A PORTOFOLIO OPTIMIZATION PROBLEM WITH A CORPORATE BOND

WITH A CORPORATE BOND

BOGDAN IFTIMIE

Communicated by the former editorial board

We consider a portfolio optimization problem for a small investor in a financial market given by a savings account, a risky stock and a defaultable asset, for which recovery of the market value scheme is valid. We consider logarithmic utility function and power utility function. We provide explicit formula for the optimal portfolio in the case of logarithmic utility. Following the approach of Jiao and Pham (2010), we decompose the original optimization problem into two sub-problems: pre-default and post-default, which are stated in complete mar- kets. We prove the existence of a solution to the portfolio optimization problem and solve explicitely the post-default problem. For power utility, under the as- sumptions of deterministic coefficients and the occurence almost surely of the default till the maturity of the investment process, we provide explicit formula for 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 of market value, martingale duality, dynamic programming.

1. INTRODUCTION

Portfolio optimization problems constitute a subject of increasing interest in the last decade. Such problems in a dynamic setting in the case of a complete market with non-defaultable assets and a Brownian filtration were studied first.

We mention here the pioneering paper of Merton (1971), who used a stochastic control approach, the so-called dynamic programming principle, consisting in the derivation of a nonlinear PDE (known as the Hamilton-Jacobi-Bellman equation) which is verified by the value function associated to the optimization problem. Also, of a huge impact was the expository article of Karatzas (1989), which studies more or less the same type of consumption/investment problems for a small investor (who cannot affect the prices of the traded assets by his actions), but in a more general framework. He uses both stochastic control approach and convex duality methods, the latter being formulated for a static problem (which is posed in a complete market and thus, a replicating portfolio can be easily obtained via a representation theorem for Brownian martingales).

MATH. REPORTS15(65),3(2013), 287–310

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

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

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

Our study is dedicated to a portfolio optimization problem in a financial market generated by a savings account, a risky asset and a corporate bond, as in the last two cited references. We work under the martingale invariance hypothesis (the so called (H) hypothesis) and we assume also, the existence of the conditional density of the default time τ. In case of occurence of the default, the holder of the defaultable asset will receive a compensation given by 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 the asset 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

performing some straightforward computations. In the case of HARA utility functions, we describe a general method of decomposition of the original opti- mization problem into two subproblems, one which is stated before the default time and a second one stated after default, both problems being formulated in 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-default problem using martingale duality. For the same utility function, under the assumptions of deterministic coefficients and the occurence almost surely of the default till the maturity T, we provide explicit formulas. We first solve the post-default optimization problem with random initial conditions (which is an optimization problem in random horizon) and afterwards we treat the pre-default problem by the stochastic control approach, by formulating a veri- fication result and establishing also the existence of a classical solution of the corresponding HJB equation. We mention that we found only a few references in the related literature on investment problems in random horizon, Blanchet- Scalliet, El Karoui, Jeanblanc and Martellini (2008) and the unpublished work El 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 (F_t)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 market filtration (orreference filtration). P is known as the historical probability. We are dealing with a portfolio optimization problem for an investor with invest- ment opportunities into a money market (savings account), a stock and a bond issued by a private corporation which may default at some random timeτ. The investment 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, and σ(t) > c, P a.s., for some positive constant c. Setθ(t) = ^µ(t)−r(t)_σ(t) the market price of risk.

We consider also a defaultable asset, for which the default (in case of occurence) cannot be predicted, but instead can only be observed (immediatly after its occurence). We adopt here the reduced form approach (also known as the intensity approach) under which the random time τ is not necesarilly a stopping time with respect to the default-free market filtrationF. We assume that

• 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 H_t := 1_(τ≤t) the default indicator process and let G be the smallest filtration containing the reference filtrationF and under whichτ is a stopping time, i.e.

G_t:=F_t∨σ(Hs;s≤t) =F_t∨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 also thefull filtration).

We assume that the financial market defined by the savings account, the stock and a corporate bond (which will be defined in the sequel) is arbitrage free, thus, leading to the existence of at least a risk-neutral probability measure (under which the discounted prices of the traded assets areG-martingales). Due to the possible arrival of the default (thus, inducing a jump in the dynamics of the wealth portfolio), the financial market is incomplete, meaning that the set of equivalent martingale measures contains more than a single element. LetQ be such a probability measure.

