Fuzzy Optimal Control with Application to Portfolio Selection Yuanguo Zhu Department of Applied Mathematics Nanjing University of Science and Technology Nanjing 210094, Jiangsu People’s Republic of China (
[email protected])
Abstract Optimal control is a very important field of study not only in theory but in applications, and stochastic optimal control is also a significant branch of research in theory and applications. Based on the concept of fuzzy process introduced by Liu, we present a fuzzy optimal control problem. Applying Bellman’s optimal principle, we obtain the principle of optimality for fuzzy optimal control, and then give a fundamental result called the equation of optimality in fuzzy optimal control. Finally, as application, we use equation of optimality to solve a portfolio selection problem. Keywords: fuzzy optimal control, fuzzy process, principle of optimality, equation of optimality, portfolio selection
1
Introduction
Since 50s of last century, optimal control theory has been an important branch of modern control theory. The study of optimal control greatly attracted the attention of many mathematician because of the necessary of strict expression form in optimal control theory. With the more use of methods and results on mathematics and computer science, optimal control theory has greatly achieved development, and been applied to many fields such as production engineering, programming, economy and management. The study of stochastic optimal control initiated in 1970s such as in Merton [7] for finance. Some researches on optimal control of Brownian motion process or stochastic differential equations and applications in finance may refer to some books such as Fleming and Rishel [3], Harrison [5] and Karatzas [6]. One of the main methods to study optimal control dues to dynamic programming. The use of dynamic programming in optimization over Ito’s process is discussed in Dixit and Pindyck [1]. The complexity of the world makes the events we face uncertain in various forms. Besides randomness, fuzziness is also an important uncertainty, which plays an essential role in the real world. Fuzzy set theory has been developed very fast since it was introduced by scientist on cybernetics Zadeh [18] in 1965. A fuzzy set was characterized with its membership function by Zadeh. For the purpose of measuring fuzzy events, Zadeh [19] presented the concept of possibility measure and the term of fuzzy variable in 1978. In order to give a self-dual measure for fuzzy events, Liu and Liu [13] introduced the concept of credibility measure in 2002. Based on credibility 1
measure, credibility theory was founded by Liu [9] in 2004 and refined by Liu [11] as a branch of mathematics for dealing with the behavior of fuzzy phenomena. A fuzzy variable may be redefined as a function from a credibility space to the set of real numbers. As fuzzy counterpart of stochastic process and Brownian motion, fuzzy process and C process was introduced by Liu [12] recently. We may call C process to be Liu process. In order to handle an optimal control problem with fuzzy process, now we will introduce and investigate fuzzy optimal control. We will deal with fuzzy optimal control by using dynamic programming. In next section, we will review some concepts such as expected value of fuzzy variable, fuzzy process, Liu process, and fuzzy deferential equation. In Section 3, we will introduce fuzzy optimal control problem, and present the principle of optimality based on Bellman’s principle of optimality in dynamic programming. In Section 4, we will obtain a fundamental result called the equation of optimality in fuzzy optimal control. In the last section, we will solve a portfolio selection problem by using the equation of optimality.
2
Preliminary
If a fuzzy variable ξ is given by a membership function µ, then we may get the credibility value via µ ¶ 1 Cr{ξ ∈ B} = sup µ(x) + 1 − sup µ(x) , B ⊂ R. 2 x∈B x∈B c Membership function represents the degree of possibility that the fuzzy variable ξ takes some prescribed value. Definition 2.1. (Liu and Liu [13]) Let ξ be a fuzzy variable. Then the expected value of ξ is defined by Z +∞ Z 0 E[ξ] = Cr{ξ ≥ r}dr − Cr{ξ ≤ r}dr (2.1) 0
−∞
provided that at least one of two integrals is finite. Definition 2.2. (Liu and Liu [13]) Let ξ be a fuzzy variable with finite expected value e. Then the variance of ξ is defined by V [ξ] = E[(ξ − e)2 ]. Theorem 2.1. (Liu and Liu [14]) Let ξ and η be independent fuzzy variables with finite expected values. Then for any numbers a and b, we have E[aξ + bη] = aE[ξ] + bE[η]. Based on the credibility space, Liu introduced the concepts of fuzzy process, Liu process, fuzzy differential equation, and etc. Definition 2.3. (Liu [12]) Let T be an index set and let (Θ, P, Cr) be a credibility space. A fuzzy process is a function from T × (Θ, P, Cr) to the set of real numbers. Definition 2.4. (Liu [12]) A fuzzy process, simply denoted by Xt , is said to have independent increments if Xt1 − Xt0 , Xt2 − Xt1 , · · · , Xtk − Xtk−1 are independent fuzzy variables for any times t0 < t1 < · · · < tk . A fuzzy process Xt is said to have stationary increments if, for any given t > 0, the increments Xs+t − Xs are identically distributed fuzzy variables for all s > 0.
