inada conditions, AE(F)(t, c) < 1

◮ C = (Ct)0≤t≤T is the agent’s cumulative consumption: ≥ 0,

increasing, adapted, left-continuous

Different answers from different people, e.g., from a . . .

mountaineer: Because it’s there!

Why should one consider such a utility functional?

EU(C) = E T F(t, Ct) dt

◮ F = F(t, c) is an agent’s felicity function: C 1,2, strictly

concave and increasing in c, Inada conditions, AE(F)(t, c) < 1

◮ C = (Ct)0≤t≤T is the agent’s cumulative consumption: ≥ 0,

increasing, adapted, left-continuous

Different answers from different people, e.g., from a . . .

mountaineer: Because it’s there! agnostic: Because it leaves open whether optimal consumption is done at rates, in gulps or other ways.

Why should one consider such a utility functional?

EU(C) = E T F(t, Ct) dt

◮ F = F(t, c) is an agent’s felicity function: C 1,2, strictly

concave and increasing in c, Inada conditions, AE(F)(t, c) < 1

◮ C = (Ct)0≤t≤T is the agent’s cumulative consumption: ≥ 0,

increasing, adapted, left-continuous

Different answers from different people, e.g., from a . . .

mountaineer: Because it’s there! agnostic: Because it leaves open whether optimal consumption is done at rates, in gulps or other ways. aesthete: Because the standard setting of felicity from consumption rates, i.e., speed of consumption does not exactly reflect savoir vivre.

Why should one consider such a utility functional?

EU(C) = E T F(t, Ct) dt

◮ F = F(t, c) is an agent’s felicity function: C 1,2, strictly

concave and increasing in c, Inada conditions, AE(F)(t, c) < 1

◮ C = (Ct)0≤t≤T is the agent’s cumulative consumption: ≥ 0,

increasing, adapted, left-continuous

Different answers from different people, e.g., from a . . .

mountaineer: Because it’s there! agnostic: Because it leaves open whether optimal consumption is done at rates, in gulps or other ways. aesthete: Because the standard setting of felicity from consumption rates, i.e., speed of consumption does not exactly reflect savoir vivre. economist: Because it allows for intertemporal substitution of consumption; see Hindy, Huang, Kreps (1992, 1993)

Problem formulation

Utility maximization

Maximize EU(C) over all consumption plans C which the agent can finance by investing the initial wealth x > 0 in the stock market: u(x) = sup

C∈C (x)

EU(C) where C (x) = {C ≥ 0, ↑, l.c., adapted, E

• [0,T)

Yt dCt ≤ x, Y ∈ Y (1)} Benth et al. (2001): exponential Levy model, HARA-felicity, HJB-approach Our view: Maximize concave functional EU(C) subject to linear constraints.

The first step: Compute the Legendre-Fenchel Transform

For deterministic Y = (Yt) ≥ 0, consider the functional V (Y ) = sup

dC≥0

• U(C) −
• [0,T)

Yt dCt

• (∗)

The first step: Compute the Legendre-Fenchel Transform

For deterministic Y = (Yt) ≥ 0, consider the functional V (Y ) = sup

dC≥0

• U(C) −
• [0,T)

Yt dCt

• (∗)

If Y > 0 and l.s.c., solution C ∗ can be computed from time-inhomogeneous convex envelope Y ∗ ≤ Y maximal with Y ∗

t =

• [t,T)

F ′(s, C ∗

s ) ds, dC ∗ ≥ 0

where F ′(t, c) =

∂ ∂c F(t, c).

In particular: U(C) = inf

Y ≥0

• V (Y ) +
• [0,T)

Yt dCt

Convex duality in the deterministic case

Corollary

For Y ∗ and C ∗ as above, the following statements are equivalent: (i) C ∗ attains supdC≥0

• U(C) −
• [0,T) Y ∗

t dCt

• (ii) Y ∗ attains infY ≥0
• V (Y ) +
• [0,T) Yt dC ∗

t

• (iii) ∇U(C ∗) = inhomogeneously convex envelope of Y ∗1[0,T)

(iv) ∇V (Y ∗) = −dC ∗

Convex duality in the stochastic case

For Y ∈ Y (y), consider the transform sup

dC≥0, adapted

E

• U(C) −
• [0,T)

Yt dCt

• (∗∗)

Duality gap?