We work under the martingale invariance property (or the immersion property), 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) we also 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 (˜λ^G(t)), with λ˜^G(t) = 0 on the set (t > τ), such that the compensated process

(2) M(t) :=˜ H_t−

Z t 0

˜λ^G(s)ds

is aG-martingale underQ. (˜λ^G(t)) is called theG-intensity of the defaultτ (or the G-hazard rate of τ) under the probability measure Q. It can be written under the form ˜λ^G(t) = 1_(t≤τ)˜λ^F(t), where the process ˜λ^F(t) isF-adapted and is called the F-intensity of default (under Q). Instead of ˜λ^F we shall simply write ˜λ.

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

(3) M(t) :=H_t− Z t∧τ

0

λ(s)ds=H_t− Z t

0

(1−H_s)λ(s)ds is aG-martingale under P.

The following remark provides a a situation in which the (H) hypothesis holds, and this is done by the so-calledcanonical constructionof a default time with a given intensity process.

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

τ := inf t≥0|

Z t 0

γ_sds≥η .

The process (γ_t) has to be interpreted as theF-intensity ofτ. It can be shown that

P(τ ≤t|F_t) =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 G_T, it admits a Radon-Nikodym density Z_T := ^dQ_dP|_GT, given by a a positive G_T-measurable random variable withE(Z_T) = 1. The Radon-Nikodym density process Z_t:=

E(ZT|G_t) is a G-martingale under P and by virtue of the Predictable Repre- sentation Theorem, which applies toG-martingales ([14]) under the immersion hypothesis, it can be represented as

(4) dZ_t=Zt−(η(t)dW(t) +γ(t)dM(t)), Z₀= 1, whereη(t), γ(t) areG-predictable processes, withγ(t)>−1.

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

E(X·)t= exp X_t^c− 1

2[X^c, X^c]t

Y

0<s≤t

(1 + ∆Xs),

where X^c is the continuous part ofX and ∆X_s=X_s−Xs− is the size of the jump of X at time s. Moreover, E(X·) is the solution of the SDE

dZ_t^X =Z_t−^XdX_t, Z₀^X = 1.

Set

Y(t) :=

Z t 0

η(s)dW(s) + Z t

0

γ(s)dM(s)

= Z t

0

η(s)dW(s)− Z t

0

(1−H_s)λ(s)γ(s)ds+ Z t

0

γ(s)dH_s.

Iftis fixed, Y has at most a single jump in the time interval [0, t] (at timeτ), of size ∆Y(τ) =γ(τ)∆H_τ =γ(τ), only ifτ ≤t. Then, ifτ ≤t,

Y

0<s≤t

(1+∆Ys) = 1+γ(τ) = exp ln(1+γ(τ))∆Hτ

= exp Z t

0

ln(1+γ(s))dHs

. Ifτ > tthe integral from the last term is equal to 0 and the discontinuous part ofY doesn’t bring any contribution toE(Y_·)_t. Henceworth, the solution of the SDE (4) is given by the stochastic exponential

Z_t=E(Y_·)_t= exp −1 2

Z t 0

η²(s)ds+ Z t

0

η(s)dW(s) exp

Z t 0

ln(1 +γ(s))dHs− Z t∧τ

0

γ(s)λ(s)ds . (5)

In addition, the process

W˜(t) :=W(t)− Z t

0

η(s)ds is a standard Brownian motion underQ and the process (6) M¯t:=Mt−

Z t∧τ 0

γ(s)λ(s)ds=Ht− Z t∧τ

0

(1 +γ(s))λ(s)ds is a discontinuousG-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 underQ, respectivelyP, are related via the formula

λ(t) = (1 +˜ γ(t))λ(t).

Set ˜S(t) := e^−R0tr(s)S(t) the discounted price of the risky asset. By Itˆo’s formula

d ˜S(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)d ˜W(t)).

Since the discounted price of the stock must be a martingale under Q, the dt term must be equal to zero and thus, η(t) = ^r(t)−µ(t)_σ(t) :=−θ(t). The dynamics of S_t underQis given by

dS(t) =S(t)(r(t)dt+σ(t)d ˜W(t))

and notice that under an equivalent martingale measure the expected rate of return of the stock is equal to theshort interest rater.