2
Definition 2.5. (Liu [12]) A fuzzy process Ct is said to be Liu process if (i) C0 = 0, (ii) Ct has stationary and independent increments, (iii) every increment Cs+t − Cs is a normally distributed fuzzy variable with expected value et and variance σ 2 t2 , whose membership function is ¶¶−1 µ µ π|x − et| √ , x ∈ R. µ(x) = 2 1 + exp 6σt The parameters e and σ are called the drift and diffusion coefficients, respectively. The Liu process is said to be standard if e = 0 and σ = 1. The Liu process plays the role of Brownian motion or Wiener process. Based on Liu process, a new kind of fuzzy differential introduced by Liu [12]. As the inverse of fuzzy differential, a kind of integral, called Liu integral, was also introduced by Liu [12]. Liu integral is different from the fuzzy integral based on fuzzy measure given by Sugeno [17]. The following concept of fuzzy differential equation is important in theory and applications, and essential in the study of this paper. Definition 2.6. (Liu [12]) Suppose Ct is a standard Liu process, and f and g are some given functions. then dXt = f (t, Xt )dt + g(t, Xt )dCt (2.2) is called a fuzzy differential equation. A solution is a fuzzy process Xt that satisfies (2.2) identically in t. Based on the concept of fuzzy differential equation, Liu [12] established a stock model for fuzzy financial market as a fuzzy counterpart of Black-Scholes stock model. Gao [4] and Peng [15] presented other stock models for fuzzy financial market, respectively. Qin and Li [16] formulated an European option pricing formula for fuzzy financial market.
3
Problem of Fuzzy Optimal Control
Fuzzy optimal control is to choose the best decision such that some objective function related to a fuzzy process provided by a fuzzy differential equation is optimized. Unless stated otherwise, we assume that Ct is a standard Liu process. We consider the following fuzzy expected value optimal control problem hR i T J(0, x ) ≡ sup E f (X , D, s)ds + G(X , T ) 0 s T 0 D (3.1) subject to dX = µ(X , D, s)ds + σ(X , D, s)dC and X = x . s
s
s
s
0
0
In the above problem, Xs is the state variable, D the decision variable (represents the function D(t, x) of time t and state x), f the objective function, and G the function of terminal reward. For a given D, dXs is defined by the fuzzy differential equation dXs = µ(Xs , D, s)ds + σ(Xs , D, s)dCs
and X0 = x0
(3.2)
where µ and σ are two functions of Xs , D and time s. The function J(0, x0 ) is the expected optimal reward obtainable in [0, T ] with the initial condition that at time 0 we are in state x0 . For 3
any 0 < t < T , J(t, x) is the expected optimal reward obtainable in [t, T ] with the condition that at time t we are in state Xt = x. That is, we have hR i T J(t, x) ≡ sup E f (X , D, s)ds + G(X , T ) s T t D (3.3) subject to dX = µ(X , D, s)ds + σ(X , D, s)dC and X = x. s
s
s
s
t
