HAL Id: hal-00743164
https://hal.archives-ouvertes.fr/hal-00743164v2
Preprint submitted on 25 Aug 2013
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires
Sequential δ-optimal consumption and investment for stochastic volatility markets with unknown parameters
Belkacem Berdjane, Sergei Pergamenshchikov
To cite this version:
Belkacem Berdjane, Sergei Pergamenshchikov. Sequentialδ-optimal consumption and investment for stochastic volatility markets with unknown parameters. 2012. �hal-00743164v2�
Sequential δ-optimal consumption and investment for stochastic volatility markets
with unknown parameters
Belkacem Berdjane∗and Serguei Pergamenshchikov† August 25, 2013
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 program- ming approach and find an optimal financial strategy which depends on the drift parameter. To estimate the drift coefficient we observe the economic factor Y 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 Consump- tion and Investment, Hamilton-Jacobi-Bellman equation.
MS Classification (2010) 62L12, 62L20, 91B28, 91G80, 93E20.
JEL Classification G11, C13.
∗Laboratoire L2CSP, de l’universit´e de Tizi-Ouzou (Alg´erie) & Laboratoire LMRS, de l’universit´e de Rouen (France); email: berdjane b@yahoo.fr
†Laboratoire de Math´ematiques Raphael Salem, Avenue de l’Universit´e, BP. 12, Uni- versit´e de Rouen, F76801, Saint Etienne du Rouvray, Cedex France and Department of Mathematics and Mechanics,Tomsk State University, Lenin str. 36, 634041 Tomsk, Russia, e-mail: Serge.Pergamenchtchikov@univ-rouen.fr
1 Introduction
We deal with the finite-time optimal consumption and investment problem in a Black-Scholes financial market with stochastic volatil- ity (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 diffu- sion process of Ornstein-Uhlenbeck type. The classical approach to this problem goes back to Merton [23].
By applying results from the stochastic control, explicit solutions have been obtained for financial markets with nonrandom coefficients (see, e.g. [13], [16] and references therein). Since then, the consump- tion and investment problems has been extended in many directions [27]. One of the important generalizations considers 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 [29]
and [26]. In these papers, authors use the dynamic programming ap- proach and show that the nonlinear HJB (Hamilton-Jacobi-Bellman) equation can be transformed into a quasilinear PDE. The similar ap- proach has been used in [17] for optimal consumption-investment prob- lems with the default risk for financial markets with non random co- efficients. Furthermore, in [5], by making use of the Girsanov measure transformation the authors study a pure optimal consumption prob- lem for stochastic volatility markets. 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 re- quire sequential analysis methods (see [24] and [20], 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 non-asymptotic upper bound for its accuracy. To overcome this difficulty we use the truncated sequential estimate from [15] which en- ables us a non-asymptotic 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 se- quential 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 parameter α for the economic factorY and use a truncated sequential method to construct its estimateα. We obtain an explicit upper for the deviationb E|bα−α| for any fixed T0 >0. Moreover considering the optimal consumption investment problem in the finite interval [T0, T], we show that the strategy calculated through this truncation procedure is δ-optimal.
Similar results are given in Section 8.3 when, in addition of using α,b we consider an 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)0≤t≤T,P) be a standard and filtered probability space with two standard independent (Ft)0≤t≤T adapted Wiener processes (Wt)0≤t≤T and (Ut)0≤t≤T taking their values inR. Our financial mar- ket consists of oneriskless money market account(S0(t))0≤t≤T and one risky stock (S(t))0≤t≤T governed by the following equations:
dS0(t) =r S0(t) dt ,
dS(t) =S(t)µdt+S(t)σ(Yt) dWt,
(2.1) with S0(0) = 1 and S(0) = s > 0. In this model r ∈ R+ is the riskless bond interest rate, µ is the stock price appreciation rate and σ(y) is stock-volatility. For all y ∈ R the coefficient σ(y) ∈ R+ is a nonrandom continuous bounded function and satisfies
y∈infRσ(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:
dYt=αYtdt+βdUt, (2.2) where the initial valueY0 is a non random constant, α <0 andβ >0 are fixed parameters. We denote by (Yst,y)s≥t the processY starts at Yt=y, i.e.
Yst,y =yeα(s−t)+ Z s
t
βeα(s−v)dUv.