Our next goal is to derive the price process of the defaultable asset. The associated dividend processDt is given by

(7) D_t:=X1_{(τ >T)}+z_τ1_(τ≤t)=X1_{(τ >T)}+ Z t

0

z_sdH_s, t≤T,

where X stands for the amount received by the investor (at time T) in the case of non-occurence of the default till time T, and (z(t)) is the process of compensations if default occurs before T. The quantity D_u−D_t represents all cash-flows between times t and u which are received by an investor which detains a defaultable bond which is purchased at timet. Then, it is well-known (see [2], Section 8.3) that the price at timetof a defaultable zero-coupon bond with maturity T is given by the formula

D(t, T) =E^Q Z T

t

e^−RtsrududD(s) G_t

=E^Q 1_{(τ >T)}e⁻

RT

t ruduX+ 1_(t<τ≤T)e⁻

Rτ t rudszτ

G_t

= 1_{(τ >t)}E^Q e⁻

RT

t (ru+˜λu)duX+ Z T

t

e⁻

Rs

t(ru+˜λu)duzsλ˜sds F_t

, (8)

whereE^Qstands for the expectation with respect to the probability measureQ.

We adopt here the recovery rate at default given by themarket value of default RMV(see [5] or [2]). It is assumed that at the timeτ of occurence of the default till timeT the bond cesses to exist and its holder receives a compensation given by a proportion of the pre-default value of the bondD(τ−, T), on the set (τ <

T). In this spirit, we consider the recovery processz(t) = (1−L(t))D(t−, T), where L(t) stands for the loss-rate. We assume that L is deterministic and 0< L(t)<1,P a.s.. Then

(9) D(t, T) = 1_{(τ >t)}E^Q

e⁻

RT

t (rs+˜λsLs)ds)X|F_t

:= 1_{(τ >t)}B(t, T) = ˜HtB(t, T), where ˜H_t:= 1−H_tandB(t, T) is the pre-default value of the defaultable bond and is equal to the value of a non-defaultable bond with default-risk adjusted interest rate r(t) :=ˆ r(t) + ˜λ(t)L(t) and credit spread given by the correction term ˜λ(t)L(t). From formula (9) and using the continuity of the pre-default value process B(t, T), we obtain the following formula for the recovery value at default

z(τ) = (1−L(τ))B(τ, T).

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

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

It is not hard to see that ^dQ_dP|F_t =Z^F(t), where Z^F(t) = exp

− Z t

0

θ(s)dW(s)−1 2

Z t 0

θ²(s)ds

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

˜

m(t) :=E^Q

e⁻

RT 0 ˆr(s)ds

F_t

= (Z^F(t))⁻¹E

e⁻

RT

0 r(s)dsˆ Z^F(T) F_t

. Setm(t) :=E

e^{−R0Tr(s)dsˆ} Z^F(T) F_t

. 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 Itˆo’s formula for the process (ln(m(t))) (for further details 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

Z t 0

q²(s)ds+ Z t

0

q(s)dW(s) . We thus, obtain

B(t, T) = exp Z t

0

ˆ r(s)ds

(Z^F(t))⁻¹mt

(10)

= exp Z t

0

ˆ r(s) +1

2(θ²(s)−q²(s)) ds+

Z t 0

(θ(s) +q(s))dW(s) . Finally, if we apply Itˆo’s rule to the exponential process (B(t, T)), we get

dB(t, T) =B(t, T) h

ˆ r(t) +1

2(θ²(t)−q²(t))

ds+ (θ(t) +q(t))dW(t) +1

2(θ(t) +q(t))²dt i

=B(t, T)[(ˆr(t) +θ(t)β(t))dt+β(t)dW(t)], (11)

whereβ(t) :=θ(t) +q(t).

Recall now that D(t, T) = ˜H_tB(t, T) and H_t = M(t) + Rt

0 H˜_sλ(s)ds.

Therefore, by Itˆo’s product rule of differentiation for jump processes, dD(t, T) = ˜Ht−dB(t, T) +B(t−, T)d ˜Ht+ d

X

0<s≤t

∆ ˜Hs∆B(s, T) (12)

= ˜H_tdB(t, T)−B(t, T)dH_t= ˜H_tdB(t, T)−B(t, T) ˜Ht−dH_t

= ˜H_tB(t, T) (ˆr(t) +θ(t)β(t))dt+β(t)dW(t)

−B(t, T) ˜Ht−(dM(t) +λ(t) ˜H_tdt)

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

−D(t−, T)dM(t)

=D(t−, T)

(ˆr(t) +θ(t)β(t)−λ(t))dt+β(t)dW(t)−dM(t) , by virtue of the continuity ofB(·, T), the formulas d ˜H_t=−dH_t and ˜H_t²= ˜H_t. We also used the identity dHt = ˜Ht−dHt, or, under integral form, Rt

