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#### This paper studies the problem of optimal investment in incomplete markets, robust with respect to stopping times. We work on a Brownian motion framework and the stopping times are adapted to the Brownian filtration. Robustness can only be achieved for logartihmic utility, otherwise a cashflow should be added to the investor s wealth. The cashflow can be decomposed into the sum of an increasing and a decreasing process. The last one can be viewed as consumption. The first one is an insurance premium the agent has to pay.Télécharger gratuit On Robust Utility Maximization pdf

On Robust Utility Maximization

Traian A. Pirvu

Department of Mathematics

The University of British Columbia

Vancouver, BC, V6T1Z2

tpirvu@math.ubc.ca

Ulrich G. Haussmanr0

Department of Mathematics

The University of British Columbia

Vancouver, BC, V6T1Z2

uhaus@math.ubc.ca

February 2, 2008

Abstract. This paper studies the problem of optimal investment in incomplete markets, robust

with respect to stopping times. We work on a Brownian motion framework and the stopping

times are adapted to the Brownian filtration. Robustness can only be achieved for logartihmic

utility, otherwise a cashflow should be added to the investor's wealth. The cashfiow can be

decomposed into the sum of an increasing and a decreasing process. The last one can be viewed

as consumption. The first one is an insurance premium the agent has to pay.

Key words: Portfolio optimization, incomplete markets, minimal martingale measure, Hauss-

mann's formula

JEL classification: Gil

Mathematics Subject Classification (2000): 91B30, 60H30, 60G44

1 Introduction

Dynamic asset allocation has been an important field in modern finance. The ground-breaking

paper in this literature is Merton ^18\. He assumed a utility function of the power type (CRRA)

^Work supported by NSERC under research grant 88051 and NCE grant 30354 (MITACS).

1

and the market consisting of a risk-free asset with constant rate of return and one or more

stocks, each with constant mean rate of return and volatihty. Merton was able to derive closed

form solution for the stochastic control problem of maximizing utility of final wealth.

Karatzas et al. |11J . Cox and Huang [4j establish the static martingale method in solving the

optimal investment and consumption problem in the complete market paradigm. In incomplete

markets perfect risk transfer is not possible, thus people selected some martingale measures

according to some risk criteria. He and Pearson [9] considered the minmax martingale measure

to transform the dynamic portfolio allocation problem into a static one. An auxiliary problem is

analyzed, where the budget constraint is turned into a static constraint using only one martingale

measure, i.e., the final wealth is feasible just under this martingale measure. The minmax

martingale measure is defined to be the martingale measure for which the solution of the auxiliary

problem, coincides with the solution of the original one. Kramkov and Schachermayer [16|

analysed the problem of maximizing utility from final wealth in a general semmimartingale

model by means of duality.

This work combines the problem of utility maximization with the problem of hedging contin-

gent claims in incomplete markets. In incomplete markets, when it comes to pricing and hedging

a crucial issue is how to choose the appropriate martingale or risk-neutral measure. There are

(among others) two main competing quadratic hedging approaches: local risk-minimization and

mean-variance. They give rise to two martingale measures: the minimal martingale measure

and the variance- optimal martingale measure (see Heath et al. |10]). These two martingale

measures coincide if the stock price has independent increments (see Grandits [8]). Moreover

the minmax martingale measure for quadratic utilities is also the variance- optimal martingale

measure (see Ex 5.3 in [5).

Our aim is to understand time consistency in an investor's optimal strategy. According to

Merton [18] an agent with CRRA preferences should invest a constant proportion of her wealth

in the risky assets. Is this strategy time consistent? The answer is NO, if due to some unforseen

events such as death, or getting fired, the agent's investment horizon is some stopping time r.

Dynamic inconsistent behavior was first formalized analytically by Strotz ^20j. The problem

arises if the investor at later dates is free to reconsider her policy. Assume that her investment

horizon is T, but later on she learns it will be changed to some stopping time r < T. Consequently

she may choose to change her investment strategy.

