On the dual problem of utility maximization Yiqing LIN Joint work - - PowerPoint PPT Presentation

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment

On the dual problem of utility maximization

Yiqing LIN Joint work with L. GU and J. YANG

University of Vienna

• Sept. 2nd 2015

Workshop “Advanced methods in financial mathematics” Angers

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment

1

Introduction Basic settings Literature review on convex duality methods

2

Utility maximization without random endowment Duality method The dual optimizer: existing result The dual optimizer: alternative method

3

Utility maximization with bounded random endowment Duality method The dual optimizer Remarks

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Basic settings

The market model

Consider a filtered probability space (Ω, F, (Ft)0≤t≤T , P) satisfying the usual conditions, where T is a finite horizon. The market consists of a bond and a stock, where the bond is

• f zero interest rate and the stock-price process S is a strictly

positive semimartingale. “No arbitrage condition”: Me(S) = ∅. For an initial value x and a predictable S-integrable trading strategy H, the value process X = (Xt)0≤t≤T is given by Xt = x + (H · S)t, 0 ≤ t ≤ T. We call H admissible if for 0 ≤ t ≤ T, the associated terminal value Xt ≥ −M, for some positive M. The agent receives an exogenous random endowment eT ∈ FT at time T, satisfying ρ := eT ∞ < ∞.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Basic settings

Utility maximization on the positive half line

A utility function U : (0, ∞) → R represents the agent’s preferences over the terminal wealth. The function U is assumed to be strictly concave, strictly increasing and continuously differentiable satisfying the Inada conditions: U ′(0) := lim

x→0 U ′(x) = ∞,

U′(∞) := lim

x→∞ U ′(x) = 0.

and the condition of reasonable asymptotic elasticity (RAE): AE(U) = lim sup

x→∞

xU ′(x) U(x) < 1. The aim of the agent is to maximize the expected utility from the terminal wealth: u(x) := sup

H adm

E[U(x + (H · S)T + eT )], x > 0,

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Literature review on convex duality methods

The Itˆ

• framework: [Karatzas-Lehoczky-Shreve-Xu, 1991],

[Cvitani´ c-Karatzas, 1992], etc...; The general semimartingale framework: U : (0, ∞) → R eT = 0 [Kramkov-Schachermayer, 1999] bounded eT [Cvitani´ c-Schachermayer-Wang, 2001] unbounded eT [Hugonnier-Kramkov, 2004] U : R → R:

• locally bounded semimartingale models:

[Schachermayer, 2001], [Owen, 2002], [Owen-ˇ Zitkovi´ c, 2009];

• general semimartingale models: [Biagini-Frittelli, 2008],

[Biagini-Frittelli-Grasselli, 2011]. Optimal consumption, with constraints, etc...

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment

1

Introduction Basic settings Literature review on convex duality methods

2

Utility maximization without random endowment Duality method The dual optimizer: existing result The dual optimizer: alternative method

3

Utility maximization with bounded random endowment Duality method The dual optimizer Remarks

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method

Duality method: eT = 0

Define X(x) := {X : X = x + (H · S), H adm, XT ≥ 0} and C(x) :=

• g ∈ L0

+(FT ) : 0 ≤ g ≤ XT , for some X ∈ X(x)

• ,

where the latter is the set of positive terminal values, which can be dominated by some admissible strategies initialed from x > 0. Then, the maximization problem (primal problem) can be rewritten into u(x) := sup

H adm

E[U(x + (H · S)T )] = sup

g∈C(x)

E[U(g)].

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method

Definition (Supermartingale deflators) We call a positive semimartingale Y a supermartingale deflator, if for each X ∈ X(1), XY is a supermartingale. Moreover, we denote by Y(y) the collection of all such processes starting from y, namely, Y(y) :=

• Y ≥ 0 : Y0 = y, XY is a supermartingale, ∀ X ∈ X(1)
• .

Note D(y) := {h ∈ L0

+(FT ) : 0 ≤ h ≤ YT , for some Y ∈ Y(y)}.

Then, the dual problem is formulated as v(y) = inf

Y ∈Y(y) E

• V (YT )
• =

inf

h∈D(y) E

• V (h)
• ,

where V (y) := sup

x>0

{U(x) − xy}, y > 0.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method

Theorem ([Kramkov-Schachermayer, 1999]) Assume “No arbitrage condition”: Me(S) = ∅, U satisfies the Inada conditions and RAE, u(x) < ∞, for some x > 0. Then The value functions u, v have the same properties as U and V . For any y > 0, there exists a unique dual optimizer h ∈ D(y). Let y := u′(x), then there exists a unique primal solution

• g ∈ C(x), which is defined by

g := (U ′)−1( h

y).