0dHs = Rt

0H˜s−dHs. Forτ > t (after the default) both integrals are equal to 0 (since there is no jump until t), and forτ ≤t

Z t 0

H˜s−dHs= ˜Hτ−∆Hτ = ∆Hτ = Z t

0

dHs(= 1), thus, proving the required identity. We are now in position to state

Proposition1. The priceD(t, T)of the corporate bond has the dynamics (13) dD(t, T) =D(t−, T)

(ˆr(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 original optimization problem (which is stated in an incomplete market) into two aux- iliary portfolio optimization problems: pre-default and post-default, which are stated in complete markets and for which standard martingale methods or BSDE techniques can be applied. In studying what happens after the default, the notion of intensity of default is not sufficient, but instead the notion of F-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 a family of random variables (α_t(s)), such that, for fixed s, the processα·(s) is F-adapted, and

P(τ ≤s|F_t) = Z s

0

α_t(u)du, or equivalentlyE(f(τ)|F_t) =R∞

0 f(s)α_t(s)ds, for every bounded and

Borel-measurable function f. Furthemore, for a F_t⊗B measurable random variable Xt(x),

E(X_t(τ)|F_t) = Z ∞

0

X_t(s)α_t(s)ds.

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

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

Ft

.

Conversely, we can only partially recover the conditional densitiesαt(s) starting from the F-intensities, and this can be done only fors≥t(only for moments t prior to default). If it holds

(14) α_T(t) =α_t(t), ∀t∈[0, T],

then P(τ ≤t|F_t) = P(τ ≤ t|F_T), which provides a a sufficient condition for the (H) hypothesis to hold. We simplify the notation by writingα(t) :=αt(t).

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

3. THE PORTOFOLIO PROCESS

We consider an investor with investment opportunities in the financial market described in the previous section. Set N_R(t), N_S(t) and N_D(t) the number 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 assumedG-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 (N_D(t)) is F-adapted. This statement is a conse- quence of the (standard) decomposition of anyG-predictable process (ψ_t)0≤t≤T,

ψt=ψ_t⁰1(t≤τ)+ψ¹_t(τ)1(t<τ <T). (15)

In this formula, the processψ_t⁰ is F-adapted, for any fixed nonnegativeu, the process ψ_·¹(u) (indexed over u) is F-adapted, and for fixed t, the mapping ψ¹_t(·,·) isF_t⊗ B([0, T]) measurable.

The wealth processX^N,x(t) is defined by

X^N,x(t) =x+N_R(t)R(t) +N_S(t)S(t) +N_D(t)D(t, T),

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

(16) dX^N,x(t) =N_R(t)dR(t) +N_S(t)dS(t) +N_D(t)dD(t, T).

Set π_R(t), π_S(t) and π_D(t) the corresponding fractions of wealth invested in each asset, i.e.

π_R(t) := N_R(t)R(t)

X^N,x(t−), π_S(t) := N_S(t)S(t)

X^N,x(t−), π_D(t) := N_D(t)D(t−, T) X^N,x(t−) .

Obviously,π_R(t)+π_S(t)+π_D(t) = 1. We identify an investment strategy with a left-continuous process π(t) := (π_R(t), π_S(t), π_D(t)) and denote by (X_t^π,x; 0 ≤ t ≤ T) the corresponding wealth process. The self-financing condition now reads

(17) (

dX^π,x(t) = X^π,x(t−) πR(t)^dR(t)_R(t) +πS(t)^dS(t)_S(t) +πD(t)_D(t−,T^dD(t,T)₎

; X^π,x(0) = x.

After default, the corporate bond is not traded anymore, so πD(t) = 0 and D(t, T) = 0 fort > τ. In this case, we make the convention ⁰₀ = 0. Taking into account the dynamics of the traded assets, the value process of the portfolio may be written as

dX^π,x(t) =X^π(t−)

r(t) +π_S(t)(µ(t)−r(t))

+π_D(t) λ(t)(γ(t)L(t) +L(t)−1) +θ(t)β(t) dt + π_S(t)σ(t) +π_D(t)β(t)

dW(t)−π_D(t)dM(t) . (18)

In fact, π = π(t)1(t≤τ)+π(t)1_{(τ >t)}, where π(t) = (π_R(t), π_S(t), πD(t)) stands for the pre-default strategy and π(t) = (π_R(t), π_S(t),0) stands for the after- default strategy.

