FBSDE approach to utility portfolio selection in a market with random parameters

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Statistics & Probability Letters 78 (2008) 426–434 www.elsevier.com/locate/stapro

FBSDE approach to utility portfolio selection in a market with random parameters$ Rene´ Ferland, Franc- ois Watier De´partement de mathe´matiques, Universite´ du Que´bec a` Montre´al, Montre´al Que., Canada H3C 3P8 Received 23 March 2006; received in revised form 14 May 2007; accepted 25 July 2007 Available online 22 August 2007

Abstract A continuous-time utility portfolio selection problem is studied in a market in which the interest rate, appreciation rates and volatility coefficients are driven by Brownian motion. We construct an optimal portfolio using results from forward–backward stochastic differential equations (FBSDE) theory. As an illustration, exact computation of the optimal strategy is done for the power and exponential type utilities. r 2007 Elsevier B.V. All rights reserved. MSC: primary 91B28; secondary 91B70; 60H10 Keywords: Expected utility maximization; Optimal portfolio; Forward–backward stochastic differential equations

1. Introduction In the field of economics, an individual’s utility curve encompasses his degree of satisfaction associated with different level of wealth attained as well as his behavior towards risk in an attempt to acquire additional wealth. Utility portfolio selection addresses the issue of allocation of wealth amongst a basket of securities (a bond and several stocks) in order to maximize the expected utility from consumption and/or terminal wealth. The first treatment of this stochastic optimization problem in a continuous-time setting originated in the seminal papers of Merton (1969, 1971). In the context of a market model with deterministic coefficients (appreciation rate, volatility and interest rate) and stock price processes driven by Brownian motion, Merton used the Hamilton–Jacobi–Bellman (HJB) equation of dynamic programming to construct an explicit optimal portfolio for log and power utility. Later on, the emergence of the concept of equivalent martingale measure in contingent claim valuation theory led Pliska (1986) and Karatzas et al. (1987) to develop an alternate method for solving utility portfolio selection problems, namely the martingale method. Basically, one uses the martingale representation theorem to guarantee the existence of a portfolio process attaining a given admissible wealth process, and further one solves a static utility maximization problem using convex analysis to obtain the optimal wealth. $

This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada.

Corresponding author.

E-mail address: [email protected] (R. Ferland). 0167-7152/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2007.07.016

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Since the 1990s, backward stochastic differential equations (BSDE) have played an important role in modern finance theory as clearly illustrated in El Karoui et al. (1997). The first results on well posedness and solvability of non-linear BSDEs were established in Pardoux and Peng (1990). Subsequently, results concerning solvability of coupled forward–backward stochastic differential equations (FBSDE) were obtained in Antonelli (1993). In the following years, general methods for solving different classes of FBSDE were developed, namely, the four step scheme introduced by Ma et al. (1994) and the method of continuation of Hu and Peng (1995). In the past years, numerical schemes were also introduced for solving FBSDE in the specific cases where the drift and diffusion terms of the FBSDE are deterministic functions of the adapted solution (see for example Cvitanic and Zhang, 2005; and Milstein and Tretyakov, 2006). As a result, BSDE very rapidly established themselves as powerful tools for solving stochastic control problems, and in particular portfolio allocation problems. A recent illustration of this fact is the general meanvariance efficient portfolio obtained in Lim and Zhou (2002). In this paper, we derive an optimal strategy that maximizes the expected utility of the terminal wealth of an investor in a continuous-time market model with random coefficients driven by Brownian motion. We do this by linking the optimal portfolio to the solution of a FBSDE. The subsequent sections present, respectively, the formulation of the problem (Section 2), the representation of the optimal portfolio (Theorem 3.1), and the computation of the optimal solution for the well-known power and exponential type utilities. 2. Problem formulation Let ðO; F; fFt gtX0 ; PÞ be a complete filtered probability space such that F0 is augmented by all the P-null sets of F. Let W ðtÞ ¼ ðW 1 ðtÞ; . . . ; W m ðtÞÞ0 be an Rm -valued standard Brownian motion, and assume that fFt gtX0 is generated by W. We consider a market with m þ 1 securities, consisting of a bond and m stocks. The bond price P0 ðtÞ satisfies the (stochastic) ordinary differential equation: ( dP0 ðtÞ ¼ rðtÞP0 ðtÞ dt; t 2 ½0; T; (1) P0 ð0Þ ¼ p0 40; where the interest rate rðtÞ40 is a uniformly bounded, fFt gtX0 -adapted, scalar-valued stochastic process. The stock price Pi ðtÞ (i ¼ 1; . . . ; m) satisfies the stochastic differential equation (SDE): 8 ( ) m > > < dPi ðtÞ ¼ Pi ðtÞ bi ðtÞ dt þ P sij ðtÞ dW j ðtÞ ; t 2 ½0; T; (2) j¼1 > > : Pi ð0Þ ¼ p 40; i