In general (∗∗) < EV (Y ) = E

• sup

dC≥0

• U(C) −
• [0,T)

Yt dCt

• because ω-wise argmax will not give an adapted consumption plan,

even if Y is adapted: non-time additive structure of U ❀ we need to relax our choice of Y to address adaptivity constraint; cf. Davis, Karatzas (1994)

Convex duality in the stochastic case

Well known: E

• [0,T)

Yt dCt = E

• [0,T)

˜ Yt dCt for all C iff EYτ = E ˜ Yτ for all stopping times τ < T: optional projection

Convex duality in the stochastic case

Well known: E

• [0,T)

Yt dCt = E

• [0,T)

˜ Yt dCt for all C iff EYτ = E ˜ Yτ for all stopping times τ < T: optional projection Should we thus enlarge the space of dual variables to all ˜ Y with

• ptional projection o ˜

Y = Y : sup

dC≥0, adapted

E

• U(C) −
• [0,T)

Yt dCt

• =

inf

• ˜

Y =Y

EV ( ˜ Y ) ?

Convex duality in the stochastic case

Well known: E

• [0,T)

Yt dCt = E

• [0,T)

˜ Yt dCt for all C iff EYτ = E ˜ Yτ for all stopping times τ < T: optional projection Should we thus enlarge the space of dual variables to all ˜ Y with

• ptional projection o ˜

Y = Y : sup

dC≥0, adapted

E

• U(C) −
• [0,T)

Yt dCt

• =

inf

• ˜

Y =Y

EV ( ˜ Y ) ? Unclear how to construct optimal C ∗ from minimzer ˜ Y ∗ of dual problem: non-uniqueness, adaptedness of argmax?

Convex duality in the stochastic case

Idea: confine dual space to ‘convex’ processes

For Y ∈ Y (y) find ˜ Y ≤ Y with inhomgeneously convex paths and adapted density ∂t ˜ Yt = −F ′(t, Ct) such that

• ˜

Yt = Yt whenever dCt > 0 . B., El Karoui (2004): Such a probabilistic analog of an inhomogeneously convex envelope for Y can be found in exactly

• ne way.

Convex duality: complete financial market

Corollary (B., Riedel (2001))

Let M = {Q}, Yt = E

• dQ

dP

• Ft
• 1[0,T)(t) and denote by ˜

Y its probabilistic inomogeneously convex envelope. Then the density C ∗ of ˜ Y is optimal in its budget set: EU(C) ≤ EU(C ∗) for all C ∈ C (x) provided x E

• [0,T) Yt dC ∗

t < ∞.

Allows to compute closed-form solutions when envelopes can be computed in closed form; see work with Riedel (2001), Föllmer (2003), Baumgarten (2009)

Convex duality: incomplete financial market

In the general incomplete case |M | = ∞ consider ˜ Y (y) = { ˜ Y ≥ 0 w/ inhomogeneously convex paths and adapted density such that o ˜ Y ≤ Y for some Y ∈ Y (y)}

Theorem

• 1. ˜

Y (y) is convex iff ∂2

tcF ◦ ∂cF −1 is concave in c.

• 2. ˜

Y (y) is Fatou-closed.

Remark:

◮ Sufficient for item 1.: F(t, c) = f (t)g(c) ◮ Fatou-closure see Kramkov (1996); Zitkovic (2002) proves

Fatou-closure of Y (y)

◮ Fatou-stability of inhomogenous convexity and densities

Convex duality: incomplete financial market

Theorem

For any y > 0, there exists a unique ˜ Y (y) ∈ ˜ Y (y) that attains v(y) = inf

˜ Y ∈ ˜ Y (y)

EV ( ˜ Y ) . The density of ˜ Y (y) yields a plan C(x) which attains u(x) = sup

C∈C (x)

EU(C) for initial capital x = −v′(y). u and v are C 1, strictly convex, and conjugate to each other: v(y) = sup

x≥0

{u(x) − xy}, u(x) = inf

y≥0{v(y) + xy}

and ∇U(C(x)) = Y (y), ∇V (Y (y)) = −dC(x) .

Conclusion

◮ consumption with intertemporal substitution: non

time-additive utility functional

◮ no interior solutions ◮ combination of variants of convexity: time-inhomogeneous,

space-inhomogeneous, probabilistic

◮ non-adapted dual variables with adapted densities

Conclusion

◮ consumption with intertemporal substitution: non

time-additive utility functional

◮ no interior solutions ◮ combination of variants of convexity: time-inhomogeneous,

space-inhomogeneous, probabilistic

◮ non-adapted dual variables with adapted densities

Herzlichen Glückwunsch Walter!

A.Ma.Me.F. workshop in Berlin

◮ September 27–30, 2010 ◮ http://sites.google.com/site/amamefberlin2010/ ◮ featuring: Walter Schachermayer ◮ limited capacity: register soon!