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A PORTOFOLIO OPTIMIZATION PROBLEMWITH A CORPORATE BOND

BOGDAN IFTIMIE

Communicated by the former editorial board

We consider a portfolio optimization problem for a small investor in a financialmarket given by a savings account, a risky stock and a defaultable asset, forwhich recovery of the market value scheme is valid. We consider logarithmicutility function and power utility function. We provide explicit formula for theoptimal portfolio in the case of logarithmic utility. Following the approach ofJiao and Pham (2010), we decompose the original optimization problem into twosub-problems: pre-default and post-default, which are stated in complete mar-kets. We prove the existence of a solution to the portfolio optimization problemand solve explicitely the post-default problem. For power utility, under the as-sumptions of deterministic coefficients and the occurence almost surely of thedefault till the maturity of the investment process, we provide explicit formulafor the optimal strategy, by using as main tool stochastic control approach.

AMS 2010 Subject Classification: 49L20, 91B28, 93E20.

Key words: portfolio optimization, HARA utility, defaultable bond, recovery ofmarket value, martingale duality, dynamic programming.

1. INTRODUCTION

Portfolio optimization problems constitute a subject of increasing interestin the last decade. Such problems in a dynamic setting in the case of a completemarket with non-defaultable assets and a Brownian filtration were studied first.We mention here the pioneering paper of Merton (1971), who used a stochasticcontrol approach, the so-called dynamic programming principle, consisting inthe derivation of a nonlinear PDE (known as the Hamilton-Jacobi-Bellmanequation) which is verified by the value function associated to the optimizationproblem. Also, of a huge impact was the expository article of Karatzas (1989),which studies more or less the same type of consumption/investment problemsfor a small investor (who cannot affect the prices of the traded assets by hisactions), but in a more general framework. He uses both stochastic controlapproach and convex duality methods, the latter being formulated for a staticproblem (which is posed in a complete market and thus, a replicating portfoliocan be easily obtained via a representation theorem for Brownian martingales).

MATH. REPORTS 15(65), 3 (2013), 287–310

288 Bogdan Iftimie 2

Korn and Kraft (2001) make use of stochastic control methods whenstudying a portfolio optimization problem with a riskless asset and one/severalrisky assets, where the interest rate of the riskless asset (savings account) isstochastic (and not deterministic, as in other models which use HJB equationsas main tool) and follows some linear SDE, such as the well known Vasicek orHo-Lee models. Blanchet-Scalliet, El Karoui, Jeanblanc and Martellini (2008)consider the case of a portfolio optimization problem with a random terminaltime (there are many situations in practice when investors can be forced toliquidate their portfolios in an unexpected manner), where a stopping time ofthe reference filtration F or an independent random time with respect to thesame filtrationis is not necessary. They use both stochastic control and convexduality techniques.

Using a convex duality approach, Kramkov and Schachermayer (1999)derive the existence of the general optimal investment problem in an incompletebut arbitrage free market (where the prices of the assets are given as generalsemimartingales and the set of equivalent martingale measures is non empty,in order to avoid arbitrage), by solving the associated dual problem defined viathe conjugate function of the utility function, but only for utility functions withthe asymptotic elasticity strictly less than 1 (so, the results hold true for thelogarithm utility and for power utilities with risk aversion coefficient strictlyless than 1). We shall use their general abstract result in order to derive theexistence of the solution to our optimization problem.

Bielecki and Jang (2007) consider a financial optimization problem withthree type of assets: a bond, a risky stock and a defaultable asset with constantcoefficients of the dynamics, while Capponi and Figueroa Lopez (2011) considerthe same type of problem, under the assumption that the coefficients of therisky assets are modelled via a multidimensional Markov process in continuoustime with a finite set of possible states, which has to be interpreted as thestates of the economy. In both papers, the dynamic programming approach isused as main tool and explicit formulas for the optimal strategy are obtained.

Our study is dedicated to a portfolio optimization problem in a financialmarket generated by a savings account, a risky asset and a corporate bond,as in the last two cited references. We work under the martingale invariancehypothesis (the so called (H) hypothesis) and we assume also, the existenceof the conditional density of the default time τ . In case of occurence of thedefault, the holder of the defaultable asset will receive a compensation givenby a fraction of the value of the asset just before the occurence of the default,called the fractional recovery of the market value RMV. After the default theasset is not traded anymore.

We obtain a formula for the dynamics of the wealth process. The op-timization problem is solved explicitely in the case of logarithmic utility by

3 A portofolio optimization problem with a corporate bond 289

performing some straightforward computations. In the case of HARA utilityfunctions, we describe a general method of decomposition of the original opti-mization problem into two subproblems, one which is stated before the defaulttime and a second one stated after default, both problems being formulatedin complete markets and for which standard martingale methods may be ap-plied. This method was widely inspired by a paper of Jiao and Pham (2010).In the case of power utility, we characterize the solution of the post-defaultproblem using martingale duality. For the same utility function, under theassumptions of deterministic coefficients and the occurence almost surely ofthe default till the maturity T , we provide explicit formulas. We first solvethe post-default optimization problem with random initial conditions (whichis an optimization problem in random horizon) and afterwards we treat thepre-default problem by the stochastic control approach, by formulating a veri-fication result and establishing also the existence of a classical solution of thecorresponding HJB equation. We mention that we found only a few referencesin the related literature on investment problems in random horizon, Blanchet-Scalliet, El Karoui, Jeanblanc and Martellini (2008) and the unpublished workEl Karoui, Jeanblanc and Huang (2004).

2. THE SETTING

2.1. THE DEFAULT-FREE MARKET

We consider a probability space (Ω,F , P ) endowded with a filtration(Ft)t≥0, given by the right continuous version of the natural filtration gen-erated by a standard one dimensional Brownian motion W (t), properly aug-mented with the P -nul sets. This filtration is called the default-free marketfiltration (or reference filtration). P is known as the historical probability. Weare dealing with a portfolio optimization problem for an investor with invest-ment opportunities into a money market (savings account), a stock and a bondissued by a private corporation which may default at some random time τ . Theinvestment process has a finite horizon T .

The dynamics of the money market account is given through

(1) dR(t) = R(t)r(t)dt.

The stock price process is a geometric Brownian motion

dS(t) = S(t)(µ(t)dt+ σ(t)dW (t)).

We assume that r ≥ 0, µ, σ ≥ 0 are bounded F-adapted procesess, andσ(t) > c, P a.s., for some positive constant c. Set θ(t) = µ(t)−r(t)

σ(t) the marketprice of risk.


