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DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS Supplement Volume 2005

Website: http://AIMsciences.org pp. 345–354

OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN

E.V. Grigorieva Department of Mathematics and Computer Sciences Texas Woman’s University Denton, TX 76204, USA

E.N. Khailov Department of Computer Mathematics and Cybernetics Moscow State Lomonosov University Moscow, 119992, Russia Abstract. We consider a controlled system of differential equations modeling a firm that takes a loan in order to expand its production activities. The objective is to determine the optimal loan repayment schedule using the variables of the business current profitability, the bank’s interest rate on the loan and the cost of reinvestment of capital. The portion of the annual profit which a firm returns to the bank and the value of the total loan taken by the firm are control parameters. We consider a linear production function and investigate the attainable sets for the system analytically and numerically. Optimal control problems are stated and their solutions are found using attainable sets. Attainable sets for different values of the parameters of the system are constructed with the use of a computer program written in MAPLE. Possible economic applications are discussed.

1. Introduction. Global competition and increasingly sophisticated business strategies are forcing businesses to make the most of every dollar. Profitable companies often take out commercial loans for expansion of production in order to remain at the forefront of their fields or in hopes of surpassing competitors. The commercial loans, unlike the individual loans, do not necessarily require monthly installments, giving to businesses an opportunity to decide the most advantageous schedule for repayment. From a mathematical point of view it is of interest to model the dynamics of such a situation and find the optimal loan repayment schedule. Let us consider a microeconomic model of a firm which takes a loan of value S at k% annual interest for T years. As a basis for our studies, we took the model created by Tokarev ([1],[2]) and investigated it respectively two control parameters: the portion of the profit which the firm decides to repay to the bank , u(t), and the total loan amount S, that we suggest the company borrows. Naturally, 0 ≤ u(t) ≤ 1 ¯ here S¯ is a maximum value of the credit. We assume that the entire and 0 ≤ S ≤ S; loan S is taken at moment t = 0, and it is immediately transformed into production funds x1 . The firm repays the debt and the interest, which is added continuously using the compound interest formula under the fixed total interest rate k. The firm has the right to chose a loan repayment schedule; the only constraint is that the total debt must be completely paid off at the moment t = T . Then the current 2000 Mathematics Subject Classification. 49J15, 58E25, 90A16, 93B03. Key words and phrases. Controlled system, attainable set, microeconomic dynamical model.

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debt, x2 (T ), becomes 0. The current net profit of the firm, P , can be determined by the production function F (x1 ) with respect to the achieved level of production funds x1 as P = rF (x1 ), F (x01 ) = x01 , F 0 (x1 ) > 0, F 00 (x1 ) ≤ 0. Here r (year −1 ) is the profitability of the firm with initial funds x01 . The firm is distributing the flow of its pure net profit P between the repayment to the bank ( uP ) and refinancing its further development ( (1 − u)P ). The proportion between these two expenses can be changed by the firm as a function of time u = u(t). Using the assumptions above, the dynamics of production funds x1 (t) of the firm with the profit P and the debt x2 (t) can be written as a nonlinear controlled system :  x˙ 1 (t) = r(1 − u(t))F (x1 (t)), t ∈ [0, T ],      x˙ 2 (t) = kx2 (t) − ru(t)F (x1 (t)), (1)  x1 (0) = x01 + S, x2 (0) = S, x01 > 0,     ¯ 0 ≤ u(t) ≤ 1, 0 ≤ S ≤ S.

We assume that the firm’s objective is maximizing its final production funds x 1 (T ), which maximizes the continuous profit, and that the debt of the firm must be completely paid off at time t = T , that is x2 (T ) = 0. For a model with the linear production function, we investigate the attainable set and its properties for different parameters of the model. We prove that the attainable set is a compact convex set and that each point of the boundary of the attainable set can be reached by exactly one control taking values {0; 1} with at most one switching. Furthermore, we formulate the optimal control problems and look for their solutions using attainable sets and their properties. This approach of solving the optimal control problems differs our work from the works of Tokarev ([1],[2]). Moreover, the model is completely investigated analytically and numerically and can be used as a study tool at the business schools. 2. Model with Linear Production Function. The model (1) with the linear production function can be rewritten as  x˙ (t) = r(1 − u(t))x1 (t), t ∈ [0, T ],   1 x˙ 2 (t) = kx2 (t) − ru(t)x1 (t), (2)   0 0 x1 (0) = x1 + S, x2 (0) = S, x1 > 0, S > 0, Here x1 , x2 are variables, and r, k, S, x01 and T are constant parameters; the function u(t) is the control function. Thus, the system (2) is a controlled bilinear system. We call D(T ) the control set, such that D(T ) is the set of all Lebesgue measurable functions u(t), satisfying the inequality 0 ≤ u(t) ≤ 1 for almost all t ∈ [0, T ]. The following lemma describes the property of the variable x1 for system (2).