We note, that for the model (2.1) the risk premium is the R → R function defined as
θ(y) = µ−r
σ(y) . (2.3)
Similarly to [14] we consider the fractional portfolio process ϕ(t), i.e. ϕ(t), is the fraction of the wealth process Xt invested in the stock at the time t. The fractions for the consumption is denoted by c = (ct)0≤t≤T. In this case the wealth process satisfies the following stochastic equation
dXt=Xt(r +πtθ(Yt)−ct) dt+XtπtdWt, (2.4)
where πt=σ(Yt)ϕt and the initial endowment X0=x.
Now we describe the set of all admissible strategies. A portfolio control (financial strategy) ϑ = (ϑt)t≥0 = ((πt, ct))t≥0 is said to be admissibleif it is (Ft)0≤t≤T - progressively measurable with values in R×[0,∞), such that
kπkT :=
Z T
0
|πt|2dt <∞ and Z T
0
ctdt <∞ a.s. (2.5) and equation (2.4) has a unique strong a.s. positive continuous solu- tion (Xtϑ)0≤t≤T on [0, T]. We denote the set of admissible portfolios controls byV.
In this paper we consider an agent using the power utility function xγ for 0< γ <1. The goal is to maximize the expected utilities from the consumption on the time interval [T0, T], for fixed T0, and from the terminal wealth at maturity T. 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
! ,
were ET0,x,y is the conditional expectation E(.|XT0 = x, YT0 = y).
Our goal is to maximize this function, i.e. to calculate J(T0, x, y, ϑ∗) = sup
ϑ∈V
J(T0, x, y, ϑ). (2.6) For the sequel we will use the notations J∗(T0, x, y) or simply JT∗0 instead of J(T0, x, y, ϑ∗), moreover we set Te= [T −T0].
Remark 2.1. Note that the same problem as (2.6)is solved in [1], but the economic factor Y 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 that Y 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 equation for the problem (2.6). To this end, for any differentiable function f we denote by Dyf(t, y) and
Dx,yf(t, x, y) its partial derivatives i.e.
Dyf(t, y) = ∂
∂yf(t, y) and Dx,yf(t, x, y) = ∂2f(t, x, y)
∂x∂y . (3.1) Moreover we denote by D2x,yf(t, x, y) the Hessian of f, that is the square matrix of second order partial derivatives with respect to x and y.
Let now (q1,q2)∈R2 be fixed parameters and M= M11; M12
M21; M22
!
, Mij ∈R.
For these parameters with q1>0 we define the Hamilton function as H(t, x, y,q1,q2,M) =x rq1+αq2+ 1
q∗ γ
q1 q∗−1
+|θ(y)q1|2 2|M11| + β2
2 M22 , (3.2)
where q∗= (1−γ)−1. The HJB equation is given by
( zt(t, x, y) +H(t, x, y,Dxz(t, x, y),Dyz(t, x, y),D2x,yz(t, x, y)) = 0 z(T, x, y) =xγ.
(3.3) To study this equation we represent z(t, x, y) as
z(t, x, y) =xγh(t, y). (3.4) It is easy to deduce that the function h satisfies the following quasi- linear PDE:
ht(t, y) +Q(y)h(t, y) +α yDyh(t, y) +β2
2 Dy,yh(t, y) +1
q∗ 1
h(t, y) q∗−1
= 0 ; h(T, y) = 1.
(3.5)
We recall that q∗= 1/(1−γ) and we define Q(y) =γ
r + |θ(y)|2 2 (1−γ)
. (3.6)
Note that, using the conditions on σ(y); the functionQ(y) is bounded differentiable and has bounded derivative. Therefore, we can set
Q∗ = sup
y∈R
Q(y) and Q∗1 = sup
y∈R |DyQ(y)|. (3.7) Our goal is to study equation (3.5). By making use of the proba- bilistic representation 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 next section.
4 Useful definitions
First, to study equation (3.5) we introduce a special functional space.
Let X be the set of continuous functions defined onK := [T0, T]×R with values in [1,∞) such that
kfk∞= sup
(t,y)∈K |f(t, y)| ≤r∗, (4.1) where
r∗ = (Te+ 1)eQ∗Te. (4.2) Now, we define a metric ̺∗(., .) inX as follows: for anyf, g inX
̺∗(f, g) =kf −gk∗, kfk∗ = sup
(t,y)∈K
e−κ(T−t)|f(t, y)|, (4.3) where
κ=Q∗+ζ + 1. (4.4)
Here ζ is any positive parameter which will be specified later.