We consider also a defaultable asset, for which the default (in case ofoccurence) cannot be predicted, but instead can only be observed (immediatlyafter its occurence). We adopt here the reduced form approach (also knownas the intensity approach) under which the random time τ is not necesarilly astopping time with respect to the default-free market filtration F . We assumethat

• P (τ = 0) = 0 (the default cannot arrive at the initial time);• P (τ > t) > 0 (default can arrive at any time till maturity).

Set Ht := 1(τ≤t) the default indicator process and let G be the smallestfiltration containing the reference filtration F and under which τ is a stoppingtime, i.e.

Gt := Ft ∨ σ(Hs; s ≤ t) = Ft ∨ σ(τ ∧ t).This procedure is known as the progressive enlargement of the filtration F(with the random variable τ), and G is called the enlarged filtration (or alsothe full filtration).

We assume that the financial market defined by the savings account, thestock and a corporate bond (which will be defined in the sequel) is arbitragefree, thus, leading to the existence of at least a risk-neutral probability measure(under which the discounted prices of the traded assets are G-martingales). Dueto the possible arrival of the default (thus, inducing a jump in the dynamics ofthe wealth portfolio), the financial market is incomplete, meaning that the setof equivalent martingale measures contains more than a single element. Let Qbe such a probability measure.

We work under the martingale invariance property (or the immersionproperty), usually called the (H) hypothesis. This means that every square-integrable F-martingale is also a square-integrable martingale under the en-larged filtration G.

In the valuation of defaultable claims (via the martingale approach) wealso assume that the default indicator process (Ht) admits, under Q, a com-pensator which is absolutely continuous with respect to the Lebesgue mea-sure. This means that it exists a nonnegative G-adapted process (λG(t)), withλG(t) = 0 on the set (t > τ), such that the compensated process

(2) M(t) := Ht −∫ t

0λG(s)ds

is a G-martingale under Q. (λG(t)) is called the G-intensity of the default τ (orthe G-hazard rate of τ) under the probability measure Q. It can be writtenunder the form λG(t) = 1(t≤τ)λ

F (t), where the process λF (t) is F-adapted and

is called the F-intensity of default (under Q). Instead of λF we shall simplywrite λ.


We also, denote by λ the F-intensity of default under P , which meansthat the process

(3) M(t) := Ht −∫ t∧τ

0λ(s)ds = Ht −

∫ t

0(1−Hs)λ(s)ds

is a G-martingale under P .

The following remark provides a a situation in which the (H) hypothesisholds, and this is done by the so-called canonical construction of a default timewith a given intensity process.

Remark 1. Assume that on some filtered probability space (Ω,F , (Ft)t≥0,P ) lies a non-negative F-predictable process (γt), and let η be an uniformlydistributed r.v. independent of the filtration F . We define the random timeτ by

τ := inft ≥ 0 |

∫ t

0γsds ≥ η

.

The process (γt) has to be interpreted as the F-intensity of τ . It can be shownthat

P (τ ≤ t|Ft) = P (τ ≤ t|F∞),

which provides a sufficient condition for the (H) hypothesis to hold.

Since Q is absolutely continuous with respect to P on GT , it admits aRadon-Nikodym density ZT := dQ

dP |GT , given by a a positive GT -measurablerandom variable with E(ZT ) = 1. The Radon-Nikodym density process Zt :=E(ZT |Gt) is a G-martingale under P and by virtue of the Predictable Repre-sentation Theorem, which applies to G-martingales ([14]) under the immersionhypothesis, it can be represented as

(4) dZt = Zt−(η(t)dW (t) + γ(t)dM(t)), Z0 = 1,

where η(t), γ(t) are G-predictable processes, with γ(t) > −1.

If X is a (right-continuous) G-semimartingale, the Doleans-Dade stochas-tic exponential process (E(X·)t) is given by

E(X·)t = exp(Xct −

1

2[Xc, Xc]t

) ∏0<s≤t

(1 + ∆Xs),

where Xc is the continuous part of X and ∆Xs = Xs −Xs− is the size of thejump of X at time s. Moreover, E(X·) is the solution of the SDE

dZXt = ZXt−dXt, ZX0 = 1.


Set

Y (t) :=

∫ t

0η(s)dW (s) +

∫ t

0γ(s)dM(s)

=

∫ t

0η(s)dW (s)−

∫ t

0(1−Hs)λ(s)γ(s)ds+

∫ t

0γ(s)dHs.

If t is fixed, Y has at most a single jump in the time interval [0, t] (at time τ),of size ∆Y (τ) = γ(τ)∆Hτ = γ(τ), only if τ ≤ t. Then, if τ ≤ t,∏0<s≤t

(1+∆Ys) = 1+γ(τ) = exp(

ln(1+γ(τ))∆Hτ

)= exp

( ∫ t

0ln(1+γ(s))dHs

).

If τ > t the integral from the last term is equal to 0 and the discontinuous partof Y doesn’t bring any contribution to E(Y·)t. Henceworth, the solution of theSDE (4) is given by the stochastic exponential

Zt = E(Y·)t = exp(− 1

2

∫ t

0η2(s)ds+

∫ t

0η(s)dW (s)

)exp

( ∫ t

0ln(1 + γ(s))dHs −

∫ t∧τ

0γ(s)λ(s)ds

).

(5)

In addition, the process

W (t) := W (t)−∫ t

0η(s)ds

is a standard Brownian motion under Q and the process

(6) Mt := Mt −∫ t∧τ

0γ(s)λ(s)ds = Ht −

∫ t∧τ

0(1 + γ(s))λ(s)ds

is a discontinuous G-martingale under Q, ortogonal to W (t).Comparing now the equations (6) and (3) we are lead to M = M and the

F-intensities of default under Q, respectively P , are related via the formula

λ(t) = (1 + γ(t))λ(t).

Set S(t) := e−∫ t0 r(s)S(t) the discounted price of the risky asset. By Ito’s

formula

dS(t) = S(t)((µ(t)− r(t)) dt+ σ(t)dW (t))

= S(t)((µ(t)− r(t) + σ(t)η(t)) dt+ σ(t)dW (t)).

Since the discounted price of the stock must be a martingale under Q, the dtterm must be equal to zero and thus, η(t) = r(t)−µ(t)

σ(t) := −θ(t). The dynamicsof St under Q is given by

dS(t) = S(t)(r(t)dt+ σ(t)dW (t))


and notice that under an equivalent martingale measure the expected rate ofreturn of the stock is equal to the short interest rate r.