Remark 3.1. Of course, we can consider the fuzzy α-optimistic optimal control problem and α-pessimistic optimal control problem. These will be our next work. Now we present the following principle of optimality. Theorem 3.1 (Principle of optimality). For any (t, x) ∈ [0, T ) × R, and ∆t > 0 with t + ∆t < T , we have "Z # t+∆t J(t, x) = sup E f (Xs , D, s)ds + J(t + ∆t, x + ∆Xt ) , (3.4) D
t
where x + ∆Xt = Xt+∆t . e x). It follows from the definition of J(t, x) Proof: We denote the right side of (3.4) by J(t, that "Z # Z T t+∆t J(t, x) ≥ E f (Xs , D|[t,t+∆t) , s)ds + f (Xs , D|[t+∆t,T ] , s)ds + G(XT , T ) (3.5) t
t+∆t
for any D, where D|[t,t+∆t) and D|[t+∆t,T ] are the variables of decision variable D restrictedly on [t, t + ∆t) and [t + ∆t, T ], respectively. Since the fuzzy processes dCs (s ∈ [t, t + ∆t)) and dCs (s ∈ [t + ∆t, T ]) are independent, we know that Z
Z
t+∆t
f (Xs , D|[t,t+∆t) , s)ds and
t
are independent. Thus "Z
"Z
t+∆t
J(t, x) ≥ E
f (Xs , D|[t,t+∆t) , s)ds + E
t
T
f (Xs , D|[t+∆t,T ] , s)ds
t+∆t
##
T
t+∆t
f (Xs , D|[t+∆t,T ] , s)ds + G(XT , T )
(3.6)
by Theorem 2.1. Taking the supremum with respect to D|[t+∆t,T ] first, and then D|[t,t+∆t) in (3.6), e x). we get J(t, x) ≥ J(t, On the other hand, for all D, we have "Z # T
E
f (Xs , D, s)ds + G(XT , T ) t
(Z
"Z
t+∆t
=E
#)
T
f (Xs , D, s)ds + E t
"Z
t+∆t
f (Xs , D|[t+∆t,T ] , s)ds + G(XT , T ) #
t+∆t
f (Xs , D, s)ds + J(t + ∆t, x + ∆Xt )
≤E t
e x). ≤ J(t, e x), and then J(t, x) = J(t, e x). The theorem is proved. Hence, J(t, x) ≤ J(t, 4
4
Equation of Optimality
Also, we assume Ct is a standard Liu process. Consider the following fuzzy optimal control problem hR i T J(t, x) ≡ sup E t f (Xs , D, s)ds + G(XT , T ) D (4.1) subject to dXs = µ(Xs , D, s)ds + σ(Xs , D, s)dCs and Xt = x. Now let us give a fundamental result called equation of optimality in fuzzy optimal control. Theorem 4.1 (Equation of optimality). Let J(t, x) be twice differentiable on [0, T ] × R. Then we have −Jt (t, x) = sup {f (x, D, t) + Jx (t, x)µ(x, D, t)} . (4.2) D
Proof: For any ∆t > 0, we have Z
t+∆t
f (Xs , D, s)ds = f (x, D, t)∆t + o(∆t)
(4.3)
t
By using Taylor series expansion, we get 1 J(t + ∆t, x + ∆Xt ) = J(t, x) + Jt (t, x)∆t + Jx (t, x)∆Xt + Jtt (t, x)∆t2 2 1 + Jxx (t, x)∆Xt 2 + Jtx (t, x)∆t∆Xt + o(∆t) 2
(4.4)
Substituting Equations (4.3) and (4.4) into Equation (3.4) yields 1 0 = sup{f (x, D, t)∆t + Jt (t, x)∆t + E[Jx (t, x)∆Xt + Jtt (t, x)∆t2 2 D 1 + Jxx (t, x)∆Xt 2 + Jtx (t, x)∆t∆Xt ] + o(∆t)} 2
(4.5)
Let ξ be a fuzzy variable such that ∆Xt = ξ + µ(x, D, t)∆t. It follows from (4.5) that 0 = sup{f (x, D, t)∆t + Jt (t, x)∆t + Jx (t, x)µ(x, D, t)∆t + E[(Jx (t, x) D
1 +Jxx (t, x)µ(x, D, t)∆t + Jtx (t, x)∆t)ξ + Jxx (t, x)ξ 2 ] + o(∆t)} 2 = sup{f (x, D, t)∆t + Jt (t, x)∆t + Jx (t, x)µ(x, D, t)∆t + E[aξ + bξ 2 ] + o(∆t)},
(4.6)
D
where a ≡ Jx (t, x) + Jxx (t, x)µ(x, D, t)∆t + Jtx (t, x)∆t, and b ≡ fuzzy differential equation dXs = µ(Xs , D, s)ds + σ(Xs , D, s)dCt
1 2 Jxx (t, x).