E[ g h] = x y.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: existing result

In general, the optimal element Y from Y( y) associated with the dual optimizer h

y is not a martingale. An example can be found in

[Kramkov-Schachermayer, 1999] even with an one-period model. However, under certain condition, Y is proved a local martingale. Theorem ([Larson-ˇ Zitkovi´ c, 2007]) In addition to the conditions for the above theorem, suppose that S is continuous, then the dual optimizer h

y is attained by a local

martingale from the set of supermartingale deflators. Outline of the proof: That S is continuous and Me(S) = ∅ implies the following representation ([Delbean-Schachermayer, 1995]): St = 1 + Mt + t λudMu, 0 ≤ t ≤ T, where M is a local martingale and λ is a predicable M-integrable process.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: existing result

Proposition For any Y ∈ Y(y), we have the following multiplicative decomposition Y = yE(−λ · M)E(L)D, where L is a c` adl` ag local martingale satisfying M, L ≡ 0, and D is a predictable, non-increasing, strictly positive, c` adl` ag process with D0 = 1. It can be verified by Itˆ

• ’s formula that yE(−λ · M)E(L) ∈ Y(y).

Then, from the fact that V is strictly decreasing, one can deduce that

• Y =

yE(−λ · M)E( L), which is a local martingale, namely, D ≡ 1.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: alternative method

We would like to provide an alternative method to prove the same theorem [Larson-ˇ Zitkovi´ c, 2007]. Based on the same idea, we could generalize this theorem to the case of bounded random endowment in the next section. The idea is as follows: we first stop the process X by a sequence

• f stopping times {τk}k∈N, such that before each τk,

X is bounded away from 0. Precisely, define a localizing sequence τk := inf{t : Xt < 1/k} ∧ T. Since X is continuous, Xτk ≥ 1/k and P(limk τk = T) = 1. Then, we shall construct a process Y , such that YT = h

y and

prove that the stopped process Y·∧τk is a martingale by means of that X Y is a uniformly integrable martingale. Finally, we prove that Y ∈ D( y).

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: alternative method

From the result in [Kramkov-Schachermayer, 1999], one can find a sequence {Qn}∞

n=1 from Me(S) such that

• ydQn

dP − → h

y, a.s..

We denote by Y n the associated density process, which is a martingale, i.e., Y n

t :=

ydQn dP

• Ft

. Then, we construct a process Y in terms of {Y n}∞

n=1, such that

• YT =

h

• y. To this end, we need the following lemma.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: alternative method

Lemma ([Czichowsky-Schachermayer, 2014]) Let {Y n}∞

n=1 be a sequence of non-negative optional strong

supermartingales Y n = {Y n

t }0≤t≤T starting at Y n 0 = y. Then

there is a sequence { Y n}∞

n=1 of convex combinations

• Y n ∈ conv(Y n, Y n+1, · · · )

and a non-negative optional strong supermartingale

• Y = {

Yt}0≤t≤1 such that for every [0, T]-valued stoppting time τ, we have convergence in probability, i.e.,

• Y n

τ −

→ Yτ. WLOG, we may choose a subsequence such that for each k,

• Y n

τk −

→ Yτk, a.s..

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: alternative method

In our case, Y n are all true martingales associated with the equivalent local martingale measure Qn defined by y d

Qn dP =

YT . Obviously, Y n

T −

→ YT = h

y.

Fixing k, by super-replication theorem, we have E[ Xτk Y n

τk] ≤ x

y. On the other hand, by applying the above lemma again, one can see that X Y is an optional strong supermatingale. Furthermore,

• X0

Y0 = E[ XT YT ] = E[ g h] = x y, then X Y is a true martingale. Therefore, E[ Xτk Yτk] = x y.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer: alternative method

Lemma Let X ∈ L0(Ω, F, P), X ≥ a > 0, a.s., and {Y n}∞

n=1 ⊂ L1 +(Ω, F, P), Y n → Y , a.s.. If

E[XY ] ≥ lim inf

n

E[XY n]. Then, {Y n}∞

n=1 is uniformly integrable.

By the above lemma, we know from

• E[

XτkY n

τk]

≤ x y; E[ Xτk Yτk] = x y. that {Y n

τk}∞ n=1 is uniformly integrable and thus, the stopped

process Y·∧τk is a true martingale. Thus, Y is a local martingale and has a c` adl` ag version. Moreover, by the lemma in [Czichowsky-Schachermayer, 2014] again, we can verify that for each X ∈ X(1), X Y is a supermatingale.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment

1

Introduction Basic settings Literature review on convex duality methods

2

Utility maximization without random endowment Duality method The dual optimizer: existing result The dual optimizer: alternative method

3

Utility maximization with bounded random endowment Duality method The dual optimizer Remarks

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method

Duality method: eT bounded

Recall the primal problem of the utility maximization u(x) := sup

H adm

E[U(x+(H·S)T +eT )] = sup

g∈C0

E[U(x+g+eT )], x > 0, where C0 := {g : g = (H · S)T , H adm}.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method

The dual problem is formulated by v(y) := inf

Q∈D

• E
• V
• ydQr

dP

• + yQ, eT
• , y > 0,

where D :=

• Q ∈ (L∞)∗

+ : Q(L∞)∗ = 1, Q, g ≤ x,

for all g ∈ C(x), for all x > 0

• .

