Sequential optimal consumption and investment for stochastic volatility markets with unknown parameters ^{†}^{†}thanks: The research is funded by the grant of the Government of Russian Federation No.14.A.12.31.0007 and by the Russian Science Fondation (research project No. 144900079) and the second author is partially supported by the International Laboratory of Statistics of Stochastic Processes and Quantitative Finance of Russian National Research Tomsk State
Abstract
We consider an optimal investment and consumption problem for a BlackScholes 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 OrnsteinUhlenbeck 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 factor in an interval for fixed , and use sequential estimation. We show that the consumption and investment strategy calculated through this sequential procedure is optimal.
Key words. Sequential analysis, Truncate sequential estimate, BlackScholes market model, Stochastic volatility, Optimal Consumption and Investment, HamiltonJacobiBellman equation.
AMS subject classifications. 62L12, 62L20, 91B28, 91G80, 93E20.
1 Introduction
We deal with the finitetime optimal consumption and investment problem in a BlackScholes 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 OrnsteinUhlenbeck 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], [30], [27] and references therein). Since then, the consumption and investment problems has been extended in many directions [28]. One of the important generalizations considers financial models with stochastic volatility, since empirical studies of stockprice returns show that the estimated volatility exhibits random characteristics (see e.g., [29] and [10]).
The pure investment problem for such models is considered in [32] and [26]. In these papers, authors use the dynamic programming approach and show that the nonlinear HJB (HamiltonJacobiBellman) equation can be transformed into a quasilinear PDE. The similar approach has been used in [17] for optimal consumptioninvestment problems with the default risk for financial markets with non random coefficients. Furthermore, in [5], by making use of the Girsanov measure transformation the authors study a pure optimal consumption problem 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 integrodifferential 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 OrnsteinUhlenbeck 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 fixedpoint theorem for this mapping and we show that the fixedpoint 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 OrnsteinUhlenbeck type we require sequential analysis methods (see [24] and [20], Sections 17.56). The drift parameter will be estimated from the observations of the process , in some interval . 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 [15] which enables us a nonasymptotic upper bound for mean accuracy estimation. After that we deal with the optimal strategy in the interval , 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 i.e. the strategy calculated through the sequential procedure is 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 LABEL:sec:MarketModelLABEL:sec:HJB we introduce the market model, state the optimization problem and give the related HJB equation. Section LABEL:sec:usefull_definitions is set for definitions. The solution of the optimal consumption and investment problem is given in Sections LABEL:sec:SolutionHJBLABEL:sec:OptimalStrategy. In Section LABEL:sec:deltaoptimal_strategy we consider unknown the drift parameter for the economic factor and use a truncated sequential method to construct its estimate . We obtain an explicit upper for the deviation for any fixed . Moreover considering the optimal consumption investment problem in the finite interval , we show that the strategy calculated through this truncation procedure is optimal. Similar results are given in Section LABEL:sec:muunknown when, in addition of using , we consider an estimate of the unknown stock price appreciation rate. A numerical example is given in Section LABEL:sec:simulation and auxiliary results are reported into the appendix.
2 Market model
Let be a standard and filtered probability space with two standard independent adapted Wiener processes and taking their values in . Our financial market consists of one riskless money market account and one risky stock governed by the following equations:
\hb@xt@.01(2.1) 
with and . In this model is the riskless bond interest rate, is the stock price appreciation rate and is stockvolatility. For all the coefficient is a nonrandom continuous bounded function and satisfies
We assume also that is differentiable and has bounded derivative. Moreover we assume that the stochastic factor valued in is of OrnsteinUhlenbeck type. It has a dynamics governed by the following stochastic differential equation:
\hb@xt@.01(2.2) 
where the initial value is a non random constant, and are fixed parameters. We denote by the process starts at , i.e.
We note, that for the model (LABEL:eq:BSmodel) the risk premium is the function defined as
\hb@xt@.01(2.3) 
For any let denote the amount of investment into bond and the amount of investment into risky assets. We recall that a trading strategy is an valued progressively measurable process and that
is called the wealth process. Moreover, an progressively measurable nonnegative process satisfying for the investment horizon
is called consumption process. The trading strategy and the consumption process are called selffinancing, if the wealth process satisfies the following equation
\hb@xt@.01(2.4) 
where is the initial endowment. Similarly to [14] we consider the fractional portfolio process , i.e. is the fraction of the wealth process invested in the stock at the time . The fraction for the consumption is denoted by . In this case the wealth process satisfies the following stochastic equation
\hb@xt@.01(2.5) 
where and the initial endowment .
Now we describe the set of all admissible strategies. A portfolio control (financial strategy) is said to be admissible if it is  progressively measurable with values in , such that
\hb@xt@.01(2.6) 
and equation (LABEL:eq:EDSX) has a unique strong a.s. positive continuous solution on . We denote the set of admissible portfolios controls by .