Lemma 1. Let u(·) ∈ D(T ) be a control function. Then, the component x 1 (t) of the solution x(t) = (x1 (t), x2 (t))> of the system (2) corresponding to this control u(t), satisfies the inequality x1 (t) > 0, t ∈ [0, T ].

(3)

Here and further, the symbol > means transpose. Proof is by direct integration of the first equation of the system (2).

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3. An Attainable Set and its Properties. We denote X(T ) an attainable set for system (2) reachable from an initial condition x0 = (x01 , x02 )> at moment T ; that is X(T ) is the set of all ends x(T ) = (x1 (T ), x2 (T ))> of trajectories x(t) = (x1 (t), x2 (t))> for system (2) under corresponding admissible controls u(·) ∈ D(T ). The set X(T ) is a compact set in R2 . ([3, pp.265-267]). Then by Lemma 1 and work [4, Theorem 1], the following Theorem holds. Theorem 1. An attainable set X(T ) is a convex set in R2 . Now, we consider the case when r = k. By subtracting from the first equation d (x1 (t) − x2 (t)) = r(x1 (t) − x2 (t)), of system (2) the second equation, we obtain dt t ∈ [0, T ]. Integrating this equality over the interval [0, T ], we have x1 (T ) = x2 (T ) + x01 erT .

(4)

The following lemma holds. Lemma 2. If r = k, then the attainable set X(T ) is a segment in R2 . Now, we consider the general case r 6= k. A point x∗ on the boundary of the attainable set X(T ) corresponds to the control u∗ (·) ∈ D(T ) and to the trajectory x∗ (t) = (x∗1 (t), x∗2 (t))> , t ∈ [0, T ] of system (2), such that x∗ = x∗ (T ). Then it follows from [3, pp.278-281] that there exists a non-trivial solution ψ(t) = (ψ1 (t), ψ2 (t))> of the adjoint system:   ψ˙ 1 (t) = −r(1 − u∗ (t))ψ1 (t) + ru∗ (t)ψ2 (t), (5) ψ˙ (t) = −kψ2 (t), t ∈ [0, T ],  2 ψ1 (T ) = ψ1∗ , ψ2 (T ) = ψ2∗ , for which

  0 u∗ (t) = [0, 1]   1

, L(t) < 0, , L(t) = 0, , L(t) > 0.

(6)

Here L(t) = −(ψ1 (t) + ψ2 (t)) is the switching function, which behavior determines the type of control u∗ (t) leading to the boundary of the attainable set X(T ). The vector ψ ∗ = (ψ1∗ , ψ2∗ )> 6= 0 is a support vector to the set X(T ) at the point x∗ . Using the adjoint system (5) and the definition of the switching function L(t), it is easy to see that the function L(t) satisfies the Cauchy problem :  ˙ L(t) = r(1 − u∗ (t))L(t) + (k − r)ψ2∗ ek(T −t) , t ∈ [0, T ], (7) L(T ) = −(ψ1∗ + ψ2∗ ). The following lemma results from the analysis of the Cauchy problem (7). Lemma 3. The switching function L(t) is a nonzero solution of the Cauchy problem (7). Lemma 3 allows us to rewrite (6) in the following form : ( 0, L(t) < 0, u∗ (t) = 1, L(t) > 0.

(8)

At points of discontinuity we define the function u∗ (t) by its left limit. Consequently, the control u∗ (t), t ∈ [0, T ] leading to the point x∗ on the boundary of the attainable set X(T ) is a piecewise constant function, taking values {0, 1}.