This issue of horizon uncertainty goes back to Levhari and Mirman . One can regard the

stopping time horizon as a major event and by time change, it can be turned into a stochastic

clock. Gol and Kallsen [7] solve the problem of logarithmic utility maximization in a general

semimartingale framework and show that the optimal strategy is robust if one uses a stochastic

clock. In a similar setup Bouchard and Pham [2j extends duality techniques to characterize

the optimal solution. Blanchet-Scalliet et al [Tj considers a random horizon not necessarily a

stopping time. Zitkovic [21] looks at the problem of utility maximization with a stochastic clock

and an unbounded random endowment. Karatzas and Wang [14j treats utility maximization

problem of mixed optimal stopping/control type.

Choulli et all. [3] address the consistency problem in a semmimartingale setup when the

preferences are CRRA. The core idea is that for a stopping time r, the optimal investment of

2

two agents (one can also think of two governments) over time intervals [0, r] and [r, T] is not the

same as the investments of one agent over [0, T] unless they are myopic. Strictly speaking this is

saying that non-logarithmic utility maximization is not robust with respect to stopping times.

As pointed by Choulli et al., the resolution is to add a correction term to investor's wealth.

In a different context Ekeland and Lazrak [5] question time consistency of optimal consump-

tion when discounting is not exponential. It turns out that the policy which is optimal from time

zero perspective will be still optimal at some later time t only if the discounting is exponential.

Thus if the investors at time zero cannot commit to the decision maker at later times t > 0, the

optimal control approach will derive a policy which is impossible to implement. Instead they

use a game-theoretic approach to derive an equilibrium strategy.

Let us notice that the time consistency required by Choulli et al. is stronger because it

involves all stopping times (not just the deterministic ones).

This work, in a Brownian framework, proposes a correction term process which needs to be

added to an investor's wealth to achieve time consistency. The investor's risk preferences are

more general than CRRA, thus it can be seen as an extension of Choulli et al. The correction

term is intimately related to the investor's coefficient of relative risk aversion and coefficient

of prudence. The latter one was introduced by Kimball [15] as a measure of the sensitivity of

choices to risk. In the complete market paradigm we establish the uniqueness of it.

Moreover we go one step farther and show how to finance it. If the coefficient of prudence is

less than twice the coefficient of relative risk aversion the correction term is negative in which

case the agent can just consume it. In general being a process of finite variation it can be

decomposed as the sum of an increasing and a decreasing process. The increasing processes

is an insurance premium the agent has to pay so that her policy is optimal if she would stop

investing at any stopping time r. The terminal value of it, which may be regarded as a contingent

claim, can be represented by the Clark-Haussmann-Ocone formula and a hedging portfolio can

be implemented.

Therefore to invest optimally and robustly with respect to stopping times, the agent should

use some of her initial wealth to finance the hedging portfolio and consume the decreasing com-

ponent and the difference between hedging portfolio and correction term. The same investment

strategy can be implemented in incomplete markets. However since perfect risk transfer is no

longer possible the agent would carry in her portfolio an intrinsic (unhedgeable) risk.

Our main contribution to the field is finding a correction term process for risk preferences

more general than CRRA and establish the uniqueness of it for the case of complete markets.

It would be interesting to characterize in incomplete market all the processes which added to

investor's wealth lead to time consistency.

The remainder of this paper is structured as follow. In Section 2 we introduce the financial

market model and section 3 presents the objective. Sections 4 and 5 treats the complete and

incomplete market case. Section 6 derives the hedging portfolio. We conclude with an appendix

containing some proofs.

3

2 Model Description

2.1 Financial Market

We adopt a model for the financial market consisting of one bond and d stocks. We work in

discounted terms, that is the price of bond is constant and the stock price per share satisfy

< t < oo, i = 1, . . . ,d.

Here W = {Wi, ■ ■ ■ , Wn)'^ is a n— dimensional Brownian motion on a filtered probability space

{0,,{Tt}o<t<T,J^,^)i where {Tt}o<t<T is the completed filtration generated by W. Here we

assume d < n, i.e., there are at least as many sources of uncertainty as assets. As usual

{o.{t)}te[o,oo) = {(Q^j(^))i=i,- - ,rf}tG[o,oo) is an M*^ valued mean rate of return process, and

{fT(t)}jg[o^oo) = {('^ii(^))i=i'... '^}te[o,oo) is an dxn— matrix valued volatility process, progressively

measurable with respect to {.^t}o<t<T-

Standing Assumption 2.1 The process a is uniformly bounded and the volatility matrix a

has full rank. Moreover aa'^ is assumed uniformly elliptic, i.e. KI^ > aa'^ > el^, for some

K>e>0.