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

dX^π(t) =X^π(t)

r(t) +π_S(t)σ(t)θ(t) +π_D(t)(λ(t)(γ(t)L(t) +L(t)−1) +θ(t)β(t)))dt+ (π_S(t)σ(t) +πD(t)β(t)) dW(t)

, fort < τ∧T, (19)

and

(20) dX^π(t) =X^π(t) [r(t) +πS(t)σ(t)θ(t)] dt+πS(t)σ(t)dW(t), fort≥τ∧T.

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

E Z _T

0

π_S²(t)<∞, E Z _T

0

π_D²(t)<∞andπ_D(t)<1, 0≤t≤T, P a.s.

Notice that the wealth will remain positive at each moment t, P a.s. It is natural to assume that the investor won’t invest all his capital in the defaultable asset, 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→0U⁰(x) = ∞ and limx→∞U⁰(x) = 0. The logarithmic utility and the power utility clearly satisfy the properties listed above. The investor is interested in maximizing his expected utility (under the historical probability) from the final wealth over the classA(x) of admissible portfolios. We are thus,

lead to the optimization problem

(21) V(x) := sup

π∈A(x)

E[U(X^π,x(T))] = sup

π∈A(x)

J^x(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+ π_S(t)σ(t) +π_D(t)β(t)

dW(t)−π_D(t)dH_t. The value of the wealth process is obviously given by the stochastic exponential of (Y^π(t)), i.e.

X^π,x(t) =E(Y^π(·))_t=xexp Z t

0

r(s) +π_S(s)σ(s)θ(s) +π_D(s)(λ(s) (γ(s)L(s) +L(s)−H_s) +θ(s)β(s))−1

2(π_S(s)σ(s) +π_D(s)β(s))²

ds exp

Z t 0

π_S(s)σ(s) +π_D(s)β(s)

dW(s)

×exp Z t

0

ln(1−π_D(s))dH_s , (22)

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

ln (X^π,x(t)) = ln(x) + Z _t

0

r(s) +π_S(s)σ(s)θ(s) +π_D(s)(λ(s)(γ(s)L(s) +L(s)−H_s) +θ(s)β(s))− 1

2(π_S(s)σ(s) +π_D(s)β(s))² + ˜Hsλ(s) ln(1−πD(s))

ds +

Z t 0

πS(s)σ(s) +πD(s)β(s)

dW(s) + Z t

0

ln(1−πD(s))dM(s), where we rewrote the integralRt

0ln(1−πD(s))dHs by taking into account the formula dH_t= dM_t+ ˜H_tλ(t)dt. Under our assumptions the stochastic integral processes appearing in the last line of the multiline formula from above are true (not only local) martingales of zero expectation. Thus,

J^x(π) = ln(x) +E nZ T

0

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

+L(t)−H_t) +θ(t)β(t))−1

2(π_S(t)σ(t) +π_D(t)β(t))² + ˜H_tλ(t) ln(1−π_D(t))

dto

= ln(x) +E Z T

0

gt(πS(t), πD(t))dt ,

with the mappingg_t(y, z) properly defined. We are thus, lead to solve a path- wise optimization problem, which is splitted into the pre-default optimization problem and the post-default one. When dealing with the former, define, for t≤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),

fory∈Rand z <1. Fort > τ∧T (after the default),πD = 0 (the defaultable asset 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

∂˜g_t

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

∂˜g_t

∂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 ˜g_t, it follows easily that ˜g_t attains 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 that our optimization problem admits a solution under the assumptions

(i) The asymptotic elasticity of the utility functionU(x) satisfies AE(U) := lim sup

x→∞

xU⁰(x) U(x) <1;

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

(iii) The value functionV(x) is finite for somex >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 the conjugate function of the value function V, denoted V^∗, is finite at the point y=V⁰(x). A sufficient condition for the last assertion to hold is that

(25) E[U^∗(yZT)]<∞, for somey >0,

where U^∗ stands for the convex conjugate function (or the Legendre-Fenchel transform) ofU, defined by

U^∗(y) = sup

x≥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 the associated optimal portfolio but no explicit formulas for the value function (or the optimal strategy).

We follow now the arguments of Jiao and Pham [9]. If π = (π, π) is an admissible strategy, we denote by (X^π,s,x(t)), s ≤t≤T, the solution of SDE (20) starting at time sfrom the stateX^π,x(s)(1−π_s).

J^x(π) =E[U(X^π,x(T))] =E

E U(X^π,x(T))1_{(τ >T)}+U(X^π,τ,x(T)1_(τ≤T)|F_T

=E

U(X^π,x(T))P τ > T|F_T +E

E U(X^π,τ,x(T))1(τ≤T)|F_T

=E

U(X^π,x(T))GT

+E Z T

0

U(X^π,s,x(T))αT(s)ds .