Our next goal is to derive the price process of the defaultable asset. Theassociated dividend process Dt is given by

(7) Dt := X1(τ>T ) + zτ1(τ≤t) = X1(τ>T ) +

∫ t

0zsdHs, t ≤ T,

where X stands for the amount received by the investor (at time T ) in thecase of non-occurence of the default till time T , and (z(t)) is the process ofcompensations if default occurs before T . The quantity Du − Dt representsall cash-flows between times t and u which are received by an investor whichdetains a defaultable bond which is purchased at time t. Then, it is well-known(see [2], Section 8.3) that the price at time t of a defaultable zero-coupon bondwith maturity T is given by the formula

D(t, T ) = EQ( ∫ T

te−

∫ st rududD(s)

∣∣Gt)= EQ

(1(τ>T )e

−∫ Tt ruduX + 1(t<τ≤T )e

−∫ τt rudszτ

∣∣Gt)= 1(τ>t)E

Q(e−

∫ Tt (ru+λu)duX +

∫ T

te−

∫ st (ru+λu)duzsλsds

∣∣Ft),(8)

where EQ stands for the expectation with respect to the probability measure Q.We adopt here the recovery rate at default given by the market value of defaultRMV (see [5] or [2]). It is assumed that at the time τ of occurence of the defaulttill time T the bond cesses to exist and its holder receives a compensation givenby a proportion of the pre-default value of the bond D(τ−, T ), on the set (τ <T ). In this spirit, we consider the recovery process z(t) = (1− L(t))D(t−, T ),where L(t) stands for the loss-rate. We assume that L is deterministic and0 < L(t) < 1, Pa.s.. Then

(9) D(t, T ) = 1(τ>t)EQ(e−

∫ Tt (rs+λsLs)ds)X|Ft

):=1(τ>t)B(t, T )=HtB(t, T ),

where Ht := 1−Ht and B(t, T ) is the pre-default value of the defaultable bondand is equal to the value of a non-defaultable bond with default-risk adjustedinterest rate r(t) := r(t) + λ(t)L(t) and credit spread given by the correctionterm λ(t)L(t). From formula (9) and using the continuity of the pre-defaultvalue process B(t, T ), we obtain the following formula for the recovery valueat default

z(τ) = (1− L(τ))B(τ, T ).

We assume without loss of generality that the face value X is equal to onemonetary unit.


2.2. DYNAMICS OF THE PRICE OF DEFAULTABLE BOND D(t, T )UNDER THE HYSTORICAL PROBABILITY

It is not hard to see that dQdP |Ft = ZF (t), where

ZF (t) = exp

(−∫ t

0θ(s)dW (s)− 1

2

∫ t

0θ2(s)ds

).

Using Bayes’ rule (see [11], Lemma 3.5.3), it follows

m(t) := EQ(e−

∫ T0 r(s)ds

∣∣Ft) = (ZF (t))−1E(e−

∫ T0 r(s)dsZF (T )

∣∣Ft) .Set m(t) := E

(e−

∫ T0 r(s)dsZF (T )

∣∣Ft). We first apply the representation theo-

rem for Brownian Martingales (see [11], Theorem 3.4.2) to the (positive) pro-cess (m(t)), followed by Ito’s formula for the process (ln(m(t))) (for furtherdetails see [15], Proposition 6.1.1.). This leads us to the existence of an F-adapted process (q(t)) s.t.

m(t) = exp(− 1

2

∫ t

0q2(s)ds+

∫ t

0q(s)dW (s)

).

We thus, obtain

B(t, T ) = exp( ∫ t

0r(s)ds

)(ZF (t))−1mt(10)

= exp( ∫ t

0

(r(s) +

1

2(θ2(s)− q2(s))

)ds+

∫ t

0(θ(s) + q(s))dW (s)

).

Finally, if we apply Ito’s rule to the exponential process (B(t, T )), we get

dB(t, T ) = B(t, T )[(r(t) +

1

2(θ2(t)− q2(t))

)ds+ (θ(t) + q(t))dW (t)

+1

2(θ(t) + q(t))2dt

]= B(t, T )[(r(t) + θ(t)β(t))dt+ β(t)dW (t)],

(11)

where β(t) := θ(t) + q(t).Recall now that D(t, T ) = HtB(t, T ) and Ht = M(t) +

∫ t0 Hsλ(s)ds.

Therefore, by Ito’s product rule of differentiation for jump processes,

dD(t, T ) = Ht−dB(t, T ) +B(t−, T )dHt + d( ∑

0<s≤t∆Hs∆B(s, T )

)(12)

= HtdB(t, T )−B(t, T )dHt = HtdB(t, T )−B(t, T )Ht−dHt

= HtB(t, T )((r(t) + θ(t)β(t))dt+ β(t)dW (t)

)−B(t, T )Ht−(dM(t) + λ(t)Htdt)


= D(t, T )((r(t) + θ(t)β(t)− λ(t))dt+ β(t)dW (t)

)−D(t−, T )dM(t)

= D(t−, T )[(r(t) + θ(t)β(t)− λ(t))dt+ β(t)dW (t)− dM(t)

],

by virtue of the continuity of B(·, T ), the formulas dHt = −dHt and H2t = Ht.

We also used the identity dHt = Ht−dHt, or, under integral form,∫ t0 dHs =∫ t

0 Hs−dHs. For τ > t (after the default) both integrals are equal to 0 (sincethere is no jump until t), and for τ ≤ t∫ t

0Hs−dHs = Hτ−∆Hτ = ∆Hτ =

∫ t

0dHs (= 1),

thus, proving the required identity. We are now in position to state

Proposition 1. The price D(t, T ) of the corporate bond has the dynamics

(13) dD(t, T ) = D(t−, T )[(r(t) + θ(t)β(t)− λ(t))dt+ β(t)dW (t)− dM(t)

].

2.3. CONDITIONAL DENSITY OF DEFAULT

In the case of a HARA utility function, we shall follow the general ap-proach of Jiao and Pham [9], which consists in the decomposition of the originaloptimization problem (which is stated in an incomplete market) into two aux-iliary portfolio optimization problems: pre-default and post-default, which arestated in complete markets and for which standard martingale methods orBSDE techniques can be applied. In studying what happens after the default,the notion of intensity of default is not sufficient, but instead the notion ofF-conditional density of the default (see [12] or [9]) will be very useful.