This implies the existence of the inverse {a{t)a'^ (t))^^ and the market price of risk process

0» = aT(t)(a(t)aT(t))-ia(t), (2.1)

which is uniformly bounded. All the processes encountered are defined on the fixed, finite

interval [0, T]. The stochastic exponential process

Z{t) = Zg{t) 4 exp | - ^ F{u) dW{u) ^(u) f dnj (2.2)

is a (true) martingale, thus by the Girsanov theorem (section 3.5 in |12j )

W{t) = W{t) + [ e{u) du (2.3)

Jo

is a Brownian motion under the equivalent martingale measure

Q(A) ^E[Z(r)lA], AeJ^T. (2.4)

Definition 2.1 We denote by A4 the set of probability measure satisfying

(i) Q<P and § e L2(P);

(ii) S is a local martingale under Q on [0,T].

In the light of boundedness of 6, it follows that Q £ ^A.

dSi{t) = Siit)

ai{t)dt + 2^aij{t)dWj{t)

4

2.2 Portfolio and wealth processes

A (self-financing) portfolio is defined as a pair (x,7r). The constant x, exogenously given, is the

initial value of the portfolio and vr = (vri, • • • , ird)'^ is a predictable S— integrable process which

specifies how many units of the asset i are held in the portfolio at time t. The wealth process of

such a portfolio is given by

X^'^(t) = x+/ TT{ufdS{u). (2.5)

^0

2.3 Utility Function

A function U : (0, oo) M strictly increasing and strictly concave is called utility function. We

restrict ourselves to utility functions which are 3— times continuous differentiable and satisfy the

Inada conditions

;7'(0+) = limC/'(x) = C50, U'{oo) = lim U'{x) = 0. (2.6)

We shall denote by /(•) the (continuous, strictly decreasing) inverse of the marginal utility

function [/'(•), and by (^Bj

1(0+) 4 hm/(x) = oo, /(oo) = lim I{x) = 0. (2.7)

xlO x^oo

Let us introduce the Legeandre transform of —U{—x)

U{y) ^ sup[[/(x) - xy] = U{I{y)) - yl{y), < y < oo. (2.8)

The function [/(•) is strictly decreasing, strictly convex and satisfies the dual relationships

U'{y) = -x iff U'{x) = -y, (2.9)

and

U{x) = mi[U{y) + xy\ = U{U'{x)) + xU'{x), < x < oo. (2.10)

j/>0

Standing Assumption 2.1

y^\l"{y)\ V {-yl'{y)) V I{y) < hy-'^ for every y G (0,oo), (2.11)

for some ki > 0, k > 0, and a\J h = max(a, h).

3 Objective

Let X be the agent's initial wealth and Vx a process progressively measurable with respect to

{•^i}o<t<T which satisfies T4(0) = 0,

E[ sup \Vx{t)\^] < oo. (3.1)

0<t<T

5

For a given initial positive wealth x and a given utility function U find a process Vx as above

such that for any stopping time t <T there exists a portfolio process tt^ such that

sup EC/(X"'-(r) + 14(r)) = EC/(X-'*(r) + ^^(t)). (3.2)

Here Av{x,t) is the set of admissible portfolios given the time horizon r. It is defined by

Av{x,t) = < vr

X^'^(r) + K(t) > 0, E[[/(X^'"(t) + T4(r))]- < oo, and

is Q- supermartingale, 0<t<T, VQgA4

Remark 3.1 Notice this class of admissible trading strategy includes the classical one

A*y{x, t) = {tt\ + Vx{t)>0, 0<t< t}. (3.3)

Indeed the process X^''^{t) is a local martingale under the probability measure Q. Let T„ | T

be a localizing sequence, u < t < t, and E"^ the expectation operator with respect to a given

martingale measure Q. Holder's inequality and (|3.1|) imp/y E'Q[supQ<^<j. |T4(t)j] < oo. Fatou's

Lemma and (j3.ip yield

■X"'"(t)+ sup |T4(s)| \Tu] < liminfEQ[X"'"(tAT„)+ sup |K(s)| |^n]

0<s<t n^oo 0<s<t

= X^'^(tx) + EQ[sup \V,is)\ I^J,

0<s<t

E^[X^'^(t) \Tu] < X^'^(u), (3.4)

so X^'^[t) is Q- supermartingale, <t <t.