We define now the process η by its dynamics
dηs =α ηsds+ βdUes with η0=Y0 (4.5) so that ηt has the same distribution as Yt. Here (Uet) is a standard Brownian motion independent of (Ut). Let’s now define the X → X Feynman-Kac mapping L:
Lf(t, y) =EG(t, T, y) + 1 q∗
Z T
t Hf(t, s, y) ds , (4.6)
where G(t, s, y) = expRs
t Q(ηt,yu ) du and Hf(t, s, y) =E f(s, ηt,ys )1−q∗
G(t, s, y). (4.7) and (ηt,ys )t≤s≤T is the processη starting atηt=y. To solve the HJB equation we need to find the fixed-point solution for the mapping L inX, i.e.
Lh =h . (4.8)
To this end we construct the following iterated scheme. We seth0 ≡1 hn(t, y) =Lhn−1(t, y) for n≥1. (4.9) and study the convergence of this sequence inK. Actually, we will use the existence argument of a fixed point, for a contracting operator in a complete metrical space.
5 Solution of the HJB equation
We give in this section the existence and uniqueness result, of a solu- tion for the HJB equation (3.5). For this, we show some properties of the Feynman-Kac operator L..
Proposition 5.1. The operator L. is ”stable” in X that is Lf ∈ X, ∀f ∈ X.
Moreover, Lf ∈C1,2(K) for any f ∈ X.
Proof. Obviously, that for any f ∈ X the mappingLf is continuous and Lf ≥1. Moreover, setting
fes=f(s, ηt,ys ), (5.1) we represent Lf(t, y) as
Lf(t, y) =EG(t, T, y) + 1 q∗
Z T
t
E
fes1−q∗
G(t, s, y)ds . (5.2) Therefore, taking into account that fes≥1 andq∗≥1 we get
Lf(t, y) ≤ eQ∗(T−t)+ Z T
t
1
q∗ eQ∗(s−t)ds ≤ r∗ , (5.3)
where the upper bound r∗ is defined in (4.2). Now we have to show thatLf ∈C1,2(K), for anyf ∈ X. Indeed, to this end we consider for any f from X the equation (3.5), i.e.
ut(t, y) +Q(y)u(t, y) + αDyu(t, y) +β2
2 Dy,yu(t, y) + 1 q∗
1 f(t, y)
q∗−1
= 0 ; u(T, y) = 1.
(5.4)
Setting hereeu(t, y) =u(T0+T−t, y) we obtain a uniformly parabolic equation for euwith initial condition eu(T0, y) = 1. Moreover, we know that Q has bounded derivative. Therefore, for any f from X Theo- rem 5.1 from [18] (p. 320) with 0 < l < 1 provides the existence of the unique solution of (5.4) belonging to C1,2(K). Applying the Itˆo formula to the process
u(s, ηst,y)e
Rs
t Q(ηt,yv ) dv
t≤s≤T
and taking into account equation (5.4) we get
u(t, y) =Lf(t, y). (5.5) Therefore, the functionLf(t, y)∈C1,2(K), i.e. Lf ∈ X for anyf ∈ X. Hence Proposition 5.1.
Proposition 5.2. The mappingLis a contraction in the metric space (X, ̺∗), i.e. for anyf, g from X
̺∗(Lf,Lg)≤λ̺∗(f, g), (5.6) where the parameter 0< λ <1 is given by
λ= 1
ζ+ 1, ζ >0. (5.7)
Actually, as shown in Corollary 6.2, an appropriate choice ofζgives a super-geometric convergence rate for the sequence (hn)n≥1 defined in (4.9), to the limit function h(t, y), which is the fixed point of the operator L.
Proof. First note that, for any f and g fromX and for any y∈R
|Lf(t, y)− Lg(t, y)| ≤ 1 q∗ E
Z T
t G(t, s, y)
fes1−q∗
−(egs)1−q∗ ds
≤γE Z T
t
G(t, s, y) efs−egsds .
We recall that fes=f(s, ηt,ys ) andges=g(s, ηt,ys ). Taking into account here that G(t, s, y)≤eQ∗(s−t) we obtain
|Lf(t, y)− Lg(t, y)| ≤ Z T
t
eQ∗(s−t)E|fes−egs|ds . Taking into account in the last inequality, that
|fes−egs| ≤ eκ(T−s)̺∗(f, g) a.s., (5.8) we get for all (t, y) in K
e−κ(T−t) Lf(t, y)− Lg(t, y)≤ 1
κ−Q∗̺∗(f, g). (5.9) Taking into account the definition of κ in (4.4), we obtain inequality (5.6). Hence Proposition 5.2.
