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SEQUENTIAL δ-OPTIMAL CONSUMPTION AND INVESTMENT FOR STOCHASTIC VOLATILITY
MARKETS WITH UNKNOWN PARAMETERS
B Berdjane, S Pergamenshchikov
To cite this version:
B Berdjane, S Pergamenshchikov. SEQUENTIAL δ-OPTIMAL CONSUMPTION AND INVEST- MENT FOR STOCHASTIC VOLATILITY MARKETS WITH UNKNOWN PARAMETERS. SIAM Theory of Probability and its Applications, Society for Industrial and Applied Mathematics, 2016, 60, pp.533-560. �10.1137/S0040585X97986588�. �hal-02334869�
Vol. 60, No. 4, pp. 000–000
SEQUENTIAL δ-OPTIMAL CONSUMPTION AND INVESTMENT FOR STOCHASTIC VOLATILITY MARKETS WITH UNKNOWN
PARAMETERS∗
B. BERDJANE† AND S. PERGAMENSHCHIKOV‡
Abstract. We consider an optimal investment and consumption problem for a Black–Scholes financial market with stochastic volatility and unknown stock price appreciation rate. The volatility parameter is driven by an external economic factor modeled as a diffusion process of Ornstein–
Uhlenbeck type with unknown drift. We use the dynamical programming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factorY in an interval [0, T0] for fixedT0>0, and use sequential estimation.
We show that the consumption and investment strategy calculated through this sequential procedure isδ-optimal.
Key words. sequential analysis, truncate sequential estimate, Black–Scholes market model, stochastic volatility, optimal consumption and investment, Hamilton–Jacobi–Bellman equation
DOI.10.1137/S0040585X97986588
1. Introduction. We deal with the finite-time optimal consumption and invest- ment problem in a Black–Scholes financial market with stochastic volatility (see, e.g., [7]). We consider the same power utility function for both consumption and terminal wealth. The volatility parameter in our situation depends on some economic factor, modeled as a diffusion process of Ornstein–Uhlenbeck type. The classical approach to this problem goes back to Merton [22].
By applying results from the stochastic control, explicit solutions have been ob- tained for financial markets with nonrandom coefficients (see, e.g., [13], [15], [29], [26], and references therein). Since then, the consumption and investment problems has been extended in many directions [27]. One of the important generalizations consid- ers financial models with stochastic volatility, since empirical studies of stock-price returns show that the estimated volatility exhibits random characteristics (see, e.g., [28] and [10]).
The pure investment problem for such models is considered in [31] and [25]. In these papers, authors use the dynamic programming approach and show that the nonlinear HJB (Hamilton–Jacobi–Bellman) equation can be transformed into a quasi- linear PDE. The similar approach has been used in [16] for optimal consumption- investment problems with the default risk for financial markets with nonrandom coef- ficients. Furthermore, in [5], by making use of the Girsanov measure transformation
∗Received by the editors March 2, 2015; revised June 15, 2015. This work was supported by the grant by the Government of Russian Federation grant No 14.A.12.31.0007, by the Russian Science Fondation project No 14-49-00079, National Research University “MPEI” (Moscow, Russia), by the International Laboratory of Statistics of Stochastic Processes and Quantitative Finance by Russian National Research Tomsk State. Originally published in the Russian journalTeoriya Veroyatnostei i ee Primeneniya, 60 (2015), No. 4, pp. 628–659.
http://www.siam.org/journals/tvp/60-4/T98757.html
†D´epartement de math´ematiques, universit´e du Qu´ebec a Montr´eal, Qu´ebec, Canada (berd- jane.belkacem@uqam.ca).
‡Laboratoire de Math´ematiques Raphael Salem, Avenue de l’Universit´e, BP. 12, Universit´e de Rouen, F76801, Saint Etienne du Rouvray, Cedex France and International Laboratory of Quantitative Finance, National Research University Higher School of Economics, Moscow, Russia (Serge.Pergamenchtchikov@univ-rouen.fr).
1
the authors study a pure optimal consumption problem for stochastic volatility mar- kets. In [2] and [9] the authors use dual methods.
Usually, the classical existence and uniqueness theorem for the HJB equation is shown by the linear PDE methods (see, for example, chapter VI.6 and appendix E in [6]). In this paper we use the approach proposed in [4] and used in [1]. The difference between our work and these two papers is that, in [4], authors consider a pure jump process as the driven economic factor. The HJB equation in this case is an integro- differential equation of the first order. In our case it is a highly non linear PDE of the second order. In [1] the same problem is considered where the market coefficients are known, and depend on a diffusion process with bounded parameters. The result therein does not allow the Gaussian Ornstein–Uhlenbeck process. Similarly to [4]
and [1] we study the HJB equation through the Feynman–Kac representation. We introduce a special metric space in which the Feynman–Kac mapping is a contraction.