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Next, we will determine the number of switchings of the control function u ∗ (t), t ∈ [0, T ]. It follows from (8) that it is sufficient to find the number of zeros of function L(t) on the interval (0, T ). Lemma 4. The switching function L(t) has at most one zero on the interval (0, T ). Proof. Let us assume that the function L(t) has at least two zeros τ1 , τ2 , such that τ1 < τ2 on the interval (0, T ). That is L(τ1 ) = 0, L(τ2 ) = 0.

(9)

Integrating the differential equation from (7) on the interval [τ1 , τ2 ] and using the equalities (9), we obtain Z τ2 R −r τs (1−u∗ (ξ))dξ ∗ 1 (k − r)ψ2 ek(T −s) e ds = 0. τ1

The expressions under the integral are positive and k 6= r. Therefore, ψ2∗ = 0 necessarily holds to satisfy the equality. Then from the differential equation in (7) and relationships (9) it follows that L(t) = 0, on [0, T ], and the initial condition from (7) leads to the equality ψ1∗ = 0. We obtain that ψ ∗ = (ψ1∗ , ψ2∗ )> = 0. This fact contradicts the nontriviality of the vector ψ ∗ . Therefore, the function L(t) has at most one zero on the interval (0, T ).

Based on the obtained results, using the inequality (3) from Lemma 1, Theorem 1, and the approach presented in [5, Theorem 6], we have the following statements. Theorem 2. Every point x∗ on the boundary of the attainable set X(T ) associates to exactly one piecewise constant control u∗ (t), t ∈ [0, T ], taking values {0; 1} and having at most one switching on (0, T ). Theorem 3. The attainable set X(T ) is strictly convex set in R2 . 4. The Problem of the Optimal Credit Repayment Plan. Now, for the system (2) and under the fixed value S > 0, we are to consider the first optimal control problem, J(u, S) = x1 (T ) → max , (10) u(·)∈D(T )

subject to x2 (T ) = 0. We denote by J∗ (S) the maximum value of the functional J(u, S). The optimal control problem (10) can be stated as : To find such optimal credit repayment schedule which results in the total debt paid off by the moment T and the total production maximized. It follows from (4) that for the situation when r = k the problem of optimization is not applicable. Therefore, further we will consider only the situation when r 6= k. It follows from Section 3 that the boundary of the attainable set X(T ) for system (2) is the union of two one-parameter curves ξ(θ) = (ξ1 (θ), ξ2 (θ))> , and χ(τ ) = (χ1 (τ ), χ2 (τ ))> , where θ and τ ∈ [0, T ]. The first curve, ξ(θ), consists of all ends x∗ (T ) of trajectories x∗ (t), t ∈ [0, T ] of the system (2), corresponding to controls ( 1, 0 ≤ t ≤ θ, v∗ (t) = (11) 0, θ < t ≤ T. The second curve, χ(τ ), is formed by the values x∗ (T ) of trajectories x∗ (t), t ∈ [0, T ] of the system (2) corresponding to the controls ( 0, 0 ≤ t ≤ (T − τ ), (12) w∗ (t) = 1, (T − τ ) < t ≤ T.

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Here (T − τ ) and θ are the moments of switchings. Direct integration of the system (2) under the controls (11) and (12) gives us the following expressions for the components of the functions ξ(θ) and χ(τ ) : ξ1 (θ) = (x01 + S)er(T −θ) , θ ∈ [0, T ], r ξ2 (θ) = SekT − (x01 + S)ek(T −θ) (ekθ − 1), k χ1 (τ ) = (x01 + S)er(T −τ ) , τ ∈ [0, T ], r χ2 (τ ) = SekT − (x01 + S)er(T −τ ) (ekτ − 1). k

(13)

It is obvious that the functions ξ(θ) and χ(τ ) satisfy ξ1 (0) = χ1 (0) = (x01 + S)erT , ξ2 (0) = χ2 (0) = SekT , (14)