Proposition 5.3. The fixed point equation Lh =h has a unique so- lution in X.
Proof. Indeed, using the contraction of the operator L inX and the definition of the sequence (hn)n≥1 in (4.9) we get, that for any n≥1
̺∗(hn, hn−1)≤λn−1̺∗(h1, h0), (5.10) 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 (4.3). Therefore, this sequence has a limit in X, i.e. there exits a function h fromX for which
n→∞lim ̺∗(h, hn) = 0.
Moreover, taking into account that hn = Lhn−1 we obtain, that for any n≥1
̺∗(h,Lh)≤̺∗(h, hn) +̺∗(Lhn−1,Lh)≤̺∗(h, hn) +λ̺∗(h, hn−1).
The last expression tends to zero asn→ ∞. Therefore̺∗(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 equa- tion:
Proposition 5.4. The HJB equation (3.5) has a unique solution which is the solution h of the fixed-point problem Lh =h.
Proof. Choosing in (5.4) the function f = u and taking into ac- count the representation (5.5) and the fixed point equation Lh = h we obtain, that the solution of equation (5.4)
u=Lh=h .
Therefore, the functionhsatisfies equation (3.5). Moreover, this solu- tion is unique sincehis the unique solution of the fixed point problem.
Remark 5.1. 1. We can find in [22] an existence and uniquness proof for a more general quasilinear equation but therein, au- thors 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.
2. The application of contraction mapping or fixed-point theorem to solve nonlinear PDE in not new see, e.g. [8] and [25] where the term ”generalised solution” is used for quasilinear/semilinear PDE, and the fixed point of the Feynamn-Kac representation is discussed.
6 Super-geometrical convergence rate
For the sequence (hn)n≥1 defined in (4.9), andh the fixed point solu- tion for h=Lh, we study the behavior of the deviation
∆n(t, y) =h(t, y)−hn(t, y).
In the following theorem we make an appropriate choice of ζ for the contraction parameter λto get the super-geometric convergence rate for the sequence (hn)n≥1.
Theorem 6.1. The fixed point problem Lh =h admits a unique so- lution h in X such that for any n≥1 and ζ >0
sup
(t,y)∈K |∆n(t, y)| ≤B∗λn, (6.1) where B∗=eκTe(1 +r∗)/(1−λ) andκ is given in (4.4).
Proof. Proposition 5.3 implies the first part of this theorem. More- over, from (5.10) it is easy to see, that for each n≥1
̺∗(h, hn)≤ λn
1−λ̺∗(h1, h0).
Thanks to Proposition 5.1 all the functions hn belong to X, i.e. by the definition of the space X
̺∗(h1, h0)≤ sup
(t,y)∈K |h1(t, y)−1| ≤1 +r∗. Taking into account that
sup
(t,y)∈K |∆n(t, y)| ≤eκTe̺∗(h, hn), we obtain the inequality (6.1). Hence Theorem 6.1.
Now we can minimize the upper bound (6.1) over ζ >0. Indeed, B∗λn=C∗ exp{gn(ζ)},
where C∗ = (1 +r∗)e(Q∗+1)Te and
gn(x) =xTe −lnx−(n−1) ln(1 +x). Now we minimize this function overx >0, i.e.
minx>0gn(x) =x∗nTe −lnx∗n−(n−1) ln(1 +x∗n), where
x∗n= q
(Te −n)2+ 4Te +n− Te
2Te .
Therefore, for
ζ =ζn∗=x∗n we obtain the optimal upper bound (6.1).
Corollary 6.2. The fixed point problem has a unique solutionh inX such that for any n≥1
sup
(t,y)∈K |∆n(t, y)| ≤U∗n, (6.2) where U∗n = C∗ exp{g∗n}. Moreover one can check directly that for any 0< δ <1
U∗n=O(n−δn) as n→ ∞.
This means that the convergence rate is more rapid than any geometric one, i.e. it is super-geometric.