where bi ðtÞ and si ðtÞ ¼ ðsi1 ðtÞ; . . . ; sim ðtÞÞ are the appreciation rate and volatilities of the ith stock, respectively, at time t. As we did for r, we assume that bi and sij are scalar-valued, fFt gtX0 -adapted, uniformly bounded processes. We also assume that 3 2 s11 ðtÞ s12 ðtÞ    s1m ðtÞ 7 6 6 s21 ðtÞ s22 ðtÞ    s2m ðtÞ 7 7 6 sðtÞ ¼ 6 . .. 7 .. .. 7 6 .. . . . 5 4 sm1 ðtÞ sm2 ðtÞ    smm ðtÞ is uniformly non-degenerate: there exists d40 such that sðtÞsðtÞ0 XdI, for all t 2 ½0; T, P-a.s. Denoting by xðtÞ the total wealth of the investor at time t, ignoring transaction costs and consumption, and assuming share trading takes place in continuous time we require: 8   m m P m P P > < dxðtÞ ¼ rðtÞxðtÞ þ ½bi ðtÞ  rðtÞui ðtÞ dt þ sij ðtÞui ðtÞ dW j ðtÞ; t 2 ½0; T; i¼1 j¼1 i¼1 (3) > : xð0Þ ¼ x 40; 0

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where ui ðtÞ is the market value of a small investor’s wealth in the ith stock at time t. We refer to uðtÞ ¼ ðu1 ðtÞ; . . . ; um ðtÞÞ0 as the portfolio of the investor. Since decisions are made without the ability to foresee future market flows, we consider portfolios which are fFt gtX0 -adapted and moreover square-integrable: Z T  E kuðtÞk2 dt o1, 0

so that the corresponding SDE (3) has a strong solution xðÞ. Finally, to capture the investor’s preferences in the spirit of the von Neumann–Morgenstern expected utility theory, let us introduce a twice differentiable utility function U defined on some domain D  R, and satisfying the conditions: UðzÞa0;

U 1 ðzÞ ¼

dU 40; dz

U 2 ðzÞ ¼

d2 U o0. dz2

The objective is to find a portfolio u whose corresponding wealth x satisfies E½Uðx ðTÞÞXE½UðxðTÞÞ with respect to any portfolio u with associated wealth x.

3. The optimal portfolio In order to exhibit an optimal portfolio, we must first introduce a carefully constructed BSDE1 for a pair of processes ðp; LÞ. Then we shall use this adjoint process p to obtain a tractable expression for E½UðxðTÞÞ which will allow maximization in a readily manner. Henceforth, for a pair of processes ðp; LÞ we consider a BSDE of the type: 8 0 > < dpðtÞ ¼ aðtÞ dt þ LðtÞ dW ðtÞ; t 2 ½0; T; pðTÞ ¼ 1; (4) > : pðtÞ40; t 2 ½0; T; in which the drift term2 a may be made to depend on x, the associated wealth process of a portfolio u; finally we introduce the following assumption (quite reasonable from a practical point of view): Assumption H. The expected value Eðpð0ÞÞ depends on the wealth process only through x0 . As it will appear clear in devising an optimal strategy, the main challenge resides in finding a suitable a that would make fpðtÞUðxðtÞÞ; t 2 ½0; Tg a diffusion process with non-positive drift. To this end, let us first rewrite the wealth Eq. (3) as follows: ( dxðtÞ ¼ frðtÞxðtÞ þ BðtÞ0 uðtÞg dt þ uðtÞ0 sðtÞ dW ðtÞ; t 2 ½0; T; (5) xð0Þ ¼ x0 40; where BðtÞ ¼ ðb1 ðtÞ  rðtÞ; . . . ; bm ðtÞ  rðtÞÞ0 . Then using Itoˆ’s formula one can easily compute dðpðtÞUðxðtÞÞÞ ¼ 1

pðtÞ U 2 ðxðtÞÞAðtÞ dt þ V ðtÞ0 dW ðtÞ, 2

See Ma and Yong (1999) for the basic definitions and results concerning BSDE. Dependence on the wealth process constitute a fundamental departure from standard Riccati BSDE used for example in Lim and Zhou (2002). 2