In this spirit, we assume that for each t ∈ [0, T ] and s ≥ 0, there exists afamily of random variables (αt(s)), such that, for fixed s, the process α·(s) isF-adapted, and

P (τ ≤ s|Ft) =

∫ s

0αt(u)du,

or equivalently E(f(τ)|Ft) =∫∞0 f(s)αt(s)ds, for every bounded and

Borel-measurable function f . Furthemore, for a Ft ⊗ B measurable randomvariable Xt(x),

E(Xt(τ)|Ft) =

∫ ∞0

Xt(s)αt(s)ds.

Define Ft := P (τ ≤ t|Ft). Then Gt := 1 − Ft = P (τ > t|Ft) stands for theconditional survival process of τ . It is easily seen that for fixed s, the process(αt(s))0≤t≤T is an Ft-martingale. It can be shown that the F-intensity process


(λ(t)) is completely determined via the conditional densities αt(s),

λ(t) =αt(t)

Ft.

Conversely, we can only partially recover the conditional densities αt(s) startingfrom the F-intensities, and this can be done only for s ≥ t (only for momentst prior to default). If it holds

(14) αT (t) = αt(t), ∀t ∈ [0, T ],

then P (τ ≤ t|Ft) = P (τ ≤ t|FT ), which provides a a sufficient condition forthe (H) hypothesis to hold. We simplify the notation by writing α(t) := αt(t).

We assume in the sequel the existence of F-conditional density of thedefault.

3. THE PORTOFOLIO PROCESS

We consider an investor with investment opportunities in the financialmarket described in the previous section. Set NR(t), NS(t) and ND(t) thenumber of units of each asset (money market, stock and respectively the de-faultable bond) detained by the investor at time t. NR(t), NS(t) and ND(t)are assumed G-predictable processes.

Since ND(t) = ND(t)1(t≤τ) (the bond cesses to exist after the time of de-fault), we may assume that (ND(t)) is F-adapted. This statement is a conse-quence of the (standard) decomposition of any G-predictable process (ψt)0≤t≤T ,

ψt = ψ0t 1(t≤τ) + ψ1

t (τ)1(t<τ<T ).(15)

In this formula, the process ψ0t is F-adapted, for any fixed nonnegative u, the

process ψ1· (u) (indexed over u) is F-adapted, and for fixed t, the mapping

ψ1t (·, ·) is Ft ⊗ B([0, T ]) measurable.

The wealth process XN,x(t) is defined by

XN,x(t) = x+NR(t)R(t) +NS(t)S(t) +ND(t)D(t, T ),

where x is the amount invested at time 0. We assume the self-financing condi-tion imposed to any strategy, which dictates

(16) dXN,x(t) = NR(t)dR(t) +NS(t)dS(t) +ND(t)dD(t, T ).

Set πR(t), πS(t) and πD(t) the corresponding fractions of wealth invested ineach asset, i.e.

πR(t) :=NR(t)R(t)

XN,x(t−), πS(t) :=

NS(t)S(t)

XN,x(t−), πD(t) :=

ND(t)D(t−, T )

XN,x(t−).


Obviously, πR(t)+πS(t)+πD(t) = 1. We identify an investment strategy with aleft-continuous process π(t) := (πR(t), πS(t), πD(t)) and denote by (Xπ,x

t ; 0 ≤t ≤ T ) the corresponding wealth process. The self-financing condition nowreads

(17)

dXπ,x(t) = Xπ,x(t−)

(πR(t)dR(t)

R(t) + πS(t)dS(t)S(t) + πD(t) dD(t,T )D(t−,T )

);

Xπ,x(0) = x.

After default, the corporate bond is not traded anymore, so πD(t) = 0 andD(t, T ) = 0 for t > τ . In this case, we make the convention 0

0 = 0. Taking intoaccount the dynamics of the traded assets, the value process of the portfoliomay be written as

dXπ,x(t) = Xπ(t−)[(r(t) + πS(t)(µ(t)− r(t))

+ πD(t)(λ(t)(γ(t)L(t) + L(t)− 1) + θ(t)β(t)

))dt

+(πS(t)σ(t) + πD(t)β(t)

)dW (t)− πD(t)dM(t)

].

(18)

In fact, π = π(t)1(t≤τ) + π(t)1(τ>t), where π(t) = (πR(t), πS(t), πD(t)) standsfor the pre-default strategy and π(t) = (πR(t), πS(t), 0) stands for the after-default strategy.

The dynamics of the pre-default and post-default wealth processes aregoverned by the equations

dXπ(t) = Xπ(t)[(r(t) + πS(t)σ(t)θ(t) + πD(t)(λ(t)(γ(t)L(t) + L(t)− 1)

+ θ(t)β(t)))dt+ (πS(t)σ(t) + πD(t)β(t)) dW (t)], for t<τ∧T,

(19)

and

(20) dXπ(t) = Xπ(t) [r(t) + πS(t)σ(t)θ(t)] dt+πS(t)σ(t)dW (t), for t ≥ τ∧T.

We denote by A(x) the set of admissible portfolios, which consists in the setof left-continuous portfolio processes (π(t); 0 ≤ t ≤ T ) such that

E

∫ T

0π2S(t) <∞, E

∫ T

0π2D(t) <∞ and πD(t) < 1, 0 ≤ t ≤ T, Pa.s.

Notice that the wealth will remain positive at each moment t, P a.s. It isnatural to assume that the investor won’t invest all his capital in the defaultableasset, due to the high probability that he’ll suffer big losses otherwise.

We consider utility functions U : (0,∞) → R, which are differentiable,strictly increasing and strictly concave, satisfying also the usual Inada condi-tions: limx→0 U

′(x) = ∞ and limx→∞ U′(x) = 0. The logarithmic utility and

the power utility clearly satisfy the properties listed above. The investor isinterested in maximizing his expected utility (under the historical probability)from the final wealth over the class A(x) of admissible portfolios. We are thus,


lead to the optimization problem

(21) V (x) := supπ∈A(x)

E[U(Xπ,x(T ))] = supπ∈A(x)

Jx(π).

4. LOGARITHMIC UTILITY

Let Y π(t) be the term appearing under square brackets in the formula(18), which we rewrite as

Y π(t) =(r(t) + πS(t)σ(t)θ(t) + πD(t)

(λ(t)(γ(t)L(t) + L(t)−Ht)

+ θ(t)β(t)))

dt+(πS(t)σ(t) + πD(t)β(t)

)dW (t)− πD(t)dHt.

The value of the wealth process is obviously given by the stochastic exponentialof (Y π(t)), i.e.