ξ1 (T ) = χ1 (T ) = x01 + S, ξ2 (T ) = χ2 (T ) = SekT −

r 0 (x + S)(ekT − 1). k 1

It follows from Section 3 that in order to solve the problem (10) it is sufficient to find the point with the largest value of the first component among the points of intersection of the curves ξ(θ) and χ(τ ) with the x1 axis. Therefore, the condition under which a solution of the optimal control problem (10) exists, is   Q = min min ξ2 (θ), min χ2 (τ ) < 0. (15) θ∈[0,T ]

τ ∈[0,T ]

Now, we will analyze the left part of the inequality (15). We find the minimum of the function ξ2 (θ) on the interval [0, T ]. Differentiating the second expression in (13) with respect to θ we obtain : ξ˙2 (θ) = −r(x01 + S)ek(T −θ) < 0. Hence the function ξ2 (θ) monotonically decreases on the interval [0, T ] and min ξ2 (θ) = ξ2 (T ).

θ∈[0,T ]

(16)

Next, we investigate the minimum of the function χ2 (τ ) on the interval [0, T ]. Differentiating the fourth expression in (13) with respect to τ we obtain : r (17) χ˙ 2 (τ ) = − (x01 + S)er(T −τ ) (r + (k − r)ekτ ). k To investigate the behavior of χ2 (τ ) we need to consider two cases. Case 1. If r < k, then χ˙ 2 (τ ) < 0 in (17). Hence the function χ2 (τ ) monotonically decreases on the interval [0, T ] and min χ2 (τ ) = χ2 (T ).

τ ∈[0,T ]

(18)

Case 2. If r > k, then  it follows from (17) that the function χ2 (τ ) reaches its 1 r minimum at τ0 = k ln r−k . r If r−k > ekT , then τ0 > T and the minimum of the function χ2 (τ ) on the interval [0, T ] is at τ = T . Therefore, the equality (18) is valid.

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r If r−k ≤ ekT , then τ0 ∈ (0, T ] and the function χ2 (τ ) reaches its minimum at an inner point τ0 of the interval [0, T ] and then

min χ2 (τ ) = χ2 (τ0 ) ≤ χ2 (T ).

(19)

τ ∈[0,T ]

Now, for analysis of the expression (15) we consider the following cases : (a) : (b) : (c) :

let let let

r < k; r>k r>k

and and

r r−k r r−k

> ekT ; ≤ ekT .

Next, we will study the cases (a)-(c). In cases (a) and (b) from (14), (16), (18) we have the relationship : Q = min ξ2 (θ) = ξ2 (T ) = χ2 (T ) = min χ2 (τ ). θ∈[0,T ]

τ ∈[0,T ]

From (15) it follows that in cases (a) and (b) the solution of the optimal control problem (10) exists if the following inequality is valid : r SekT − (x01 + S)(ekT − 1) < 0. (20) k In case (c) from (14), (16) and (19) we have the relationship : Q = min χ2 (τ ) = χ2 (τ0 ) ≤ χ2 (T ) = ξ2 (T ) = min ξ2 (θ). τ ∈[0,T ]

θ∈[0,T ]

It is easy to show that in this case the following inequalities hold : χ2 (τ0 ) < 0, χ2 (T ) < 0.

(21)

Therefore, the inequality (15) is valid and the solution of the optimal control problem (10) exists. Further, we will find the solutions of the optimal control problem (10) in the cases (a)-(c). We consider the cases (a) and (b). The relationships ξ2 (0) = χ2 (0) > 0 and ξ2 (T ) = χ2 (T ) < 0 follow from (14) and (20). Moreover, the derivatives ξ˙2 (θ) < 0 and χ˙ 2 (τ ) < 0 for all θ, τ ∈ [0, T ]. This means that the curve ξ(θ) has precisely one point of intersection with x1 axis as does the curve χ(τ ). Thus, there exist the values θ∗ , τ∗ ∈ (0, T ), such that ξ2 (θ∗ ) = 0, χ2 (τ∗ ) = 0.