We used the fact that any G_T-measurable r.v. Y can be represented as Y = E Y1_{(τ >T)}|F_T

GT

1_{(τ >T)}+Y(τ)1(τ≤T), whereY(τ) isF_T ⊗σ(τ)-measurable. We also have

(26) E Z T

0

U(X^π,s,x(T))α_T(s)ds

=E Z T

0

E U(X^π,s,x(T))α_s|F_s ds

. We define the after-default optimization problem

(27) V¯_s(x) = sup

πS

E U(X_s^πS,x(T))α_s F_s

= sup

πS

J_s(π_S, x),

where the after-default value process (Xs^πS,x(t)), s ≤ t ≤ T leaves at time s from the statex. This problem is not a classical portfolio optimization problem since we maximize the utility of the final wealth, weighted by some random function. Using Jiao and Pham [9], Theorem 3.1, the following dynamic pro- gramming type formula is valid

(28) V(x) = sup

π_S,πD

E

U(X^π,x(T))G_T + Z T

0

V¯_s(X^π,x(s)(1−π_D(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 being stated in complete markets. It is easy to see that we have to solve first the problem after-default (27), and afterwards we insert the value function of this problem in the formula (28). Resolution of the latter poses many technical difficulties 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

Z^F(t) = exp −1 2

Z t 0

θ²(u)du− Z t

0

θ(u)dW(u) ,

and set Z_s^t := _Z^ZF^F(s)^(t), for t ≥ s. Let s be fixed and define the probability measure Q^s on F_T by ^dQ_dP^s|_FT :=Z_s^T. Obviously, ^dQ_dP^s|_Ft =Z_s^t. Set also

H_s^t=e^−Rstr(u)duZ_s^t= exp − Z t

s

(r(u) +1

2θ²(u))du− Z t

s

θ(u)dW(u)

, fort≥s.

H_s^tis calleddeflator. It is easy to see that the discounted value of the wealth is obviously a G- (local) martingale (under Q^s) which takes positive values, and hence, it is a supermartingale. It follows

(29) E(H_s^TX_s^πS,x(T)|F_s)≤E(H_s^sX_s^πS,x(s)) =x.

LetU^∗ be the convex conjugate function ofU. If we denoteI(y) := (U⁰)⁻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) = ^x_p^p. An elementary computation shows thatI(y) =y^p−11 and U^∗(y) =

1−p

p y^p−1p . We can also write U^∗(y) = −^y_q^q, where q is the dual conjugate of p (i.e. _p¹+¹_q = 1).

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

L(X, λ) :=αsU(X) +λ(x−H_s^TX)

“Differentiation” with respect to X yields

α_sU⁰(X)−λH_s^T = 0, from which we get

X =I λH_s^T αs

.

We impose now thatX from the above formula satisfies the restriction (29) as an equality. We obtain

(30) E H_s^TI λH_s^T

αs

|F_s

=x, and we want to solve this equation with respect toλ. Set

g(λ) :=E H_s^TI λH_s^T αs

F_s

= λ^p−11 α

1

sp−1

E (H_s^T)^q F_s

.

We prove now thatgtakes finite values. Letδbe an arbitrary real number and set Z_s^T(δ) := E R·

s(−δθ(t))dW(t)dt

T. Set also Q^s_δ the probability measure

equivalent to P given by the Radon-Nikodym derivative ^dQ_dP^sδ

F_T = Z_s^T(δ).

Then

E (H_s^T)^δ

=E

Z_s^T(δ) exp (−δ 2)

Z T s

(r(t) + (1−δ)θ²(t))dt

=E^Qsδ

exp (−δ 2)

Z T s

(r(t) + (1−δ)θ²(t))dt

≤exp |δ|

2 T(krk_T +|1−δ|kθk²_T)

<+∞,

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 limλ&0g(λ) =∞, limλ→∞g(λ) = 0.It follows that the equation (30) admits an unique positive solution λ^∗ =g⁻¹(x). Set

(31) X^∗ =I λ^∗H_s^T

αs

.