4 The complete market solution

In this section we prove the existence and uniqueness of the process Vx if d = n (complete

market). It will be helpful first to solve the utility maximization of the corrected final wealth at

some stopping time horizon r, i.e.,

sup EUi^r), (4.1)

where

Bv{t,x) = {Cr ■■ measurable, positive, E^^ < x + ET4(r), E[[/(^^)]- < oo\. (4.2)

6

From now on we fix the stopping time r. According to ()2.11h

E sup Z{t)I{\Z{t)) < fciE sup Z{t)

0<t<T 0<t<T

1-K

If K < 1 Burholder-Davis-Gundy (see p. 166 in [12]) in conjunction with boundedness of 6 and

Holder's inequahty yields

E sup Zit)^-"" < oo.

0<t<T

Otherwise ^

Z{t)^-^ = M{tY-^ exp 1^ (k - 1) II e{u) f dnj ,

where M{t) is the local martingale

M{t) = exp 1^ F{u) dW{u) -\ j^W hu) \? du

and the same argument applies. Therefore

E sup Z{t)I{\Z{t)) < oo. (4.3)

0<t<T

Similarly by (j3.ip

E sup Z{t)\V^{t)\ < oo,

0<t<T

whence

E sup Z{t)[I{XZ{t)) - K(t)] < oo, (4.4)

0<t<T

for every A > 0. Therefore the function Xr : (0, oo) — > (— E[Z(r)V^(r)], oo) given by

Xr{X) ^ EZ{t)[I{XZ{t)) - 14(t)], (4.5)

is continuous and strictly decreasing. We shall denote by 3^t-(') its continuous, strictly decreasing

inverse.

Lemma 4.1 The random variable

l^I{yr{x)Z{T)), (4.6)

belongs to Bv{t,x). Moreover for any S,t G Sv{t,x)

Ec/(e.) <Ec/(e;), (4.7)

and ^T- is unique with this property.

Proof: See the Appendix.

□

The above Lemma solves the optimization problem at the level of claims; to obtain the optimal

portfolio we proceed as follows. The martingale E[(^t- — Vx{T))\J^t] admits the stochastic integral

representation

n{ir-Vx{T))\Tt]=x+ [ i:^{u)dW{u), 0<t<T, (4.8)

Jo

for some .Fj— adapted process ip^-) that satisfies \\ip{u)\\'^ du < oo a.s. (e.g., [T3|, Lemma

1.6.7). Let

TTccit) = ia^it))-^m 0<t<T, (4.9)

and notice that

X-.-(r) + V,{t) = t = I{yAx)Z{T)). (4.10)

For any vr € Avi^, t) the corrected final wealth X^''^{t) + Vx{t) G 13v{t, x) and by Lemma l4.ll

sup EU{X^^^t) + Vx{t))=EU{X^'^t) + Vx{t)). (4.11)

Theorem 4.2 There exists a unique process which satisfies V^(0) = and (j3.ip . for which

there exists a portfolio process 7rx{-) such that for any stopping time r,

sup E[/(X^'^(t) + Vx{t)) = E;7(X^'*(r) + V;.(t)). (4.12)

It is given by

Vx{t)= I F{U'{x)Z{u))\\e{u)\\^ du, 0<t<T, (4.13)

where

F{z) = ^l"{z)z^ + r{z)z. (4.14)

The optimal portfolio process frx{-) is

i^xm = -^{{a^{t)r'l'{U'{x)Z{t))U'{x)Z{t)9{t)\, < t < T, i = 1, • • • , d. (4.15)

The proof relies on the following Lemma.

Lemma 4.3 Let Vx be a process satisfying V^(0) =0 and (jS.ip . There exists a portfolio process

7r(-) such that for any stopping time r,

sup EC/(X"'-(r) + Vx{t)) = E[/(X"'*(r) + Vx{t)), (4.16)

if and only if

X^'*(t) + Vx{t) = I{U'{x)Z{t)), 0<t<T. (4.17)

8

Proof: See the Appendix.