(22)

Moreover, the following equalities hold : ξ1 (θ∗ ) = (x01 + S)er(T −θ∗ ) , r ξ2 (θ∗ ) = SekT − (x01 + S)ek(T −θ∗ ) (ekθ∗ − 1) = 0, k χ1 (τ∗ ) = (x01 + S)er(T −τ∗ ) , r χ2 (τ∗ ) = SekT − (x01 + S)er(T −τ∗ ) (ekτ∗ − 1) = 0. k Next, we compare the values ξ1 (θ∗ ) and χ1 (τ∗ ). We estimate the value r χ2 (θ∗ ) = SekT − (x01 + S)er(T −θ∗ ) (ekθ∗ − 1). k From (23) we have the equality : r χ2 (θ∗ ) − ξ2 (θ∗ ) = (x01 + S)(ekθ∗ − 1)(ek(T −θ∗ ) − er(T −θ∗ ) ). k

(23)

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It follows from this relationship that in the case (a) the inequality χ2 (θ∗ ) > 0 is valid and in the case (b) the inequality χ2 (θ∗ ) < 0 holds. Hence in the case (a) we obtain χ2 (θ∗ ) > 0 = χ2 (τ∗ ). The function χ2 (τ ) decreases on the interval [0, T ] and therefore, we have the inequality θ∗ < τ∗ . In the case (b) we obtain χ2 (θ∗ ) < 0 = χ2 (τ∗ ). The function χ2 (τ ) decreases on the interval [0, T ] and thus, we have the inequality θ∗ > τ∗ . Then we consider the equality ξ1 (θ∗ ) − χ1 (τ∗ ) = χ1 (θ∗ ) − χ1 (τ∗ ). We find from this relationship and the previous arguments that in the case (a) the inequality ξ1 (θ∗ ) > χ1 (τ∗ ) is valid, and in the case (b) the inequality ξ1 (θ∗ ) < χ1 (τ∗ ) holds. Summarizing all previous arguments, we conclude that : in case (a) the maximum value of functional J(u, S) is J∗ (S) = ξ1 (θ∗ ), where the expression ξ1 (θ∗ ) and the value θ∗ are defined by the first and second expressions in (23). The corresponding optimal control u∗ (t), t ∈ [0, T ] has the type (11) with the switching θ∗ ∈ (0, T ); in case (b) the maximum value of functional J(u, S) is J∗ (S) = χ1 (τ∗ ), where the expression χ1 (τ∗ ) and the value τ∗ are defined by the third and fourth expressions in (23). The corresponding optimal control u∗ (t), t ∈ [0, T ] has the type (12) with the switching (T − τ∗ ) ∈ (0, T ). In cases (a) and (b) the relationship (20) is always true. Next, we consider the case (c). In this case for the function ξ2 (θ) it follows from (14) and (21) that ξ2 (0) > 0 and ξ2 (T ) < 0. Moreover, on the interval [0, T ] we have the inequality ξ˙2 (θ) < 0. Hence the curve ξ(θ) has only one point of intersection with the x1 axis. For the function χ2 (τ ) it follows from (14) and (21) that χ2 (0) > 0 and χ2 (τ0 ) < 0. Moreover, we have the inequality χ˙ 2 (τ ) < 0 on the interval [0, τ0 ). Hence the curve χ(τ ) has only one point of the intersection with x1 axis as well. Therefore, we define the values θ∗ ∈ (0, T ) and τ∗ ∈ (0, τ0 ) such that the equalities (22) and (23) are valid. Then we compare the values ξ1 (θ∗ ) and χ1 (τ∗ ) by arguments similar to cases (a) and (b). We obtain the inequality ξ1 (θ∗ ) < χ1 (τ∗ ). Hence the conclusion in the case (c) is similar to the case (b). 5. The Problem of the Optimal Maximum Repayable Credit. In this section we want to find the value of the maximum repayable credit, which is equal to or less than the bank cutoff S¯ (the value of the maximum credit offered by the bank) and the most advantageous for the firm. This can be written as the second optimal control problem : J(u, S) →

max

(24)

¯ u(·)∈D(T ),S∈[0,S]

subject to x2 (T ) = 0. Using the results of Section 4 we can rewrite (24) as   max J(u, S) = max J∗ (S). max J(u, S) = max ¯ u(·)∈D(T ),S∈[0,S]

¯ S∈[0,S]

¯ S∈[0,S]

u(·)∈D(T )

(25)