In this paper we consider an agent using the power utility function for . The goal is to maximize the expected utilities from the consumption on the time interval , for fixed , and from the terminal wealth at maturity . Then for any , and the value function is defined by
were is the conditional expectation . Our goal is to maximize this function, i.e. to calculate
\hb@xt@.01(2.7) 
For the sequel we will use the notations or simply instead of .
Remark 2.1
Note that the same problem as (LABEL:eq:OptimisationProblem) is solved in [1], but the economic factor considered there is a general diffusion process with bounded coefficients. In the present paper is an OrnsteinUhlenbeck process, so the drift is not bounded, but we take advantage of the fact that is Gaussian and not correlated to the market, which is not the case in [1].
3 HamiltonJacobiBellman equation
Now we introduce the HJB equation for the problem (LABEL:eq:OptimisationProblem). To this end, for any two times differentiable function we denote by and the following vectors of the partial derivatives i.e.
and
Here the prime denotes the transposition. Let now , and be fixed parameters. For these parameters we set
Now we define the Hamilton function as
Note that, in this case for , and
where . The HJB equation is given by
\hb@xt@.01(3.1) 
To study this equation we represent as
\hb@xt@.01(3.2) 
It is easy to deduce that the function satisfies the following quasilinear PDE:
\hb@xt@.01(3.3) 
where
\hb@xt@.01(3.4) 
Note that, using the conditions on ; the function is bounded differentiable and has bounded derivative. Therefore, we can set
\hb@xt@.01(3.5) 
Our goal is to study equation (LABEL:eq:HJBh). By making use of the probabilistic representation for the linear PDE (the FeynmanKac formula) we show in Proposition LABEL:Pr.sec:Prl.5, that the solution of this equation is the fixedpoint solution for a special mapping of the integral type which will be introduced in the next section.
4 Useful definitions
First, to study equation (LABEL:eq:HJBh) we introduce a special functional space. Let be the set of uniformly continuous functions on with values in such that
\hb@xt@.01(4.1) 
where and . Now, we define a metric in as follows: for any in
\hb@xt@.01(4.2) 
Here and is some positive parameter which will be specified later. We define now the process by its dynamics
\hb@xt@.01(4.3) 
So, has the same distribution as . Here is a standard Brownian motion independent of . Let’s now define the FeynmanKac mapping :
\hb@xt@.01(4.4) 
where and
\hb@xt@.01(4.5) 
and is the process starting at . To solve the HJB equation we need to find the fixedpoint solution for the mapping in , i.e.
\hb@xt@.01(4.6) 
To this end we construct the following iterated scheme. We set
\hb@xt@.01(4.7) 
and study the convergence of this sequence in . 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 (LABEL:eq:HJBh). For this, we show some properties of the FeynmanKac operator .
Proposition 5.1
The operator is ”stable” in that is
Proof. Obviously, that for any we have . Moreover, setting
\hb@xt@.01(5.1) 
we represent as
\hb@xt@.01(5.2) 
Therefore, taking into account that and we get
\hb@xt@.01(5.3) 
where the upper bound is defined in (LABEL:def:norme_in_cX0). Now we have to show that is a uniformly continuous function on for any . For any we consider equation (LABEL:eq:HJBh), i.e.
\hb@xt@.01(5.4) 
Setting here we obtain a uniformly parabolic equation for with initial condition . Moreover, we know that has bounded derivative. We deduce that for any from , Theorem 5.1 from [18] (p. 320) with provides the existence of the unique solution of (LABEL:sec:PrL.4) belonging to . Applying the Itô formula to the process
and taking into account equation (LABEL:sec:PrL.4) we get
\hb@xt@.01(5.5) 
Therefore, the function , i.e. for any .
Moreover, for any there exists a sequence from such that
This implies
So is uniformly continuous on i.e. . Hence Proposition LABEL:Pr.sec:PrL.2.
Proposition 5.2
The mapping is a contraction in the metric space , i.e. for any , from
\hb@xt@.01(5.6) 
where
\hb@xt@.01(5.7) 
Actually, as shown in Corollary LABEL:Corolaire:supergeometricrate, an appropriate choice of gives a supergeometric convergence rate for the sequence defined in (LABEL:def:hnsequence), to the limit function , which is the fixed point of the operator .
Proof. First note that, for any and from and for any
We recall that and . Taking into account here that we obtain
Taking into account in the last inequality, that
\hb@xt@.01(5.8) 
we get for all in
\hb@xt@.01(5.9) 
Taking into account the definition of in (LABEL:def:norme_in_cX), we obtain inequality (LABEL:sec:PrL.6). Hence Proposition LABEL:Pr.sec:Prl.3.
Proposition 5.3
The fixed point equation has a unique solution in .