□

Proof of Theorem 14.21 Ito's Lemma gives

I{U'{x)Z{t)) = X - [ U'{x)l'{U'{x)Z{u))Z{u)e{u)dW{u) (4.18)

+ / F{U'ix)Z{u))\\9{u)\\'^du.

Jo

For any tt, this and (|2.5p yield

VS) ^ I{U'{x)Z{t))-X'-''^t)

r-t

Tt{u)dS{u)+ F{U'{x)Z{u))\\e{u)f du, (4.19)

for the process

vfi(t) =

^^{t) + ^^{{a^{t))-'i\u'{x)m)zit)em

with -Kx of (j4.15p . If we set vr = vTa;, then tt is the zero vector so Vx{t) = T4(t) of (|4.13p . and

(j4.17p holds. Therefore by Lemma 14.31

sup EC/(X"'-(r) + y,(r)) = EU{X^'^t) + V,{t)), (4.20)

with K and vr^. defined by (fiT3]l and KIM . From (fiT8]l and (|i39l) follows

X^'*(t) =x- f U'{x)I{U'{x)Z{u))Z{u)e{u)dW{u). (4.21)

The inequality

U{I{y))>U{x) + y[I{y)-x\ Vx > 0, y > 0,

follows from ()2.8p and in conjunction with ()2.1ip give

U-{I{y)) <k2 + hy^-^ (4.22)

for some positive /c2, ^3- This and the boundedness of 9 imply that for any r stopping time

E[C/(X"'*(r) + K(t))]- = EC/[/(C/'(x)Z(t))]- < oo.

Moreover by Remark 3.1, t^x £ Av{x.,t). The assumption (j2.1ip show that the process Vx of

(|4.13p satisfies (|3.ip . The uniqueness of Vx is up to translations by S'-integrals, i.e., the process

Vx{t) = Vx{t) + Jq Tt{u)dS{u) is another solution and the corresponding optimal portfolio is

Vr = TTx — TT.

□

9

5 The incomplete market solution

When markets are incomplete we establish existence of the process Vx, but we can no longer

prove uniqueness.

Theorem 5.1 There exists a process Vx which satisfies Vx{0) = 0, and (I3.ip . for which there

exists a portfolio process tTx{-) such that for any stopping time r,

sup EC/(X"'-(r) + Vxir)) = EUiX^'^r) + T4(r)). (5.1)

It is given by

Vx{t)= [ F{U'ix)Z{u))\\e{u)\\'^du, 0<t<T, (5.2)

Jo

where F is the function of ()4.14p . The optimal portfolio fcx is

i^xm = -^{{a{t)a^{t))'^l'{U'{x)Z{t))Z{t)a{t))^, < t < T, i = I, ■ ■ ■ ,d, (5.3)

where a is the mean rate of return process.

Proof: It follows from Ito's Lemma applied to I{U'{x)Z{t)) that

X^'*(t) + Vx{t) = I{U'{x)Z{t)), 0<t<T, (5.4)

with Vx defined in ()5.2|) and tTx of (jS.Sp . The same argument as in the proof of Lemma 14.31

concludes.

□

Remark 5.2 // the utility is logarithmic, i.e., U{x) = logx, then Vx{t) = 0. Being optimal for

the logarithmic utility, the vector vr^ satisfies

Mt))i = , ^ = l,...,d, (5.5)

with CAf(i) — {(^{'t)(^'^ lJ-{t) the Merton proportion. The future evolution of S does not enter

in the formula 115.'^) and 115. 5\) . hence we refer to tTx as the myopic component.

Remark 5.3 Direct computation shows that F of (|4.14p is in fact

FM = \_e::vm , . u'^mr

[C/"(/(y))]2 [ U"{I{y)) ^ U'{I{y))_

and it has the following economic interpretation. It is the difference between the coefficient of

prudence,— jpjjj^^, and twice the coefficient of relative risk aversion, ~jjr^^^- The coefficient

of prudence reflects an individual's propensity to take precautions when faced with risk. The

coefficient of relative risk aversion goes back to Arrow-Pratt and reflects the tendency to avoid

risk altogether.