Xπ,x(t) = E(Y π(·))t = x exp( ∫ t

0

(r(s) + πS(s)σ(s)θ(s) + πD(s)(λ(s)

(γ(s)L(s) + L(s)−Hs) + θ(s)β(s))− 1

2(πS(s)σ(s)

+ πD(s)β(s))2)ds)

exp( ∫ t

0

(πS(s)σ(s) + πD(s)β(s)

)dW (s)

)× exp

( ∫ t

0ln(1− πD(s))dHs

),

(22)

since obviously, for fixed t, the size of the jump of Y π in the point τ (jumparises only if τ ≤ t) is ∆Y π(τ) = −πD(τ). We further proceeded in the sameway we derived the explicit formula (5) for the Radon-Nikodym density process(Zt). We get

ln (Xπ,x(t)) = ln(x) +

∫ t

0

[r(s) + πS(s)σ(s)θ(s) + πD(s)(λ(s)(γ(s)L(s)

+ L(s)−Hs) + θ(s)β(s))− 1

2(πS(s)σ(s) + πD(s)β(s))2

+ Hsλ(s) ln(1− πD(s))]ds

+

∫ t

0

(πS(s)σ(s) + πD(s)β(s)

)dW (s) +

∫ t

0ln(1− πD(s))dM(s),

where we rewrote the integral∫ t0 ln(1− πD(s))dHs by taking into account the

formula dHt = dMt+ Htλ(t)dt. Under our assumptions the stochastic integralprocesses appearing in the last line of the multiline formula from above aretrue (not only local) martingales of zero expectation. Thus,

Jx(π) = ln(x) + E∫ T

0

[r(t) + πS(t)σ(t)θ(t) + πD(t)(λ(t)(γ(t)L(t)


+ L(t)−Ht) + θ(t)β(t))− 1

2(πS(t)σ(t) + πD(t)β(t))2

+ Htλ(t) ln(1− πD(t))]dt

= ln(x) + E( ∫ T

0gt(πS(t), πD(t))dt

),

with the mapping gt(y, z) properly defined. We are thus, lead to solve a path-wise optimization problem, which is splitted into the pre-default optimizationproblem and the post-default one. When dealing with the former, define, fort ≤ τ ∧ T , the random function

gt(y, z) = r(t) + σ(t)θ(t)y + (λ(t)(γ(t)L(t) + L(t)) + θ(t)β(t))z

− 1

2(σ(t)y + β(t)z)2 + λ(t) ln(1− z),

for y ∈ R and z < 1. For t > τ ∧T (after the default), πD = 0 (the defaultableasset is not traded anymore, once default occured) and also Ht = 0. We thus,define

gt(y) = r(t) + σ(t)θ(t)y − 1

2σ(t)2y2.

We compute now the (absolute) maximum point of the (random) function

gt(y, z) = gt(y, z)1(t≤τ∧T ) + gt(y)1(t>τ∧T ).

For the pre-default problem, first order optimality conditions read

∂gt∂y

(y, z) = σ(t)θ(t)− (σ(t)y + β(t)z)σ(t) = 0,

and

∂gt∂z

(y, z) = λ(t)(γ(t)L(t) + L(t)) + θ(t)β(t)− (σ(t)y + β(t)z)β(t)− λ(t)

1− z= 0.

First equation implies

σ(t)y∗ + β(t)z∗ = θ(t),

and inserting in the second equation we get

z∗(t) = 1− 1

L(t)(γ(t) + 1)< 1.

The term (γ(t) + 1) can be interpreted as the ratio between the default inten-sities with respect to Q, respectively P . We also obtain

y∗(t) =θ(t) + q(t)(1− L(t)− L(t)γ(t))

σ(t)L(t)(1 + γ(t)).

Writing down the Hessian matrix associated to gt, it follows easily that gtattains its maximum value at the point (y∗(t), z∗(t)). Obviously, the point


y∗(t) = θ(t)σ(t) is the maximum point of gt. We are now in position to state the

main result of this section.

Theorem 1. Assume that U(x) = ln(x). Then the strategy π∗t = (π∗S(t),π∗D(t)) given by

(23) π∗S(t) =(θ(t) + q(t)(1− L(t)− L(t)γ(t))

σ(t)L(t)(1 + γ(t))

)1(t≤τ∧T ) +

θ(t)

σ(t)1(t>τ∧T ),

and

(24) π∗D(t) =(

1− 1

L(t)(1 + γ(t))

)1(t≤τ∧T )

is optimal for the problem (21).

5. EXISTENCE OF A SOLUTION TO THE OPTIMIZATION PROBLEM

Using Theorem 2.2 Kramkov and Schachermayer (1999), we know thatour optimization problem admits a solution under the assumptions

(i) The asymptotic elasticity of the utility function U(x) satisfies

AE(U) := lim supx→∞

xU ′(x)

U(x)< 1;

(ii) There exist at least an equivalent (local) martingale measure;

(iii) The value function V (x) is finite for some x > 0 .

The assumption (i) is obviously satisfied for our choices of utility func-tions. We assumed the existence of an EMM Q so (ii) is also satisfied. Next,according to the result we just cited, V (x) is finite for some positive x if theconjugate function of the value function V , denoted V ∗, is finite at the pointy = V ′(x). A sufficient condition for the last assertion to hold is that

(25) E[U∗(yZT )] <∞, for some y > 0,

where U∗ stands for the convex conjugate function (or the Legendre-Fencheltransform) of U , defined by

U∗(y) = supx≥0

(U(x)− yx), y > 0.

We assume in the subsequent that formula (25) holds tue. Theorem 2.2 from[13] allows us to provide a dual characterization of the value function and theassociated optimal portfolio but no explicit formulas for the value function (orthe optimal strategy).


We follow now the arguments of Jiao and Pham [9]. If π = (π, π) is anadmissible strategy, we denote by (Xπ,s,x(t)), s ≤ t ≤ T , the solution of SDE(20) starting at time s from the state Xπ,x(s)(1− πs).

Jx(π) = E[U(Xπ,x(T ))] = E[E(U(Xπ,x(T ))1(τ>T ) + U(Xπ,τ,x(T )1(τ≤T )|FT

)]= E

[U(Xπ,x(T ))P

(τ > T |FT

)]+ E

[E(U(Xπ,τ,x(T ))1(τ≤T )|FT

)]= E

[U(Xπ,x(T ))GT

]+ E

[ ∫ T

0U(Xπ,s,x(T ))αT (s)ds

].

We used the fact that any GT -measurable r.v. Y can be represented as

Y =E(Y 1(τ>T )|FT

)GT

1(τ>T ) + Y (τ)1(τ≤T ),

where Y (τ) is FT ⊗ σ(τ)-measurable. We also have

(26) E[ ∫ T

0U(Xπ,s,x(T ))αT (s)ds

]= E

[ ∫ T

0E(U(Xπ,s,x(T ))αs|Fs

)ds].