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where   U 1 ðxðtÞÞ LðtÞ uðtÞ0 BðtÞ þ uðtÞ0 sðtÞ AðtÞ ¼ uðtÞ0 sðtÞsðtÞ0 uðtÞ þ 2 U 2 ðxðtÞÞ pðtÞ   2 aðtÞ þ U 1 ðxðtÞÞrðtÞxðtÞ þ UðxðtÞÞ , U 2 ðxðtÞÞ pðtÞ

ð6Þ

V ðtÞ ¼ pðtÞU 1 ðxðtÞÞuðtÞ0 sðtÞ þ UðxðtÞÞLðtÞ0 . Subsequently, since U 2 ðxðtÞÞo0 and pðtÞ40, we are lead to choose a such that AðtÞ is positive. Now observe that AðtÞ is a quadratic expression and by astutely selecting a of the form aðtÞ ¼ KðtÞpðtÞ þ LðtÞ þ

MðtÞ pðtÞ

this should give us enough latitude on the selection of the processes K, L and M in order to envisage applying a completion of square technique. Indeed, by letting   U 1 ðxðtÞÞ2 U 1 ðxðtÞÞ rðtÞxðtÞ, KðtÞ ¼ ksðtÞ1 BðtÞk2  UðxðtÞÞ 2U 2 ðxðtÞÞUðxðtÞÞ   U 1 ðxðtÞÞ2 LðtÞ ¼ LðtÞ0 sðtÞ1 BðtÞ, U 2 ðxðtÞÞUðxðtÞÞ   U 1 ðxðtÞÞ2 MðtÞ ¼ kLðtÞk2 , 2U 2 ðxðtÞÞUðxðtÞÞ AðtÞ becomes, after some tedious calculations, a squared norm:   2  U 1 ðxðtÞÞ LðtÞ  0 1   . sðtÞ AðtÞ ¼ sðtÞ uðtÞ þ BðtÞ þ sðtÞ U 2 ðxðtÞÞ pðtÞ 

(7)

Thus, by assuming that Eqs. (4) and (5) can be solved, we observe that UðxðTÞÞ ¼ pðTÞUðxðTÞÞ

Z

¼ pð0ÞUðxð0ÞÞ þ 0

T

pðtÞ U 2 ðxðtÞÞAðtÞ dt þ 2

Z

T

V ðtÞ0 dW ðtÞ, 0

which leads to Z E½UðxðTÞÞ ¼ E½pð0ÞUðxð0ÞÞ þ 0

T

  pðtÞ U 2 ðxðtÞÞAðtÞ dt. E 2

Then, under assumption H, E½UðxðTÞÞ is clearly maximized when AðtÞ ¼ 0 for all t 2 ½0; T. The optimal portfolio u is therefore the one for which   U 1 ðxðtÞÞ LðtÞ 0 1  ðsðtÞsðtÞ Þ u ðtÞ ¼  BðtÞ þ sðtÞ . (8) U 2 ðxðtÞÞ pðtÞ Finally, replacing u by its optimal value in (4) and (5) one gets a system of non-linear coupled FBSDE with random coefficients:    8 U 1 ðxðtÞÞ 1 1 0 LðtÞ 2 > > ksðtÞ BðtÞk þ ½sðtÞ BðtÞ dxðtÞ ¼ rðtÞxðtÞ  dt > > U 2 ðxðtÞÞ pðtÞ > <  0 U 1 ðxðtÞÞ LðtÞ (9) sðtÞ1 BðtÞ þ  dW ðtÞ; t 2 ½0; T; > > > U pðtÞ ðxðtÞÞ 2 > > : xð0Þ ¼ x0 ;