Taking this into account we show the fixed-point theorem for this mapping and we show that the fixed-point solution is the classical unique solution for the HJB equation in our case.
In the second part of our paper, we consider both the stock price appreciation rate and the drift of the economic factor to be unknown. To estimate the drift of a process of Ornstein–Uhlenbeck type we require sequential analysis methods (see [23]
and [19, sections 17.5-6]). The drift parameter will be estimated from the observations of the processY, in some interval [0, T0]. It should be noted that in this case the usual likelihood estimator for the drift parameter is a nonlinear function of observations and it is not possible to calculate directly a nonasymptotic upper bound for its accuracy.
To overcome this difficulty we use the truncated sequential estimate from [14] which enables us a nonasymptotic upper bound for mean accuracy estimation. After that we deal with the optimal strategy in the interval [T0, T], under the estimated parameter.
We show that the expected absolute deviation of the objective function for the given strategy is less than some known fixed level δ, i.e., the strategy calculated through the sequential procedure is δ-optimal. Moreover, in this paper we find the explicit form for this level. This allows to keep small the deviation of the objective function from the optimal one by controlling the initial endowment.
The paper is organized as follows: in sections 2, 3 we introduce the market model, state the optimization problem and give the related HJB equation. Section 4 is set for definitions. The solution of the optimal consumption and investment problem is given in sections 5–7. In section 8 we consider unknown the drift parameterαfor the economic factorY and use a truncated sequential method to construct its estimateα.b We obtain an explicit upper for the deviationE|bα−α|for any fixedT0>0. Moreover, considering the optimal consumption investment problem in the finite interval [T0, T], we show that the strategy calculated through this truncation procedure isδ-optimal.
Similar results are given in section 8.3 when, in addition of using α, we consider anb estimate μb of the unknown stock price appreciation rate. A numerical example is given in section 9 and auxiliary results are reported into the appendix.
2. Market model. Let (Ω,FT,(Ft)05t5T,P) be a standard and filtered prob- ability space with two standard independent (Ft)05t5T adapted Wiener processes (Wt)05t5T and (Ut)05t5T taking their values in R. Our financial market con- sists of one riskless money market account S0 = (S0(t))05t5T and one risky stock
S= (S(t))05t5T governed by the following equations:
(2.1) dS0(t) =r S0(t) dt,
dS(t) =S(t)μdt+S(t)σ(Yt) dWt,
withS0(0) = 1 andS(0) =s >0. In this modelr ∈R+ is theriskless bond interest rate, μis the stock price appreciation rate andσ(y) is stock-volatility. For ally ∈R the coefficientσ(y)∈R+ is a nonrandom continuous bounded function and satisfies
yinf∈Rσ(y) =σ1>0.
We assume also thatσ(y) is differentiable and has bounded derivative. Moreover, we assume that the stochastic factor Y valued in R is of Ornstein–Uhlenbeck type. It has a dynamics governed by the following stochastic differential equation:
(2.2) dYt=αYtdt+βdUt,
where the initial value Y0 is a nonrandom constant, α < 0 and β > 0 are fixed parameters. We denote by (Yst,y)s=tthe processY starts atYt=y, i.e.,
Yst,y=yeα(s−t)+ Z s
t
βeα(s−v)dUv.
In this paper we consider the optimization problem on the time interval [T0, T], where 0 5 T0 < T are fixed time limits. Let Xt be the investor capital at a time t ∈[T0, T]. We denote by ϕt ∈R the fraction of the capital invested in stocks (S) and 1−ϕtis the share of the capital invested in the risk-free asset (S0).
The strategy of the investor at the timet∈[T0, T] consists, firstly, in the choice of the proportionϕt, and secondly, in the choice consumption rate ςt(ςt=0). Then, according to the model (2.1) the capital evolution is given by the following equation:
(2.3) dXt = μ ϕtXtdt+σ(Yt)ϕtXtdWt+r(1−ϕt)Xtdt−ςtdt, T05t5T, whereXT0 =x >0 is the initial endowment. Note, that for the model (2.1) the risk premium is theR→R function defined as
(2.4) θ(y) = μ−r
σ(y).