We define the after-default optimization problem

(27) Vs(x) = supπS

E(U(XπS ,x

s (T ))αs∣∣Fs) = sup

πS

Js(πS , x),

where the after-default value process (XπS ,xs (t)), s ≤ t ≤ T leaves at time s

from the state x. This problem is not a classical portfolio optimization problemsince we maximize the utility of the final wealth, weighted by some randomfunction. Using Jiao and Pham [9], Theorem 3.1, the following dynamic pro-gramming type formula is valid

(28) V (x) = supπS ,πD

E[U(Xπ,x(T ))GT +

∫ T

0Vs(X

π,x(s)(1− πD(s)))ds].

Remark 2. From the last equation we deduce that in order to solve the op-timization problem (21), it is sufficient to solve two optimization sub-problems:the after-default problem and the pre-default problem, both problems beingstated in complete markets. It is easy to see that we have to solve first theproblem after-default (27), and afterwards we insert the value function of thisproblem in the formula (28). Resolution of the latter poses many technicaldifficulties and we shall restrict ourselves only to the resolution of the former.

5.1. THE AFTER-DEFAULT OPTIMIZATION PROBLEM

Recall also the formula of the Radon-Nykodim density

ZF (t) = exp(− 1

2

∫ t

0θ2(u)du−

∫ t

0θ(u)dW (u)

),


and set Zts := ZF (t)ZF (s)

, for t ≥ s. Let s be fixed and define the probability

measure Qs on FT by dQs

dP |FT := ZTs . Obviously, dQs

dP |Ft = Zts. Set also

Hts = e−

∫ ts r(u)duZts = exp

(−∫ t

s(r(u)+

1

2θ2(u))du−

∫ t

sθ(u)dW (u)

), for t ≥ s.

Hts is called deflator. It is easy to see that the discounted value of the wealth is

obviously a G- (local) martingale (under Qs) which takes positive values, andhence, it is a supermartingale. It follows

(29) E(HTs X

πS ,xs (T )|Fs) ≤ E(Hs

sXπS ,xs (s)) = x.

Let U∗ be the convex conjugate function of U . If we denote I(y) := (U ′)−1(y),then

U∗(y) = U(I(y))− yI(y).

The function I maps (0,∞) onto (0,∞) and is strictly decreasing.

Through this section, we consider only the case of the power utility

U(x) = xp

p . An elementary computation shows that I(y) = y1p−1 and U∗(y) =

1−pp y

pp−1 . We can also write U∗(y) = −yq

q , where q is the dual conjugate of p

(i.e. 1p + 1

q = 1).

We take now into account the completeness of the after-default problemand transform it into a static optimization problem. Heuristically, we definethe Lagrangian

L(X,λ) := αsU(X) + λ(x−HTs X)

“Differentiation” with respect to X yields

αsU′(X)− λHT

s = 0,

from which we get

X = I(λHT

s

αs

).

We impose now that X from the above formula satisfies the restriction (29) asan equality. We obtain

(30) E(HTs I(λHT

s

αs

)|Fs)

= x,

and we want to solve this equation with respect to λ. Set

g(λ) := E(HTs I(λHT

s

αs

)∣∣Fs) =λ

1p−1

α1p−1s

E((HT

s )q∣∣Fs).

We prove now that g takes finite values. Let δ be an arbitrary real number andset ZTs (δ) := E

( ∫ ·s(−δθ(t))dW (t)dt

)T

. Set also Qsδ the probability measure


equivalent to P given by the Radon-Nikodym derivativedQsδdP

∣∣FT = ZTs (δ).Then

E((HT

s )δ)

= E[ZTs (δ) exp

((−δ

2)

∫ T

s(r(t) + (1− δ)θ2(t))dt

)]= EQ

sδ[

exp((−δ

2)

∫ T

s(r(t) + (1− δ)θ2(t))dt

)]≤ exp

( |δ|2T (‖r‖T + |1− δ|‖θ‖2T )

)< +∞,

due to the boundedness assumptions imposed on the coefficients of the model.

It can be shown in a standard way that g is strictly decreasing and con-tinuous on (0,∞) and limλ0 g(λ) =∞, limλ→∞ g(λ) = 0. It follows that theequation (30) admits an unique positive solution λ∗ = g−1(x). Set

(31) X∗ = I(λ∗HT

s

αs

).

Let now Xπ be an admissible wealth portfolio (which satisfies the restric-tion (29)). Applying the (standard) inequality which is satisfied by any convexdifferentiable function h

h(y)− h(x) ≤ (y − x)h′(x),

to the utility function U and the points x = X∗, y = Xπ, taking afterwardsthe expectation and considering also the admissibility formula (29) will leadus to the optimality of X∗.

We are now in position to state

Theorem 2. The after-default optimization problem (27) admits the op-timal final wealth given by

X∗ = I(g−1(x)HT

s

αs

)=

(g−1(x))1p−1

α1p−1s

(HTs )

1p−1 .

Moreover, an optimal portfolio π∗ has the form

(32) π∗(t) =θ(t)

σ(t)+η∗(t)

σ(t), for t ≥ s,

with some F-adapted process η∗ which will be specified below.

Proof. The proof of the theorem is complete if we show now that theclaim X∗ defined in (31) is hedgeable, i.e. if it exists a portfolio π∗S s.t. the the

associated wealth process (Xπ∗S ,xs (t))s≤t≤T satisfies X

π∗S ,xs (T ) = X∗. Assume

for the moment the existence of a replicating portfolio. We know that the


discounted value of the associated wealth process is an F-martingale underQs. Hence,

e−∫ ts r(u)duX

π∗S ,xs (t) = EQ

s(e−

∫ Ts r(u)duX∗

∣∣Ft) =1

ZtsM∗s (t),

where the process M∗s (t) := E(e−

∫ Ts r(u)duX∗ZTs

∣∣Ft), defined for t ≥ s is

an F-martingale under P . Using Lamberton and Lapeyre ([15]), Proposi-tion 6.1.1. we deduce the existence of an F-adapted square integrable pro-cess η∗ s.t. M∗s (t) can be written as the Doleans-Dade stochastic exponentialE( ∫ ·

0 η∗(s)dW (s)

)t, i.e. M∗s (t) is the solution of the SDE

dM∗s (t) = M∗s (t) η∗(t)dW (t).