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8     U 1 ðxðtÞÞ2 U 2 ðxðtÞÞ > 1 2 > rðtÞxðtÞ pðtÞ dpðtÞ ¼ BðtÞk  2 ksðtÞ > > > U 1 ðxðtÞÞ 2U 2 ðxðtÞÞUðxðtÞÞ > > >  < kLðtÞk2 þ2LðtÞ0 sðtÞ1 BðtÞ þ dt þ LðtÞ0 dW ðtÞ; t 2 ½0; T; > pðtÞ > > > > pðTÞ ¼ 1; > > > : pðtÞ40; t 2 ½0; T:

(10)

We have therefore established the following: Theorem 3.1. Suppose assumption H holds. If the coupled system of FBSDE (9)–(10) has a solution and if the portfolio u given by (8) is square-integrable, then u maximizes the expected utility of the terminal wealth. 4. Explicit utility cases In economic theory, a higher curvature of a utility function translates into a higher aversion towards risk. As a common criterion of the latter concept, we naturally encounter the famed Arrow–Pratt measures of absolute and relative risk aversion defined, respectively, by rU ðzÞ ¼ 

U 2 ðzÞ ; U 1 ðzÞ

RU ðzÞ ¼ z

U 2 ðzÞ . U 1 ðzÞ

The following section is devoted to explicit computation of the optimal portfolio in the presence of the wellknown constant relative risk aversion (CRRA) and constant absolute risk aversion (CARA) utility functions. More precisely, we will solve the coupled system (9)–(10) for the corresponding main representative of each of the above utility classes, namely the power-type utility and exponential-type utility. 4.1. Power-type (CRRA) utility Suppose U : ð0; 1Þ ! R is of power-type, that is let UðzÞ ¼

zb b

ð0obo1Þ.

In this case, (10) reduces to 8   1 b > > dpðtÞ ¼  ðksðtÞ1 BðtÞk2 þ 2ð1  bÞrðtÞÞpðtÞ > > > 2 1b > >  > < kLðtÞk2 þ2LðtÞ0 sðtÞ1 BðtÞ þ dt þ LðtÞ0 dW ðtÞ; pðtÞ > > > > > pðTÞ ¼ 1; > > > : pðtÞ40; t 2 ½0; T;

t 2 ½0; T;

(11)

a BSDE which does not involve the wealth xðtÞ. Therefore, it is possible to solve (9) and (10) separately and assumption H trivially holds. Eq. (11) is a stochastic Riccati equation which can be transformed into a linear BSDE by letting b¯ ¼ ð1  bÞ1 and setting ¯

Y ðtÞ ¼ pðtÞb ;

¯

b1 ¯ ZðtÞ ¼ bpðtÞ LðtÞ.

Using Ito’s formula, one sees that ðp; LÞ solves (11) if and only if ðY ; ZÞ solves the linear BSDE : 8 0 0 1 ¯ ¯ > < dY ðtÞ ¼ 2ð1  bÞ½bF ðtÞY ðtÞ þ GðtÞ ZðtÞ dt þ ZðtÞ dW ðtÞ; t 2 ½0; T; Y ðTÞ ¼ 1; > : Y ðtÞ40; t 2 ½0; T;

(12)

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where F ðtÞ ¼ ksðtÞ1 BðtÞk2 þ 2ð1  bÞrðtÞ, GðtÞ ¼ 2sðtÞ1 BðtÞ. The coefficients F and G are uniformly bounded since r and B are uniformly bounded as well and s is uniformly non-degenerate. According to Lim (2004) we conclude the following: 1. The BSDE (12) has a unique (positive) solution. 2. The solution ðY ; ZÞ is such that Z is square integrable and there exists a constant k40 such that kpY p1. Statements 1 and 2 also hold for (11) and the pair ðp; LÞ. This enables us to solve (9). Indeed, for the powerutility function the latter becomes    8 1 1 1 0 LðtÞ 2 > > dxðtÞ ¼ xðtÞ rðtÞ þ ksðtÞ BðtÞk þ ½sðtÞ BðtÞ dt > > 1b pðtÞ > <   xðtÞ LðtÞ 0 (13) 1 sðtÞ þ BðtÞ þ dW ðtÞ; t 2 ½0; T; > > > 1  b pðtÞ > > : xð0Þ ¼ x0 ; a solvable linear SDE with square integrable coefficients. The unique solution being an exponential function is clearly positive, showing a posteriori that xðtÞ is in the domain of U. For the power-type utility, the portfolio u is therefore   xðtÞ LðtÞ ðsðtÞsðtÞ0 Þ1 BðtÞ þ sðtÞ u ðtÞ ¼ . 1b pðtÞ We now prove that u is square integrable. Using Ito’s formula one can show that ¯