It follows from (25) that we can use formulas for J∗ (S) from Section 4. Also we denote by J∗∗ the maximum value of the function J∗ (S). We consider the cases (a)-(c) as we did in Section 4. Now, we study case (a). It follows from (25) and results of Section 4 that it is sufficient to consider the problem : J∗ (S) = (x01 + S)er(T −θ∗ (S)) → max

¯ S∈[0,S]

(26)

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subject to the condition (20), which transforms to the inequality S
0 for all S ∈ [0, S]. ˆ Therefore, the maximum value of the function J∗ (S) in (33) on the interval [0, S]. ˆ ˆ Hence the company is J∗∗ = J∗ (S) and the corresponding value of S is S∗ = S. ˆ must take the entire available credit S. Next, we study case (c). It follows from (25) and results of Section 4 that it is sufficient to consider the problem (31), (32). We use arguments similar to the arguments from case (b) to find the solution of the problem. The maximum value ¯ and the corresponding value of S is S∗ = S. ¯ of the function J∗ (S) is J∗∗ = J∗ (S) ¯ Here again the company must take the entire available credit S. J˙∗ (S) =

12

10

8

Debt

6

4

2

0 2

3

4

5

6

7

8

9

Production

Figure 1. r = 0.1, k = 0.5, x01 = 1, S = 1, T = 5.

50

100

200

Production 300

400

500

600

0

–50

Debt –100

–150

Figure 2. r = 0.5, k = 0.1, x01 = 1, S = 50, T = 5.

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6. Computer Modeling. Knowing how to get to the boundary of an attainable set, we wrote a computer program in MAPLE that builds the attainable set X(T ) as the region in R2 , bounded by the union of two one-parameter curves, ξ(θ) and χ(τ ). The program solves the Cauchy problems (2) for bang-bang controls v(t) defined by (11) and w(t), defined by (12). The moments of switching θ and (T − τ ) change within the interval [0, T ]. Both curves are displayed on the same graph forming the boundary of the attainable set X(T ). The curve ξ(θ) is shown as a thin dashed curve and χ(τ ) as a thick solid curve. Attainable sets X(T ) for the cases (a), (c) are shown on Figures 1 and 2, respectively. As we proved in Section 3, all attainable sets X(T ) are strictly convex. For the first optimal control problem (10) the maximum value of the functional and the switching of the corresponding optimal control are obtained analytically. Figure 1 represents the situation when r = 0.1, k = 0.5, x01 = 1, S = 1, and T = 5. The maximum value of the functional J(u, S) is J∗ (S) = 3.06 and comes from control of type v(t) with the moment of switching θ = 3.58. It looks like the firm must pay the debt during the first 3.6 years and then spend the remaining time for development. However, it follows from Section 5 that, if r < k, then it is not beneficial for a firm to take any credit from the bank. This fact completely agrees with any book on financial management ([6]). The attainable set in Figure 2 is constructed for the following parameters : r = 0.5, k = 0.1, x01 = 1, S = 50, and T = 5. Since r > k, then the maximum value of the functional J(u, S) is J∗ (S) = 533.61 and comes from control of type w(t) with the switching at (T − τ ) = 4.7. Therefore, in order to maximize production and eliminate the debt the company must take the maximum possible credit from the bank and first develop during 4.7 years and then spend the remaining time to pay the debt. This work is supported TWU REP Grant, RFFI Grant 01-03-00737 and RFFI Support Grant of the Leading Scientific Schools SS-1846.2003.1. REFERENCES [1] V.V. Tokarev, Unimproving extension and the structure of extremals in control of the credit, Automation and Remote Control, 62 (2001) no. 9, 1433–1444. [2] V.V. Tokarev, Optimal and admissible programs of credit control, Automation and Remote Control, 63(2002), no. 1, 1–13. [3] E.B. Lee, L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, New York, 1967. [4] R.W. Brockett, On the reachable set for bilinear systems, Lecture Notes in Economics and Mathematical Systems, 111 (1975), 54–63. [5] O. Hajek, Bilinear control : rank-one inputs, Funkcialaj Ekvacioj, 34 (1991), 355–374. [6] A. Allen, Financial Risk Management, John Wiley & Sons, New York, 2003.

Received September, 2004; revised March, 2005. E-mail address: [email protected] E-mail address: [email protected]