If instead of the pair (ϕt, ςt) one considers the strategy (πt, ct), where πt=σ(Yt)ϕt
andct=ςt/Xt, then the wealth process satisfies the following stochastic differential equation
(2.5) dXt=Xt(r+πtθ(Yt)−ct) dt+XtπtdWt, XT0 =x.
Now we describe the set of all admissible strategies. A portfolio control (financial strategy)ϑ= (ϑt)T05t5T = ((πt, ct))T05t5T is said to beadmissibleif it is (Ft)T05t5T is progressively measurable with values in R×[0,∞), such that the equation (2.5) has a unique strong a.s. positive continuous solution (Xtϑ)T05t5T on [0, T]. We denote the set ofadmissible portfolios controls byV.
In this paper we consider the power utility functions xγ for 0 < γ <1 for the consumption and for the terminal wealth. The goal is to maximize the expected
utilities from the consumption on the time interval [T0, T], for fixedT0, and from the terminal wealth at maturityT. Then for anyx, y ∈R, andϑ∈ V the value function is defined by
J(T0, x, y, ϑ) :=ET0,x,y
Z T T0
cγt(Xtϑ)γdt+ (XTϑ)γ
,
whereET0,x,y is the conditional expectationE(∙ |XT0 =x, YT0 =y). Our goal is to maximize this function, i.e., to calculate
(2.6) J(T0, x, y, ϑ∗) = sup
ϑ∈V
J(T0, x, y, ϑ).
For the sequel we will use the notationJ∗(T0, x, y) or simplyJT∗0instead ofJ(T0, x, y, ϑ∗).
In the case when the parametersαandμare unknown we assume thatα25α5 α1and|μ|5μ∗, where α2< α1<0 andμ∗>0 are known fixed constants.
Our goal is to find a strategy for approaching in a some sense, to the optimal, i.e., we will seek δ-optimal strategies in the sense of the following definition.
Definition 2.1. A strategyϑe∈ V isδ optimal if sup
α25α5α1
sup
|μ|5μ∗
E
J(T0, x, YT0,ϑ)e −J∗(T0, x, YT0) 5δ.
Remark 2.1. Note that if the parametersαandμare known, then we consider the problem (2.6) with T0= 0. It should be note also that for known theses parameters the same problem as (2.6) is solved in [1], but the economic factorY considered there is a general diffusion process with bounded coefficients. In the present paper Y is an Ornstein–Uhlenbeck process, so the drift is not bounded, but we take advantage of the fact thatY is Gaussian and not correlated to the market, which is not the case in [1].
3. Hamilton–Jacobi–Bellman equation. Now we introduce the HJB equa- tion for the problem (2.6). To this end, for any two times differentiable [0, T]×R+× R→Rfunctionf we denote by Df(t, x, y) andD2f(t, x, y) the following vectors of the partial derivatives:
Df(t, x, y) = ∂
∂xf(t, x, y), ∂
∂yf(t, x, y) 0
,
D2f(t, x, y) = ∂2
∂x2f(t, x, y), ∂2
∂y2f(t, x, y) 0
Here the prime denotes the transposition. Let nowq= (q1, q2)∈R2,M= (M1, M2)∈ R2 andν= (ν1, ν2)∈R×R+ be fixed parameters. For these parameters we set
H0(x, y,q,M, ν) := (r+ν1θ(y)−ν2)xq1+αyq2+1
2M1ν12x2+β2
2 M2+ (ν2x)γ. In this case the HJB equation has the following form:
(3.1)
∂
∂tz(t, x, y) + sup
ν∈R×R+
H0(x, y, Dz(t, x, y), D2z(t, x, y), ν) = 0, z(T, x, y) =xγ.
Note, that for x >0,q1>0, andM1<0 sup
ν∈R×R+
H0(x, y,q,M, ν) =xrq1+αyq2+ 1 q∗
γ q1
q∗−1
+|θ(y)q1|2 2|M1| +β2
2 M2, whereq∗= (1−γ)−1. To study this equation we representz(t, x, y) as
(3.2) z(t, x, y) =xγh(t, y).
It is easy to deduce that the functionhsatisfies the following quasi-linear PDE:
(3.3)
∂
∂th(t, y) +Q(y)h(t, y) +αy ∂
∂yh(t, y) +β2
2
∂2
∂y2h(t, y) + 1 q∗
1 h(t, y)
q∗−1
= 0, h(T, y) = 1,
where
(3.4) Q(y) =γ
r+ θ2(y) 2(1−γ)
.