Proof. Indeed, using the contraction of the operator in and the definition of the sequence in (LABEL:def:hnsequence) we get, that for any
\hb@xt@.01(5.10) 
i.e. the sequence is fundamental in . The metric space is complete since it is included in the Banach space , and is equivalent to defined in (LABEL:def:norme_in_cX). Therefore, this sequence has a limit in , i.e. there exits a function from for which
Moreover, taking into account that we obtain, that for any
The last expression tends to zero as . Therefore , i.e. . Proposition LABEL:Pr.sec:Prl.3 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 (LABEL:eq:HJBh) has a unique solution which is the solution of the fixedpoint problem .
Proof. First, note that in view of Lemma LABEL:lemma_4.19*, the function is Hlölderian with respect to on for any . Therefor, choosing in (LABEL:sec:PrL.4) the function (where is the projection of into ) we obtain through Theorem 5.1 from [18] (p. 320) and Lemma LABEL:lemma_4.19*, that the equation (LABEL:sec:PrL.4) has a unique solution . It is clear that the function
is the solution to equation (LABEL:sec:PrL.4) for . Taking into account the representation (LABEL:eq:u=Lf) and the fixed point equation we obtain, that the solution of equation (LABEL:sec:PrL.4)
Therefore, the function satisfies equation (LABEL:eq:HJBh). Moreover, this solution is unique since is the unique solution of the fixed point problem.
Choosing in (LABEL:sec:PrL.4) the function and taking into account the representation (LABEL:eq:u=Lf) and the fixed point equation we obtain, that the solution of equation (LABEL:sec:PrL.4)
Therefore, the function satisfies equation (LABEL:eq:HJBh). Moreover, this solution is unique since is the unique solution of the fixed point problem.
Remark 5.5

We can find in [22] 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 FeynmanKac 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.
6 Supergeometric convergence rate
For the sequence defined in (LABEL:def:hnsequence), and the fixed point solution for , we study the behavior of the deviation
In the following theorem we make an appropriate choice of for the contraction parameter to get the supergeometric convergence rate for the sequence .
Theorem 6.1
The fixed point problem admits a unique solution in such that for any and
\hb@xt@.01(6.1) 
where and is given in (LABEL:def:norme_in_cX).
Proof. Proposition LABEL:Pr.sec:Prl.4 implies the first part of this theorem. Moreover, from (LABEL:sec:PrL.10) it is easy to see, that for each
Thanks to Proposition LABEL:Pr.sec:PrL.2 all the functions belong to , i.e. by the definition of the space
Taking into account that
we obtain the inequality (LABEL:sec:Mr.10). Hence Theorem LABEL:Th.sec:Mr.1.
Now we can minimize the upper bound (LABEL:sec:Mr.10) over . Indeed,
where and
Now we minimize this function over , i.e.
where
Therefore, for
we obtain the optimal upper bound (LABEL:sec:Mr.10).
Corollary 6.2
The fixed point problem has a unique solution in such that for any
\hb@xt@.01(6.2) 
where . Moreover one can check directly that for any
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 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 in [1] must be bounded and so does not allow the OrnsteinUhlenbeck process. Moreover the economic factor is correlated to the market by the Brownian motion , which is not the case in the present paper, since we consider the process independent of .
Theorem 7.1
The optimal value of for the optimization problem (LABEL:eq:OptimisationProblem) is given by
where is the unique solution of equation (LABEL:eq:HJBh). Moreover, for all an optimal financial strategy is of the form
\hb@xt@.01(7.1) 
The optimal wealth process satisfies the following stochastic equation
\hb@xt@.01(7.2) 
where
\hb@xt@.01(7.3) 
The solution can be written as
\hb@xt@.01(7.4) 
where
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 BlackScholes market with unknown stock price appreciation rate and the unknown drift parameter of the economic factor . We observe the process in the interval , and use sequential methods to estimate the drift. After that, we will deal with the consumptioninvestment optimization problem on the finite interval and look for the behavior of the optimal value function under the estimated parameters. We define the value function the estimate of
\hb@xt@.01(8.1) 
is the conditional expectation . is a simplified notation for and from LABEL:def:X*_s we write
\hb@xt@.01(8.2) 
where The functions and are defined as
\hb@xt@.01(8.3) 
The estimated consumption process is and is the unique solution for . The operator is defined by:
\hb@xt@.01(8.4) 
where . The process has the following dynamics:
\hb@xt@.01(8.5) 
Here and are some estimates for the parameters and which will be specified later.
8.1 Sequential procedure
We assume the unknown parameter taking values in some bounded interval , with . We define as the projection onto the interval of the sequential estimate .
\hb@xt@.01(8.6) 
where . Furthermore, we introduce the function , which will serve later for the optimality:
\hb@xt@.01(8.7) 
Here