Let nowX^π be an admissible wealth portfolio (which satisfies the restric- tion (29)). Applying the (standard) inequality which is satisfied by any convex differentiable functionh

h(y)−h(x)≤(y−x)h⁰(x),

to the utility function U and the points x =X^∗, y = X^π, taking afterwards the expectation and considering also the admissibility formula (29) will lead us 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)H_s^T αs

= (g⁻¹(x))^p−11 α

1 p−1

s

(H_s^T)^p−11 . Moreover, an optimal portfolio π^∗ has the form

(32) π^∗(t) = θ(t)

σ(t) +η^∗(t)

σ(t), fort≥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 the claimX^∗ defined in (31) is hedgeable,i.e. if it exists a portfolioπ^∗_S s.t. the the associated wealth process (X^π

∗ S,x

s (t))s≤t≤T satisfies X^π

∗ S,x

s (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 under Q^s. Hence,

e⁻

Rt

sr(u)duX^π

∗ S,x

s (t) =E^Qs

e⁻

RT

s r(u)duX^∗ F_t

= 1

Z_s^tM_s^∗(t), where the process M_s^∗(t) := E

e^−RsTr(u)duX^∗Z_s^T F_t

, 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 exponential E R·

0η^∗(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 Itˆo’s formula of integration by parts, we deduce dM_s^∗(t) = d Z_s^te⁻

Rt

sr(u)duX^π

∗ S,x s (t)

=Z_s^te^−Rstr(u)duX^π

∗ S,x

s (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 bounded deterministic functions, the utility function is of power type and also that the default 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 horizon T. Since, by the conditional density assumption, it holds

P(0< τ < T) =E

E 1(0<τ <T)|F_T

=E Z T

0

α_T(u)du

= Z T

0

α_udu, we notice that formula (33) is equivalent with

(34)

Z T 0

αudu= 1.

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

X^π(T) =X^π(T) =X^π(τ) exp Z T

τ

r(s) +πS(s)σ(s)θ(s)−1

2π²_S(s)σ²(s) ds (35)

×exp Z T

τ

π_S(s)σ(s)dW(s) , where

X^π(τ) =X^π(τ)(1−π_D(τ)).

6.1. THE POST-DEFAULT OPTIMIZATION PROBLEM

We consider the dynamic version of the value function of post-default optimization problem

V(t, x) = sup

π∈A(t,x)

E

(X^π(T))^p|X^π(t) =x .

Since we are interested in solving this problem with random initial conditions (for t=τ and x=X^π(τ)), we shall solve instead the optimization problem (36) V(τ, η) = sup

π∈A(τ,η)

E

(X^π,τ,η(T))^p

= sup

π∈A(τ,η)

J(τ, η, π),

where η is a positiveG_τ-measurable random variable and X^π,τ,η(t), τ ≤t≤T stands for the solution of the equation (20), starting at the random time τ from the random stateη. Obviously,

(X^π(T))^p=η^pexp

p Z T

τ

r(s) +πS(s)σ(s)θ(s)−1−p

2 π²_S(s)σ²(s) ds

× EZ · τ

pπS(s)σ(s)dW(s)

T. (37)

Setπ^∗_t the maximizer of the second order polynomial function h_t(y) =r(t) +σ(t)θ(t)y− 1−p

2 σ²(t)y², which is obviously given by

(38) π^∗_t = 1

1−p θ(t) σ(t).

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

Proof. Letπ an arbitrary chosen element of A(τ, ξ). Set M_t^π =E

Z ·

τ

pπS(s)σ(s)dW(s)

t, τ ≤t≤T.

The process of the stochastic integral R·

τpπ_S(s)σ(s)dW(s)

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

thus, the stochastic exponential (M_t^π;t≥ τ) is also a martingale, with prime element M_τ^π = 1. Henceworth,

J(τ, η, π) =Eh

η^pexp p

Z T τ

h_s(π_s)ds M_T^πi

≤E h

η^pexp

p Z T

τ

hs(π^∗_s)ds

M_T^π i

=En Eh

η^pexp p

Z T 0

h_s(π^∗_s)ds− Z τ

0

h_s(π^∗_s)ds M_T^π

G_τio

=En

η^pexp p

Z _T

0

h_s(π^∗_s)ds− Z _τ

0

h_s(π^∗_s)ds Eh

M_T^π G_τio

=E h

η^pexp

p Z T

τ

hs(π^∗_s)ds M_τ^π

i

=Eh

η^pexp p

Z _T

τ

h_s(π^∗_s)ds E

M_T^π∗ G_τi

=E h

η^pexp

p Z T

τ

hs(π^∗_s)ds M_T^π∗

i

=J(τ, η, π^∗).

(39)

We used Doob’s optional sampling theorem, the fact that the strategyπ^∗ and the functions ht are deterministic and also, M_τ^π = 1 =M_τ^π∗.