Note that, using the conditions on σ(y), the functionQ(y) is bounded differentiable and has bounded derivative. Therefore, we can set
(3.5) Q∗= sup
y∈RQ(y) and Q∗1= sup
y∈R
dQ(y)
dy .
Our goal is to study equation (3.3). By making use of the probabilistic represen- tation for the linear PDE (the Feynman–Kac formula) we show in Proposition 5.4, that the solution of this equation is the fixed-point solution for a special mapping of the integral type which will be introduced in the following section.
4. Useful definitions. First, to study equation (3.3) we introduce a special functional space. LetX be the set of uniformly continuous functions onK:= [T0, T]× Rwith values in [1,∞) such that
(4.1) kfk∞= sup
(t,y)∈K|f(t, y)| 5r∗,
where r∗ = (Te+ 1)eQ∗Teand Te =T−T0. Now, we define a metric %∗(∙,∙) inX as follows: for anyf, gin X
(4.2) %∗(f, g) =kf−gk∗= sup
(t,y)∈K
e−κ(T−t)|f(t, y)−g(t, y)|.
Here κ=Q∗+ζ+ 1 andζ is some positive parameter which will be specified later.
We define now the processη by its dynamics
(4.3) dηs=αηsds+βdUes, η0=Y0,
So, (ηt)t=0has the same distribution as (Yt)t=0. Here (Uet)t=0is a standard Brownian motion independent of (Ut)t=0. Let us now define theX → X Feynman–Kac mapping L:
(4.4) Lf(t, y) =EG(t, T, y) + 1 q∗
Z T
t Hf(t, s, y) ds,
whereG(t, s, y) = exp{Rs
t Q(ηut,y) du} and (4.5) Hf(t, s, y) =Eh
(f(s, ηst,y))1−q∗G(t, s, y)i
and (ηst,y)t5s5T is the process η starting at ηt =y. To solve the HJB equation we need to find the fixed-point solution for the mappingL inX, i.e.,
(4.6) h=Lh.
To this end we construct the following iterated scheme. We set h0≡1, (4.7) hn(t, y) =Lhn−1(t, y), n=1,
and study the convergence of this sequence in K. Actually, we will use the existence argument of a fixed point, for a contracting operator in a complete metric space.
5. Solution of the HJB equation. We give in this section the existence and uniqueness result, of a solution for the HJB equation (3.3). For this, we show some properties of the Feynman–Kac operatorL..
Proposition 5.1. The operatorL. is“stable”in X that is Lf∈ X ∀f ∈ X.
Proof. Obviously, that for anyf ∈ X we have Lf =1. Moreover, setting
(5.1) fes=f(s, ηt,ys ),
we represent Lf(t, y) as
(5.2) Lf(t, y) =EG(t, T, y) + 1 q∗
Z T t
Eh
(fes)1−q∗G(t, s, y)i ds.
Therefore, taking into account thatfes=1 andq∗=1 we get (5.3) Lf(t, y)5eQ∗(T−t)+
Z T t
1
q∗eQ∗(s−t)ds5r∗,
where the upper bound r∗ is defined in (4.1). Now we have to show that Lf is a uniformly continuous function on K for any f ∈ X. For any f ∈ XT
C1,1(K) we consider equation (3.3), i.e.,
(5.4)
∂
∂tu(t, y) +Q(y)u(t, y) +αy ∂
∂yu(t, y) +β2
2
∂2
∂y2u(t, y) + 1 q∗
1 f(t, y)
q∗−1
= 0, u(T, y) = 1.
Setting hereu(t, y) =e u(T0+T−t, y) we obtain a uniformly parabolic equation forue with initial conditioneu(T0, y) = 1. Moreover, we know thatQhas bounded derivative.
We deduce that for any f from XT
C1,1(K), Theorem 5.1 from [17, p. 320] with
0 < l <1 provides the existence of a unique solution of (5.4) belonging to C1,2(K).
Applying the Itˆo formula to the process
u(s, ηt,ys ) exp Z s
t
Q(ηt,yv ) dv
t5s5T
and taking into account equation (5.4) we get
(5.5) u(t, y) =Lf(t, y).
Therefore, the functionLf(t, y)∈C1,2(K), i.e.,Lf ∈ X for anyf ∈ X ∩C1,1(K).