10

6 The hedging portfoUo

We have learnt from the previous sections that we have to top up investor's wealth with the

process Vx of (|5.2p in order to achieve time consistency of the optimal investment. Let us

compute the process Vx for different utility functions using (|5.2|) . In the case of an exponential

utility, i.e., U{x) = —e~°-^:

yM = - fmu)\\'dn.

a Jo

If the utility is CRRA, i.e., U{x) =

= fo ^^^""^^ "^^^""^

This shows that for different utility functions the process Vx can be either positive or negative.

In the general case it can be decomposed as the difference of two increasing processes Vx{t) =

V+{t) - V-{t), as follows:

V+{t)= f F-'{U'{x)Z{u))\\e{u)\\^du, 0<t<T, (6.1)

Jo

Vx{t)= f F-{U'{x)Z{u))\\e{u)\\^du, 0<t<T, (6.2)

Jo

where as usual a~ = max{— a, 0}, and = max{a, 0}. As we have already seen the agent has

to have her wealth adjusted by the process Vx at all times. The natural question is how can she

achieve this ? We answer this question under the assumption that she starts with some initial

wealth, she does not receive extra funds in the future and the only investment instruments are the

stocks and the bond. In order to hold — {V^~(T')}o<t<T in her portfolio the agent should consume

at rate {F~ {U' (x) Z {t))\\9 (t)\\'^}o<t<T ■ The process {14+(r)}o<4<T is positive and increasing and

in order to finance it, the agent should allocate some of her initial wealth to its generation. One

approach is to create a portfolio, which at the final time T replicates V^{T). The value of

such portfolio at any time t will exceed Vj~{t) and the agent can consume the difference. The

drawback of this methodology is that incomplete markets makes perfect hedging impossible,

hence the replication of V^{T) is impossible. The resolution we propose is to consider a risk-

minimizing strategy as in [6j. Strictly speaking it is the strategy for which the remaining risk

(in hedging the contingent claim V^ (T) ) is minimal under all infinitesimal perturbations of the

strategy at some intertemporal time t (see ^). It can be determined using the Kunita-Watanabe

projection technique and is intimately related to minimal martingale measure which in our setup

is Q. Let us consider the process {V^ {t)}Q<t<T , ^^ti'^) — By Corollary 1 page

181 in [19j or Proposition 4.14 page 181 in [12j one gets the Kunita-Watanabe decomposition of

{V+{t)}o<t<T,

t7+(t)= [ Ttxiu)dS{u)+M{t)+EV^{T), (6.3)

Jo

11

where M is a process orthogonal to S, i.e., (M, S) = and M(0) = 0. Moreover the process M

is a martingale under both P and Q. According to [B], tTx is the risk-minimizing strategy and

we refer to it as the hedging portfolio. The initial cost to implement it is ET4^(T), and requires

some of the initial wealth. Let

X* = mf{z > 0\z + EV^{T) = x},

which by (|2.1ip exists and is positive.

Theorem 6.1 Starting with the initial wealth x, the agent should invest fc^^ + tTx, in stocks,

where

(^x,(t))i = -^(KiV^(t))-'/'(f/'(a:*)Z(t))Z(t)a(t))i, 0<t<T, i = l,--- ,d,

is the myopic portfolio, and tt^^, of (|6.3p is the hedging portfolio. She can consume +

E[T4+(T) — V^^{t)\Tt]- This investment strategy is time consistent up to the intrinsic risk M of

Proof: At any time t, up to M(t) the agent's wealth following the above policy is

x^"^{t) + VxM,

so Theorem 15.11 concludes.

□

In the reminder of this section we show how to compute tTx, explicitly. The martingale V^{t) =

K[V^^{T)\J^t], admits the stochastic integral representation

V+{t)=EV+{T)+ [ (3^{u)dW{u), 0<t<T, (6.4)

Jo

for some .Ft— adapted process /?(•) which satisfies \\P{u)\\'^ du < oo a.s. (e.g., [I3j, Lemma

1.6.7). In light of this we want

t rt

l3{u)dW{u) = / Ttx,{u)dS{u) + M{t), 0<t<T.

JO

We are looking for the orthogonal process M of the form M = 5 ■ W. Let A be the dx n matrix

with the entries Aij = SiGij. Since the volatility matrix a has linearly independent rows, the

matrix A has linearly independent rows, i.e., rank A = d. The processes itx^ and 5 must solve

A^Ttx, + 6 = p, and A6 = 0, (6.5)

12

where the second equahty comes from

(M, Si)= [ V] 6kAk dt= [ {A5)i dt = 0.