On the other hand, by virtue of Ito’s formula of integration by parts, we deduce

dM∗s (t) = d(Zts e

−∫ ts r(u)duX

π∗S ,xs (t)

)= Zts e

−∫ ts r(u)duX

π∗S ,xs (t)(π∗S(t)σ(t)− θ(t))dW (t).

Finally, comparing the last two equations we obtain the desired formula (32)for the optimal strategy π∗S , which is obviously admissible.

6. EXPLICIT FORMULAS IN A PARTICULAR CASE

Throughout this section, we assume that r, µ, σ, λ, γ, q, α are boundeddeterministic functions, the utility function is of power type and also that thedefault will occur almost surely till the maturity T of the investment process,i.e.

(33) P (0 < τ < T ) = 1.

Last assumption seems natural in a period of crisis, for a far enough horizonT . Since, by the conditional density assumption, it holds

P (0 < τ < T ) = E[E(1(0<τ<T )|FT

)]= E

( ∫ T

0αT (u)du

)=

∫ T

0αudu,

we notice that formula (33) is equivalent with

(34)

∫ T

0αudu = 1.

If π is an admissible strategy, π = (π, π), then the terminal value of the wealthis given by

Xπ(T ) = Xπ(T ) = Xπ(τ) exp( ∫ T

τ

(r(s) + πS(s)σ(s)θ(s)− 1

2π2S(s)σ2(s)

)ds)(35)


× exp( ∫ T

τπS(s)σ(s)dW (s)

),

where

Xπ(τ) = Xπ(τ)(1− πD(τ)).

6.1. THE POST-DEFAULT OPTIMIZATION PROBLEM

We consider the dynamic version of the value function of post-defaultoptimization problem

V (t, x) = supπ∈A(t,x)

E[(Xπ(T ))p|Xπ(t) = x

].

Since we are interested in solving this problem with random initial conditions(for t = τ and x = Xπ(τ)), we shall solve instead the optimization problem

(36) V (τ, η) = supπ∈A(τ,η)

E[(Xπ,τ,η(T ))p

]= sup

π∈A(τ,η)J(τ, η, π),

where η is a positive Gτ -measurable random variable and Xπ,τ,η(t), τ ≤ t ≤ Tstands for the solution of the equation (20), starting at the random time τfrom the random state η. Obviously,

(Xπ(T ))p = ηp exp(p

∫ T

τ

(r(s) + πS(s)σ(s)θ(s)− 1− p

2π2S(s)σ2(s)

)ds)

× E(∫ ·

τpπS(s)σ(s)dW (s)

))T.

(37)

Set π∗t the maximizer of the second order polynomial function

ht(y) = r(t) + σ(t)θ(t)y − 1− p2

σ2(t)y2,

which is obviously given by

(38) π∗t =1

1− pθ(t)

σ(t).

Proposition 2. The (deterministic) portfolio given in the formula (38)is optimal for the optimization problem (36).

Proof. Let π an arbitrary chosen element of A(τ, ξ). Set

Mπt = E

( ∫ ·τpπS(s)σ(s)dW (s)

)t, τ ≤ t ≤ T.

The process of the stochastic integral( ∫ ·

τ pπS(s)σ(s)dW (s))

is a G-martingale(due to our boundedness assumptions on the coefficients of the model) and


thus, the stochastic exponential (Mπt ; t ≥ τ) is also a martingale, with prime

element Mπτ = 1. Henceworth,

J(τ, η, π) = E[ηp exp

(p

∫ T

τhs(πs)ds

)MπT

]≤ E

[ηp exp

(p

∫ T

τhs(π

∗s)ds

)MπT

]= E

E[ηp exp

(p( ∫ T

0hs(π

∗s)ds−

∫ τ

0hs(π

∗s)ds

))MπT

∣∣Gτ]= E

ηp exp

(p( ∫ T

0hs(π

∗s)ds−

∫ τ

0hs(π

∗s)ds

))E[MπT

∣∣Gτ]= E

[ηp exp

(p( ∫ T

τhs(π

∗s)ds

))Mπτ

]= E

[ηp exp

(p( ∫ T

τhs(π

∗s)ds

)E(Mπ∗T

∣∣Gτ))]= E

[ηp exp

(p( ∫ T

τhs(π

∗s)ds

))Mπ∗T

]= J(τ, η, π∗).

(39)

We used Doob’s optional sampling theorem, the fact that the strategy π∗ andthe functions ht are deterministic and also, Mπ

τ = 1 = Mπ∗τ .

Remark 3. Notice that in solving the post-default optimization problem(36) the assumption of existence of conditional density of τ is not required. It isalso sufficient to assume that only the coefficients of the assets are deterministic.

For an admissible strategy π ∈ A(x) define

J(x, π) := E [(Xπ,x(T ))p] .

We saw that π can be represented as πt = πt1(t<τ) + πt1(t≥τ). Notice that

J(x, π) = J(τ,Xπ(τ)(1− πD(τ)), π) ≤ J(τ,Xπ(τ)(1− πD(τ)), π∗).

We obtained in the multi-lines formula from above, by conditioning with re-spect to Gτ

J(τ, η, π∗) = E[ηp exp

(p(G∗T −G∗τ )

)],

where

G∗t :=

∫ t

0hs(π

∗s)ds =

∫ t

0

(r(s) +

1

2(1− p)θ2(t)

σ2(t)

)ds.


6.2. THE PRE-DEFAULT OPTIMIZATION PROBLEM

In order to solve the optimization problem (21), it is sufficient to find thevalue function of the pre-default optimization problem

(40) V (x) = supπJ(τ,Xπ(τ)(1− πD(τ)), π∗),

where by an abuse of notation we set π(t) = (πS(t), πD(t)).We are thus dealing with an optimization portfolio investment problem

in a random horizon. We found in the literature only a few references dealingwith this subject, from which we mention Blanchet-Scalliet et al. [3] and theunpublished paper El Karoui, Jeanblanc and Huang [6]. In the second citation,the authors use the BSDE approach, a crucial assumption being P (τ ≤ T ) < 1,hence, the results obtained there are not usefull in our setting. In the first ci-tation, the authors are dealing with a portfolio optimization problem involvingseveral risky assets with prices modelled by geometric Brownian Motions, un-der the main assumptions of deterministic bounded coefficients of the modeland deterministic density of the conditional distribution function of the ran-dom horizon, using dynamic programming approach and also the martingaleapproach. As in this paper, we shall state a verification result by writingproperly the Hamilton-Jacobi-Bellman nonlinear PDE which is satisfied by thevalue function and provide also an optimal strategy.