Moreover, for anyf ∈ X there exists a sequence (fn)n=1 fromX ∩C1,1(K) such that
sup
(t,y)∈K|fn(t, y)−f(t, y)| →0, n→ ∞. This implies
sup
(t,y)∈K|Lfn(t, y)− Lf(t, y)| →0, n→ ∞.
SoLf(t, y) is uniformly continuous onK, i.e.,Lf ∈ X. Proposition 5.1 is proved.
Proposition 5.2. The mappingL is a contraction in the metric space (X, %∗), i.e.,for anyf,g from X
(5.6) %∗(Lf,Lg)5λ%∗(f, g), where
(5.7) λ= 1
ζ+ 1, ζ >0.
Actually, as shown in Corollary 6.1, an appropriate choice of ζgives a supergeo- metric convergence rate for the sequence (hn)n=1defined in (4.7), to the limit function h(t, y), which is the fixed point of the operator L.
Proof. First note, that for anya > b=1 andq >0 b−q−a−q5q(b−a).
Using this bound one can obtain that for anyf andg fromX and for anyy∈R
|Lf(t, y)− Lg(t, y)|5 1 q∗E
Z T
t G(t, s, y)
(fes)1−q∗−(egs)1−q∗ ds 5γE
Z T
t G(t, s, y)|fes−egs|ds.
We recall that fes = f(s, ηst,y) and egs = g(s, ηt,ys ). Taking into account here that G(t, s, y)5eQ∗(s−t)we obtain
|Lf(t, y)− Lg(t, y)|5Z T t
eQ∗(s−t)E|fes−egs|ds.
Taking into account in the last inequality, that
(5.8) |fes−egs|5 eκ(T−s)%∗(f, g) a.s.
we get for all (t, y) inK
(5.9) e−κ(T−t)(Lf(t, y)− Lg(t, y))5 1
κ−Q∗%∗(f, g).
Taking into account the definition of κ in (4.2), we obtain inequality (5.6). Hence Proposition 5.2 is proved.
Proposition 5.3. The fixed point equationLh=hhas a unique solution inX. Proof. Indeed, using the contraction of the operatorLinX and the definition of the sequence (hn)n=1in (4.7) we get, that for anyn=1
(5.10) %∗(hn, hn−1)5λn−1%∗(h1, h0),
i.e., the sequence (hn)n=1 is fundamental in (X, %∗). The metric space (X, %∗) is complete since it is included in the Banach space C0,0(K), andk ∙ k∞ is equivalent tok ∙ k∗defined in (4.2). Therefore, this sequence has a limit in X, i.e., there exits a function hfromX for which
nlim→∞%∗(h, hn) = 0.
Moreover, taking into account thathn =Lhn−1 we obtain, that for anyn=1
%∗(h,Lh)5%∗(h, hn) +%∗(Lhn−1,Lh)5%∗(h, hn) +λ%∗(h, hn−1).
The last expression tends to zero as n→ ∞. Therefore, %∗(h,Lh) = 0, i.e.,h=Lh. Proposition 5.2 implies immediately that this solution is unique.
We are ready to state the result about the solution of the HJB equation.
Proposition 5.4. The HJB equation (3.3) has a unique solution which is the solution hof the fixed-point problem Lh=h.
Proof. First, note that in view of Lemma 10.5, the function Lh(t, y) is 1/2- Hl¨olderian with respect toton|y|< nfor anyn=1. Therefore, choosing in (5.4) the functionf =fn(t, y) =u(t,eyn) (whereeynis the projection ofyinto [−n, n]) we obtain through Theorem 5.1 from [17, p. 320] and Lemma 10.5, that the equation (5.4) has a unique solution un(t, y). It is clear that the function
u(t, y) =X
n=1
un(t, y)1{n−1<|y|5n}
is the solution to equation (5.4) for f = u(t, y). Taking into account the represen- tation (5.5) and the fixed point equation Lh = h we obtain, that the solution of equation (5.4) is
u=Lh=h.
Therefore, the function h satisfies equation (3.3). Moreover, this solution is unique sincehis a unique solution of the fixed point problem.
Choosing in (5.4) the function f = u and taking into account the representa- tion (5.5) and the fixed point equationLh=hwe obtain, that the solution of equa- tion (5.4) is
u=Lh=h.
Therefore, the function h satisfies equation (3.3). Moreover, this solution is unique sincehis the unique solution of the fixed point problem. Proposition 5.4 is proved.