Jo Jo

Since Im(^'^) ® Ker(A) = R", we can uniquely decomposed (3 = zi + Z2, with zi G lm.{A^)

and Z2 G Ker(74). Set 5 = Z2 and = y, where zi = A^y ( the uniqueness of tt^^ is due to

V+ = 7t,,-S + M + EV+{T), (6.6)

7f,. =y, and M= Z2 dW. (6.7)

rank A = d). Hence

with

Let us notice that the representation formula in (j6.6p takes place under the probability measure

Q. However by Theorem 14 page 60 in [19] this takes place under P, since P ~ Q. Moreover

{M, S) = and M(0) = under P. The process /3(t) of (16. 4p can be computed explicitly by

Clark-Haussmann-Ocone formula.

7 Appendix

Proof of Lemma 14.11 Let us notice that

E[Z{T){ir - V^{t))] = XriyAx)) = X. (7.1)

Boundedness of 9 and ()4.22p give

whence £ i3y(r, x). Let £ 'Bv(t, x), then by (|7.ip

EZ(T)er < X + EV^{t) = EZ{T)ir.

This and the inequality ()2.10p give

< EU{yrix)Z{T)) + yrix)EZ{T)ir = E[/(|r),

for any ^t- G (r, x) . Uniqueness of is a consequence of the concavity of U.

Proof of Lemma 14.31 Let us first establish the sufficiency. For any stopping time r, and

vr G Avix,T), the process X^'^ is a Q super martingale and Problem 3.26, p. 20 in [12] yields

E[Z(t)X^'^(t)] < X. (7.2)

13

Recall that

X--*(t) = /([7'(x)Z(t))-K(t),

cf (I4.17p . The process X^''^, being a stochastic integral with integrator S, is a local Q— martingale.

Arguing as in (j4.3p one gets Esupo<i<T|I(C/'(x)Z(t))p < cxd, and hence by (|3.ip

EsupQ<^<ji|X^''^(t)P < oo. Holder's inequality implies E,supQ^^^rp\X^''^ {t)\ < oo. Since for any

< t < r, X^'^{t) < supQ<j<2^|X^''^(t)|, by Dominated Convergence Theorem, the process X^''^

is a (true) Q— martingale. Hence for any stopping time r

E[Z(t)X^'"(t)] = X, (7.3)

by Problem 3.26, p. 20 in [12j. Due to concavity of utility function

f/(X"'^(T) + T4(r)) - C/(X"'*(t) + y.(r)) < [/'(X"'*(t) + K(T))(X"'^(r) - X^'*(r))

= C/'(x)Z(r)(X^'"(T) - X^'*(t)).

Taking expectation and using (j7.2p and (j7.3p we get the sufficiency.

As for necessity of (]4.17p . let us assume the existence of a portfolio process 7r(-) such that

(jiT6]) holds. According to (|i30|)

X-'*(r) + K(T) =/(3^.(x)Z(t)),

or

[/'(X-'-(t) + K(r)) = 3^.(x)Z(r), (7.4)

for every stopping time r. In particular

U'{X'''^{t) + V.{t))=yt{x)Z{t), 0<t<T. (7.5)

We claim that yrix) = 3^0(2:^) = U'{x) and this concludes the proof. We establish first ytiix) =

{x) , for every < ti < t2 < T. Arguing by contradiction assume there is < s < t < T, such

that yt{x) 7^ ys{x). Let then A & J^g, and the stopping time f defined by

t ii LO e A

s if a; G A''.

In the light of (fT^I)

where

u'{x^^-{f) + v{f))=y,{x)z{f),

Mx)

yt{x) if a; G ^

y^ix) if ueA'^

However by (|7.4p 3^f(a;) = 3^f(x), a constant, hence 3^s(a;) = 3^t(x) = A, a contradiction so

3^t^(x) = yt2ix) = A, for every < ti < t2 < T. Since t ^ 'Yt(A) is continuous and At (A) = x,

then Afo(A) = x, i.e., 3^t(x) = A = 3^o(3;)-

14

o

Acknowledgements

The authors would hke to thank Professors Tahir Choulh and Ivar Ekeland for helpful dis-

cussions and comments.

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