By virtue of the conditional density assumption

E[(Xπ(τ)(1− πD(τ))

)pexp

(p(G∗T −G∗τ )

)]= E

E[(Xπ(τ)

)p(1− πD(τ))p exp

(p(G∗T −G∗τ )

)∣∣FT ]= E

∫ T

0(Xπ(t))p (1− πD(t))p exp (p(G∗T −G∗t ))αtdt.

We define a dynamic version of the value function V (x) by setting

V (t, x) = supπE

∫ T

t(Xπ(s))p (1− πD(s))p exp (p(G∗T −G∗s))αsds,

where (Xπ(s)) is the solution of the equation (19) starting at time t from thestate x. Thus, (Xπ) is the solution of the controlled linear SDE

dX(t) = f(t,X(t), πt)dt+ σ(t,X(t), πt)dW (t), πt = (πS(t), πD(t)),

with coefficients f(t, x, π) and σ(t, x, π) given by

f(t, x, π) = x[r(t) + σ(t)θ(t)π1 +

(λ(t)(γ(t)L(t) + L(t)− 1) + θ(t)β(t)

)π2],

andσ(t, x, π) = x

(σ(t)π1 + β(t)π2

).


Our optimization problem has the running cost functional

F (t, x, π) = xp(1− π2)p exp (p(G∗T −G∗t ))αt.

The admissible region for the control variable π is given by U = R× (−∞, 1).Obviously, |F (t, x, π)| ≤ K(1 + |x|) (since p ∈ (0, 1)).

Set δ(t) := exp (p(G∗T −G∗t ))αt and ρ(t) := λ(t)(γ(t)L(t) +L(t)−1). Westate now a verification theorem, which is based on a general result (see [7]),Chapter IV, Theorem 3.1). All the required assumptions are clearly fulfilledin our setting. We establish also the existence of a classical solution of thecorresponding HJB equation.

Theorem 3. Consider the following Hamilton-Jacobi-Bellman nonlinearPDE

(41)

∂W∂t (t, x) + supπ∈U H(t, x, π) = 0;

W (T, x) = 0.

where the Hamiltonian H is defined through

H(t, x, π) := f(t, x, π)∂W

∂x(t, x) +

1

2σ2(t, x, π)

∂2W

∂x2(t, x) + F (t, x, π)

= x[r(t) + σ(t)θ(t)π1 +

(ρ(t) + θ(t)β(t)

)π2] ∂W∂x

(t, x)

+1

2x2(σ(t)π1 + β(t)π2

)2 ∂2W∂x2

(t, x) + xp(1− π2)pδ(t).

(42)

Let K(t) be the unique solution of the ODE of Bernoulli’s type(43)

K ′(t) + p(r(t) + ρ(t) +

θ2(t)

2(1− p))K(t) + (1− p)ρ(t)

pp−1

δ(t)1p−1

K(t)pp−1 = 0, t ∈ [0, T ),

with terminal Cauchy condition K(T ) = 0. K(t) has the explicit form given by

K(t) = exp(− p

1− p

∫ t

0(r(s) + ρ(s) +

θ2(s)

2(1− p)ds))

×∫ T

t

[ρ(s)pp−1

δ(s)1p−1

exp( p

1− p

∫ s

0(r(u) + ρ(u) +

θ2(u)

2(1− p))du)]

ds

=

∫ T

t

[ρ(s)pp−1

δ(s)1p−1

exp( p

1− p

∫ s

t(r(u) + ρ(u) +

θ2(u)

2(1− p))du)]

ds.

(44)

Then(a) The function W (t, x) := xpK(t) ∈ C1,2([0, T ] × R) is a solution of the

equation (41) and

(45) V (t, x) = W (t, x),∀t ∈ [0, T ], x ∈ R.


(b) In addition, we assume that L(t) > 11+γ(t) , for any t ∈ [0, T ]. Then the

optimal post-default strategy π∗ is given by

(46) π∗S(t) =1

1− pθ(t)

σ(t)− β(t)

σ(t)+β(t)

σ(t)

(ρ(t)

δ(t)K(t)

) 1p−1

and

(47) π∗D(t) = 1−(ρ(t)

δ(t)K(t)

) 1p−1 .

Proof. In order to determine the optimal strategy π∗ we write the firstorder optimality equations

∂H

∂π1= xσ(t)θ(t)

∂W

∂x(t, x) + x2

(σ(t)π1 + β(t)π2

)σ(t)

∂2W

∂x2(t, x) = 0,

and

∂H

∂π2= x

(ρ(t) + θ(t)β(t)

)∂W∂x

(t, x) + x2(σ(t)π1 + β(t)π2

)β(t)

∂2W

∂x2(t, x)

− pxp(1− π2)p−1δ(t) = 0.

We assume for the moment that the function W (·, t) is strictly increasing andstrictly concave, for any t ∈ [0, T ]. Elementary computations lead us to thefollowing candidates for the optimal strategies

π∗D(t) = 1− 1

x

(1

p

ρ(t)

δ(t)

∂W

∂x(t, x)

) 1p−1 ,

and

π∗S(t) = − θ(t)σ(t)

∂W∂x (t, x)

x∂2W∂x2

(t, x)− β(t)

σ(t)+

1

x

β(t)

σ(t)

(1

p

ρ(t)

δ(t)

∂W

∂x(t, x)

) 1p−1 .

It follows, that the Hamiltonian H is strictly concave with respect to π, andthus π∗(t) := (π∗S(t), π∗D(t)) is the maximum point for the mapping π →H(t, x, π), for fixed t, x.

We prove now that the equation (41) has a solution W (t, x) of the formW (t, x) = xpK(t) (for which separation of variables holds). Replacing in theequation (41) π1 and π2 with π∗S(t) and π∗D(t) we get, after straightforwardcomputations the ODE (43), which admits the unique solution K(t) given bythe formula (44). Notice that K(t) takes positive values on the time interval[0, T ). Hence, W is strictly increasing and strictly concave with respect to xand has the required regularity properties. The assertions (a) si (b) follow nowas a direct consequence of Theorem 3.1 from [7].

Acknowledgments. The research of the author was supported by the Sectorial Op-erational Programme Human Resources Development (SOP HRD), financed from theEuropean Social Fund and by the Romanian Government under the contract numberSOP HRD/89/1.5/S/62988.


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Received 25 July 2013 Academy of Economic Studies,Department of Applied Mathematics,

010874 Bucharest, Romania,“Simion Stoilow” Institute of Mathematics

of Romanian Academy,015700 Bucharest, Romania,

[email protected]

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