From the result in [Yosida-Hewitt, 1952], for any Q ∈ (L∞)∗

+, Q

can be uniquely decomposed into Q = Qr + Qs, where Qr is countably additive and Qs is purely finitely additive.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Duality method

Theorem ([Cvitani´ c-Schachermayer-Wang, 2001])

Assume “No arbitrage condition”: Me(S) = ∅, U satisfies the Inada conditions and RAE, u(x) < ∞, for some x > ρ. Then The value functions u, v have the same properties as U and V . The dual solution Qy ∈ D exists for all y > 0 and Qr

y is unique.

For all x > x0 := supQ∈DQ, −eT , g := I

• y

d Qr

• y

dP

• − x − eT ∈ C0

is the solution to the primal problem, where y = u′(x). Denote by H the corresponding optimal strategy, then Qr

• y, x + (

H · S)T + eT = Q

y, x + (

H · S)T + eT = x + Q

y, eT .

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer

Theorem (Main result) In addition to the conditions for the above theorem, we assume that the filtration is Brownian, then the regular part of the dual

• ptimizer

Qr

• y can be attained by some local martingale

Y ∈ Y(1). Outline of the proof: for simplicity of notation, we drop the subscript y in Q

y.

We prove that the dual optimizer Q can be “approximated” by a sequence {Qn}n∈N from Me(S) such that dQn dP − → d Qr dP , a.s., and Qn, eT − → Q, eT .

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer

For each n, denote by Y n the density process associated with

• Qn. We choose a sequence {

Y n}∞

n=1 of convex combinations

• Y n ∈ conv(Y n, Y n+1, · · · ),

and a non-negative optional strong supermartingale Y , such that Y n

τ −

→ Yτ in probability, for any finite stopping time τ. For each n, denote by Qn the equivalent martingale measure determined by Y n. We define a fictional wealth process by

• W n

t := x + (

H · S)t + E

• Qn[eT |Ft] = x + (

H · S)t + en

τ .

Then, the fictional optimal wealth process can be construct in a similar way as the step above:

• Wt := x + (

H · S)t + et, where for any finite stopping time τ, en

τ −

→ eτ in probability.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer

The process W Y can be proved a martingale. Thanks to the assumption on the filtration, one can find a sequence of stopping times {τk}k∈N such that on 0, τk, W stays above 1/k. Consider a cluster point Q∗ of { Qn}n∈N, which is still a dual

• ptimizer and Q∗r ≡

Qr by the uniqueness. We can prove that

• Y· = d(Q∗|F·)r

dP . Fixing k, (Q∗|Fτk)r, x + ( H · S)τk + eτk = x + Q∗, eT .

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment The dual optimizer

It also can be shown that ( H · S) and e is a “martingale” under the finitely additive measure Q∗, namely, Q∗|Fτk, x + ( H · S)τk + eτk = x + Q∗, eT . Because x + ( H · S)τk + eτk ≥ 1/k, we can compare the above equality with (Q∗|Fτk)r, x + ( H · S)τk + eτk = x + Q∗, eT , and deduce that (Q∗|Fτk)s ≡ 0, which implies that E[ Yτk] = 1. By Scheff´ e’s lemma, we conclude that { Y n

τk}n∈N is uniformly

integrable and thus, Y is a local martingale from Y(1).

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Remarks

In the case that eT = 0, we need a condition on the filtration instead of only assuming that S is continuous. That is because we do not have enough information on the fictional process e so that it is difficult to stop the fictional optimal wealth process W and let it stay away from 0. if we could do better? namely, could we find a martingale associated with the dual optimizer?

• If eT = 0, [Kramkov-Weston, 2015] have a positive answer

under some (Ap) condition over the dual domain.

• If eT is uniformly bounded, [Larsen-Soner-ˇ

Zitkovi´ c, 2015] have a counterexample with a geometric brownian motion stock price process.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Remarks

In the case that U supports the whole real line and S is locally bounded, Bellini, Frittelli, Owen, Schachermayer, ˇ Zitkovi´ c, observe that the dual optimizer does not lose any

• mass. However, it may not be equivalent to P (only

absolutely continuous). If we consider the numeraire based model in a market with proportional transaction cost, we can deduce a similar result when eT = 0, i.e., if S is continuous and satisfying (NUPBR), the dual optimizer is attained by some local martingale from the set of supermatingale deflators. The case that eT is uniformly bounded is under consideration.

Introduction Utility maximization without random endowment Utility maximization with bounded random endowment Remarks

Thank you for your attention!