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

For an admissible strategy π ∈ A(x) define J(x, π) :=E[(X^π,x(T))^p].

We saw thatπ can be represented asπ_t=π_t1_(t<τ)+π_t1_(t≥τ). Notice that J(x, π) =J(τ, X^π(τ)(1−π_D(τ)), π)≤J(τ, X^π(τ)(1−π_D(τ)), π^∗).

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

J(τ, η, π^∗) =E

η^pexp p(G^∗_T −G^∗_τ) , where

G^∗_t :=

Z t 0

h_s(π^∗_s)ds= Z 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 the value function of the pre-default optimization problem

(40) V(x) = sup

π

J(τ, X^π(τ)(1−π_D(τ)), π^∗), where by an abuse of notation we setπ(t) = (π_S(t), π_D(t)).

We are thus dealing with an optimization portfolio investment problem in a random horizon. We found in the literature only a few references dealing with this subject, from which we mention Blanchet-Scalliet et al. [3] and the unpublished paper El Karoui, Jeanblanc and Huang [6]. In the second citation, the authors use the BSDE approach, a crucial assumption beingP(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 involving several risky assets with prices modelled by geometric Brownian Motions, un- der the main assumptions of deterministic bounded coefficients of the model and deterministic density of the conditional distribution function of the ran- dom horizon, using dynamic programming approach and also the martingale approach. As in this paper, we shall state a verification result by writing properly the Hamilton-Jacobi-Bellman nonlinear PDE which is satisfied by the value function and provide also an optimal strategy.

By virtue of the conditional density assumption E

X^π(τ)(1−π_D(τ))p

exp p(G^∗_T −G^∗_τ)

=E E

X^π(τ)p

(1−π_D(τ))^pexp p(G^∗_T −G^∗_τ) F_T

=E Z T

0

(X^π(t))^p(1−π_D(t))^pexp (p(G^∗_T −G^∗_t))α_tdt.

We define a dynamic version of the value function V(x) by setting V(t, x) = sup

π

E Z T

t

(X^π(s))^p(1−π_D(s))^pexp (p(G^∗_T −G^∗_s))αsds, where (X^π(s)) is the solution of the equation (19) starting at time t from the statex. Thus, (X^π) is the solution of the controlled linear SDE

dX(t) =f(t, X(t), π_t)dt+σ(t, X(t), π_t)dW(t), π_t= (π_S(t), πD(t)), with coefficientsf(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

, and

σ(t, x, π) =x σ(t)π₁+β(t)π₂ .

Our optimization problem has the running cost functional F(t, x, π) =x^p(1−π₂)^pexp (p(G^∗_T −G^∗_t))α_t.

The admissible region for the control variableπ is given byU =R×(−∞,1).

Obviously,|F(t, x, π)| ≤K(1 +|x|) (since p∈(0,1)).

Setδ(t) := exp (p(G^∗_T −G^∗_t))α_tand ρ(t) :=λ(t)(γ(t)L(t) +L(t)−1). We state now a verification theorem, which is based on a general result (see [7]), Chapter IV, Theorem 3.1). All the required assumptions are clearly fulfilled in our setting. We establish also the existence of a classical solution of the corresponding HJB equation.

Theorem 3. Consider the following Hamilton-Jacobi-Bellman nonlinear PDE

(41)

_∂W

∂t (t, x) + sup_π∈UH(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) +x^p(1−π2)^pδ(t).

(42)

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

K⁰(t) +p r(t) +ρ(t) + θ²(t) 2(1−p)

K(t) + (1−p)ρ(t)

p p−1

δ(t)^p−11 K(t)

p

p−1 = 0, t∈[0, T), with terminal Cauchy conditionK(T) = 0. K(t) has the explicit form given by

K(t) = exp − p 1−p

Z t 0

(r(s) +ρ(s) + θ²(s) 2(1−p)ds)

× Z T

t

hρ(s)^p−1p δ(s)^p−11

exp p 1−p

Z s 0

(r(u) +ρ(u) + θ²(u)

2(1−p))dui ds

= Z T

t

hρ(s)

p p−1

δ(s)^p−11

exp p 1−p

Z s t

(r(u) +ρ(u) + θ²(u)

2(1−p))dui ds.

(44)

Then

(a) The function W(t, x) := x^pK(t) ∈ C^1,2([0, T]×R) is a solution of the equation (41) and

(45) V(t, x) =W(t, x),∀t∈[0, T], x∈R.