xðtÞ ¼ pðtÞb Y 2 ðtÞ, where Y 2 is the unique square-integrable solution of the linear SDE:      8 Y ðtÞ 1 b > < dY 2 ðtÞ ¼ 2 rðtÞ þ ksðtÞ1 BðtÞk2 dt þ ðsðtÞ1 BðtÞÞ0 dW ðtÞ ; 1b 2 1b > : Y ð0Þ ¼ pð0Þb¯ x : 2

0

It follows that x is square integrable since p is uniformly bounded. Moreover, xðtÞ being continuous on ½0; T, u is locally square integrable. One then applies Lemma 3.1 of Lim (2004) to conclude that u is square integrable. Remark. When the coefficients of (1) and (2) are deterministic, the process   1 b ~ ¯ ðtÞ F ðtÞ ¼ bF 2 b1 turns out to be deterministic. Straightforward computations yield   Z T   exp  F~ ðsÞ ds ; 0 t

as a solution of (12), which is unique. Consequently, the optimal portfolio is just   xðtÞ u ðtÞ ¼ ðsðtÞsðtÞ0 Þ1 BðtÞ, 1b a formula obtained, for example, by Karatzas (1997).

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4.2. Exponential-type (CARA) utility Suppose U : R ! R is of exponential-type, that is let UðzÞ ¼ ebz

ðb40Þ.

In this case (9) becomes    8 1 LðtÞ 1 1 0 2 > > ksðtÞ BðtÞk þ ½sðtÞ BðtÞ dxðtÞ ¼ rðtÞxðtÞ þ dt > > b pðtÞ > <  0 1 LðtÞ þ sðtÞ1 BðtÞ þ dW ðtÞ; t 2 ½0; T; > > > b pðtÞ > > : xð0Þ ¼ x0 ; while (10) reduces to 8  1 > > dpðtÞ ¼ ðksðtÞ1 BðtÞk2 þ 2brðtÞxðtÞÞpðtÞ > > > 2 > >  > < kLðtÞk2 þ2LðtÞ0 sðtÞ1 BðtÞ þ dt þ LðtÞ0 dW ðtÞ; pðtÞ > > > > > pðTÞ ¼ 1; > > > : pðtÞ40; t 2 ½0; T:

t 2 ½0; T;

(14)

(15)

Contrary to the previous case, the BSDE (15) does involve the wealth xðtÞ. As before, we make a change of variables: Y ðtÞ ¼ log pðtÞ;

ZðtÞ ¼ pðtÞ1 LðtÞ.

The BSDE for ðY ; ZÞ is then given by ( dY ðtÞ ¼ 12½F ðtÞ þ GðtÞ0 ZðtÞ dt þ ZðtÞ0 dW ðtÞ;

t 2 ½0; T;

Y ðTÞ ¼ 0;

(16)

with F ðtÞ ¼ ksðtÞ1 BðtÞk2 þ 2brðtÞxðtÞ, GðtÞ ¼ 2sðtÞ1 BðtÞ. Case 1: In a case in which r; B and s are deterministic, one may apply the ‘‘four step scheme’’ (see Ma and Yong, 1999) and get the solution: Z T 1 ksðsÞ1 BðsÞk2 ds, Y ðtÞ ¼ bð1  f ðtÞÞxðtÞ  t 2 ZðtÞ ¼ sðtÞ1 BðtÞðf ðtÞ1  1Þ,  Z t  Z t

0 1 xðtÞ ¼ f ðtÞ1 f ð0Þx0 þ ksðsÞ1 BðsÞk2 ds þ sðsÞ1 BðsÞ dW ðsÞ , b 0 0 where Z f ðtÞ ¼ exp

T

 rðsÞ ds .

t

The portfolio u is u ðtÞ ¼

1 ðsðtÞsðtÞ0 Þ1 ðBðtÞf ðtÞ1 Þ, b

a well-known result in CARA utility portfolio theory.