Remark 5.1. (1) We can find in [21] an existence and uniquness proof for a more general quasilinear equation but therein, authors did not give a way to calculate this solution, whereas in our case, the solution is the fixed point function for the Feynman–
Kac operator. Moreover our method allows to obtain the super geometric convergence rate for the sequence approximating the solution, which is a very important property in practice. In [3] author shows an existence and uniquness result where the global result is deduced from a local existence and uniqueness theorem.
The application of contraction mapping or fixed-point theorem to solve nonlinear PDE in not new see, e.g., [8] and [24] where the term “generalised solution” is used for quasilinear/semilinear PDE, and the fixed point of the Feynamn–Kac representation is discussed.
6. Supergeometric convergence rate. For the sequence (hn)n=1 defined in (4.7), andhthe fixed point solution forh=Lh, we study the behavior of the deviation
Δn(t, y) =h(t, y)−hn(t, y).
In the following theorem we make an appropriate choice ofζ for the contraction parameterλto get the super-geometric convergence rate for the sequence (hn)n=1.
Theorem 6.1. The fixed point problemLh=hadmits a unique solution hin X such that for anyn=1 andζ >0
(6.1) sup
(t,y)∈K|Δn(t, y)|5B∗λn, whereB∗=eκTe(1 +r∗)/(1−λ)andκ is given in (4.2).
Proof. Proposition 5.3 implies the first part of this theorem. Moreover, from (5.10) it is easy to see, that for eachn=1
%∗(h, hn)5 λn
1−λ%∗(h1, h0).
Thanks to Proposition 5.1 all the functions hn belong toX, i.e., by the definition of the spaceX
%∗(h1, h0)5 sup
(t,y)∈K|h1(t, y)−1|51 +r∗. Taking into account that
sup
(t,y)∈K|Δn(t, y)|5eκTe%∗(h, hn), we obtain the inequality (6.1). Hence Theorem 6.1 is proved.
Now we can minimize the upper bound (6.1) over ζ >0. Indeed, B∗λn=C∗exp{gn(ζ)},
whereC∗= (1 +r∗)e(Q∗+1)eT and
gn(x) =xTe−logx−(n−1) log(1 +x).
Now we minimize this function over x >0, i.e.,
minx>0gn(x) =x∗nTe−logx∗n−(n−1) log(1 +x∗n), where
x∗n= q
(Te−n)2+ 4Te+n−Te
2Te .
Therefore, forζ=ζn∗=x∗n we obtain the optimal upper bound (6.1).
Corollary 6.1. The fixed point problem has a unique solution hinX such that for any n=1
(6.2) sup
(t,y)∈K|Δn(t, y)|5U∗n,
whereU∗n =C∗exp{g∗n}. Moreover,one can check directly that for any0< δ <1 U∗n =O(n−δn), n→ ∞.
This means that the convergence rate is more rapid than any geometric one,i.e.,it is supergeometric.
7. Known parameters. We consider our optimal consumption and investment problem in the case of markets with known parameters. The following theorem is the analogous of Theorem 3.4 in [1]. The main difference between the two results is that the drift coefficient of the processY in [1] must be bounded and so does not allow the Ornstein–Uhlenbeck process. Moreover, the economic factor Y is correlated to the market by the Brownian motion U, which is not the case in the present paper, since we consider the processUindependent ofW.
Theorem 7.1. The optimal value of J(T0, x, y, ϑ) for the optimization prob- lem (2.6)is given by
JT∗0=J(T0, x, y, ϑ∗) = sup
ϑ∈V
J(T0, x, y, ϑ) =xγh(T0, y),
whereh(t, y)is a unique solution of equation (3.3). Moreover, for allT05t5T an optimal financial strategy ϑ∗= (π∗, c∗)is of the form
(7.1) π∗t =π∗(Yt) = θ(Yt)
1−γ, c∗t =c∗(t, Yt) = (h(t, Yt))−q∗.
The optimal wealth process(Xt∗)T05t5T satisfies the following stochastic equation:
(7.2) dXt∗ =a∗(t, Yt)Xt∗dt+Xt∗b∗(Yt) dWt, XT∗0=x, where
(7.3) a∗(t, y) = |θ(y)|2
1−γ +r−(h(t, y))−q∗, b∗(y) = θ(y) 1−γ. The solutionXt∗ can be written as
(7.4) Xs∗=Xt∗exp
Z s t
a∗(v, Yv) dv
Et,s, whereEt,s= exp{Rs
t b∗(Yv) dWv−(1/2)Rs
t |b∗(Yv)|2dv}.
The proof of the theorem follows the same arguments, as Theorem 3.4 in [1], so it is omitted.