7 Known parameters
We consider our optimal consumption and investment problem in the case of markets with known parameters. The next theorem is the analogous of theorem 3.4 in [1]. The main difference between the two results is that the drift coefficient of the process Y 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 processU independent of W.
Theorem 7.1. The optimal value ofJ(T0, x, y, ϑ)for the optimization problem (2.6) is given by
JT∗0 =J(T0, x, y, ϑ∗) = sup
ϑ∈V
J(T0, x, y, ϑ) =xγh(T0, y)
where h(t, y) is the unique solution of equation (3.5). Moreover, for all T0 ≤ t ≤ T an optimal financial strategy ϑ∗ = (π∗, c∗) is of the
form
π∗t =π∗(Yt) = θ(Yt) 1−γ ; c∗t =c∗(t, Yt) = (h(t, Yt))−q∗ .
(7.1) The optimal wealth process (Xt∗)T0≤t≤T satisfies the following stochas- tic equation
dXt∗ =a∗(t, Yt)Xt∗dt+Xt∗b∗(Yt) dWt, XT∗
0 = x , (7.2)
where
a∗(t, y) = |θ(y)|2
1−γ +r −(h(t, y))−q∗ ; b∗(y) = θ(y)
1−γ .
(7.3)
The solution Xt∗ can be written as Xs∗ =Xt∗e
Rs
t a∗(v,Yv) dv
Et,s, (7.4)
where Et,s= expnRs
t b∗(Yv) dWv−12Rs
t |b∗(Yv)|2dvo .
The proof of the theorem follows the same arguments, as Theorem 3.4 in [1], so it is omitted.
8 Unknown parameters
In this section we consider the Black-Scholes market with unknown stock price appreciation rate µ. Moreover, we consider unknown the drift parameterα of the economic factorY. We observe the processY in the interval [0, T0], and use sequential methods to estimate the drift.
After that, we will deal with the consumption-investment optimization problem on the finite interval [T0, T] and look for the behavior of the optimal value function J∗(T0, x, y) under the estimated parameters.
8.1 Sequential procedure
We assume the unknown parameter α taking values in some bounded interval [α2, α1], withα2 < α < α1 <0. We define the function ǫ(.), which will serve later for theδ-optimality:
ǫ(T0) = sβ2
H + α22 β12
k(3) T02
. (8.1)
Here H=β2(T0−T0ε), β2=β2/2|α2|,ε= 5/6 and
k(m) = 32m−1 Y02m+ (1 + (m(2m−1))m(2β)2m)k1(m) , with k1(m) = 22m−1 Y02m+ (2m−1)!!β1m
and β1 =β2/2|α1|. The proposition bellow gives αb the truncated sequential estimate ofα and gives a bound for the expected deviation E|bα−α|. We set for the sequel α=αb−α.
Proposition 8.1. We can find αb an estimate forα, such that E|bα−α| ≤ǫ(T0).
More precisely we define αb as the projection onto the interval [α2, α1] of the sequential estimate α∗.
b
α=P roj[α2,α1]α∗, α∗ = RτH
0 YtdYt H
1{τH≤T0} (8.2) where τH = infn
t≥0,Rt
0Ys2ds≥Ho .
Proof. Note first that E|bα−α| ≤E|α∗−α|, so it is enough to show thatE|α∗−α| ≤ǫ(T0). Moreover, we know from [20] chapter 17, that the maximum likelihood estimate ofα is given by
RT0 0 YtdYt RT0
0 Yt2dt with Z ∞
0
Yt2dt= +∞ a.s.
We define byαetheα-sequential that is e
α= RτH
0 YtdYt RτH
0 Yt2dt =α+β RτH
0 YtdUt
H ,
so that αe N(α, β2/H) and henceE|eα−α|2 =β2/H.
The problem with the previous estimate is that τH may be greater thanT0. To overcome this difficulty we define the truncated sequential estimate α∗ as in the theorem ie: α∗ =αe1{τH≤T0}. We observe that
α∗−α = (α∗−α)1{τH≤T0}+ (α∗−α)1{τH>T0}
= β
RτH
0 YtdUt
H 1{τH≤T0}−α1{τH>T0}. So
E(α∗−α)2 = Hβ22 E Z τH
0
YtdUt1(τH≤T0) 2
+α2P(τH > T0)
≤ β2 H2 E
Z τH
0
YtdUt
2
+α2P(τH > T0)
≤ β2
H +α2P(
Z T0
0
Yt2dt < H). (8.3)