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Case 2: When r; B and s are stochastic, the process ðY ; ZÞ as defined in case 1, is not adapted and therefore cannot be a solution to (16). However, if only r is deterministic, we may look for a solution of the form: Y ðtÞ ¼ bð1  f ðtÞÞxðtÞ þ Y 2 ðtÞ, ZðtÞ ¼ gðZ 2 ðtÞÞ,

ð17Þ

where ðY 2 ; Z 2 Þ solve the BSDE: dY 2 ðtÞ ¼ a2 ðtÞ dt þ Z 2 ðtÞ0 dW ðtÞ;

Y 2 ðTÞ ¼ 0

(18)

for a suitably chosen coefficient a2 . The motivation for trying such a solution comes from the deterministic case. By (14) and (16) we have dY ðtÞ ¼ b dxðtÞ  12ksðtÞ1 BðtÞk2 dt  ðsðtÞ1 BðtÞÞ0 dW ðtÞ, while (17) gives dY ðtÞ ¼ bð1  f ðtÞÞ dxðtÞ þ bxðtÞrðtÞf ðtÞ dt þ a2 ðtÞ dt þ Z 2 ðtÞ0 dW ðtÞ. Equating these two leads to dxðtÞ ¼ ðbf ðtÞÞ1 ½ðbxðtÞrðtÞf ðtÞ þ a2 ðtÞ þ 12ksðtÞ1 BðtÞk2 Þ dt þ ðsðtÞ1 BðtÞ þ Z 2 ðtÞÞ0 dW ðtÞ.

ð19Þ

Matching the coefficients of (19) with those of (14), we obtain a2 ðtÞ ¼ 12ksðtÞ1 BðtÞk2 þ ðsðtÞ1 BðtÞÞ0 Z 2 ðtÞ, ZðtÞ ¼ sðtÞ1 BðtÞðf ðtÞ1  1Þ þ f ðtÞ1 Z 2 ðtÞ. For this choice of a2 , the BSDE (18) has a (unique) square-integrable solution by Theorem 4.2 of Ma and Yong (1999). Finally, since log pð0Þ ¼ bð1  f ð0ÞÞx0 þ Y 2 ð0Þ, assumption H holds and the optimal portfolio is 1 1 u ðtÞ ¼ ðsðtÞsðtÞ0 Þ1 ðBðtÞ þ sðtÞZðtÞÞ ¼ ðsðtÞsðtÞ0 Þ1 ðBðtÞ þ sðtÞZ 2 ðtÞÞf ðtÞ1 , b b which is clearly square integrable. Acknowledgement The authors wish to thank an anonymous referee, Sorana Froda and Manzoor Ahmad Khan whose useful comments have helped improve the presentation. References Antonelli, F., 1993. Backward–forward stochastic differential equations. Ann. Appl. Probab. 3, 777–793. Cvitanic, J., Zhang, J., 2005. The steepest descent method for forward–backward SDEs. Electron. J. Probab. 45, 1468–1495. El Karoui, N., Peng, S., Quenez, M.C., 1997. Backward stochastic differential equation in finance. Math. Finance 7, 1–71. Hu, Y., Peng, S., 1995. Solution of forward–backward stochastic differential equations. Probab. Theory Related Fields 103, 273–283. Karatzas, I., 1997. Lectures on the Mathematics of Finance, CRM Monograph Series, vol. 8. American Mathematical Society, Providence, RI. Karatzas, I., Lehoczky, J.P., Shreve, S.E., 1987. Optimal portfolio and consumption decisions for a small investor on a finite horizon. SIAM J. Control Optim. 25, 1157–1586. Lim, A.E.B., 2004. Quadratic hedging and mean-variance portfolio selection with random parameters in an incomplete market. Math. Oper. Res. 29, 132–161. Lim, A.E.B., Zhou, X.Y., 2002. Mean-variance portfolio selection with random parameters in a complete market. Math. Oper. Res. 27, 101–120. Ma, J., Yong, J., 1999. Forward–Backward Stochastic Differential and their Applications. Lecture Notes in Mathematics, vol. 1702. Springer, New York. Ma, J., Protter, P., Yong, J., 1994. Solving forward–backward stochastic differential equations explicitly—a four step scheme. Probab. Theory Related Fields 98, 339–359. Merton, R.C., 1969. Lifetime portfolio selection under uncertainty: the continuous-time case. Rev. Econom. Statist. 51, 247–257.

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