It follows from the
and Xt = x
that ξ = ∆Xt − µ(x, D, t)∆t is a normally distributed fuzzy variable with expected value 0 and variance σ 2 (x, D, t)∆t2 . Simply denote σ = σ(x, D, t) in sequel. Let x0 =
|a| , 2|b|
z=√
πx , 6σ∆t
5
π|a| z0 = √ . 2 6|b|σ∆t
Formula (6.1) in the Appendix implies that Z
µ
+∞
2
E[aξ + bξ ] =
µ
(a + 2bx) 1 + exp 0
Z
x0
− 0
√ =
πx √ 6σ∆t
¶¶−1 dx
µ µ ¶¶−1 πx (a − 2bx) 1 + exp √ dx 6σ∆t Z
6aσ∆t2 π
+∞
Z
1 12bσ 2 ∆t2 dz + z 1+e π2
z0
z0
0
z 12bσ 2 ∆t2 dz + z 1+e π2
Z
+∞
0
z dz 1 + ez
= o(∆t). Formula (6.2) in the Appendix implies that Z
x0
E[aξ + bξ 2 ] = 0
µ µ ¶¶−1 πx (a + 2bx) 1 + exp √ dx 6σ∆t Z
−
µ
+∞
µ
(a − 2bx) 1 + exp 0
√ =−
6aσ∆t π
Z
+∞
z0
πx √ 6σ∆t
¶¶−1
1 12bσ 2 ∆t2 dz + z 1+e π2
dx Z
z0
0
z 12bσ 2 ∆t2 dz + z 1+e π2
Z 0
+∞
z dz 1 + ez
= o(∆t). Hence √ 2
E[aξ + bξ ] = ±
6aσ∆t π
+
Z
+∞
z0
12bσ 2 ∆t2 π2
Z
1 12bσ 2 ∆t2 dz + z 1+e π2 +∞
0
Z 0
z0
z dz 1 + ez
z dz 1 + ez
= o(∆t).
(4.7)
Substituting Equation (4.7) into Equation (4.6) yields −Jt (t, x)∆t = sup {f (x, D, t)∆t + Jx (t, x)µ(x, D, t)∆t + o(∆t)} .
(4.8)
D
Dividing Equation (4.8) by ∆t, and letting ∆t → 0, we can obtain the result (4.2). Remark 4.1. The equation of optimality gives a necessary condition for an extremum. If the equation has solutions, then the optimal decision and optimal value of objective function are determined. If function f is convex in its arguments, then the equation will produce a minimum, and if f is concave in its arguments, then it will produce a maximum. We note that the boundary condition for the equation is J(T, XT ) = G(XT , T ). Remark 4.2. We note that in the equation of optimality for stochastic optimal control (called Hamilton-Jacobi-Bellman equation), there is an extra term 12 Jxx (t, x)σ 2 (x, D, t).
6
Example 4.1. Consider the following fuzzy optimization problem: hR i T J(t, x) ≡ min E 0 e−βs (aXs2 + bXs D + cD2 )ds D subject to dXs = (µD + αXs )ds + σXs dCs , where a > 0, c > 0, σ > 0, b2 − 4ac ≤ 0, µ 6= 0, α ∈ R, β is a discount factor, T is the terminal time. We see that f (Xs , D, s) = e−βs (aXs2 + bXs D + cD2 ). It follows from Equation (4.2) that −Jt = min{e−βt (ax2 + bxD + cD2 ) + Jx (µD + αx)} = min L(D), D
D
(4.9)
where L(D) denotes the term in the braces. The optimal decision D satisfies dL(D) = bxe−βt + 2cDe−βt + Jx µ = 0 or dD
D=−
Jx eβt µ + bx . 2c
Equation (4.9) becomes " −βt
−Jt = e
µ
Jx eβt µ + αx ax + bx − 2c 2
¶
µ +c
Jx e2βt µ + bx 2c
¶2 #
¶ µ Jx eβt µ2 + bµx +Jx − + αx . 2c That is
µ −Jt eβt =
a−
b2 4c
¶ x2 −
µ ¶ Jx2 e2βt µ2 bµ + α− xJx eβt . 4c 2c
(4.10)
We conjecture that J(t, x) = kx2 e−βt . Thus, we have Jt = −βkx2 e−βt , Jx = 2kxe−βt . Substituting them into Equation (4.10) yields µ ¶ b2 µ2 bµ 2 βkx = a − x2 − (2kx)2 − (2kx2 ) + α(2kx2 ), 4c 4c 2c or
µ 2 µ ¶ ¶ b µ2 2 bµ k+ k + β − 2α + − a = 0. c c 4c
The solution of the parameter k (selecting k such that J(t, x) ≥ 0) is p (2cα − cβ − bµ) + (2cα − cβ − bµ)2 + µ2 (4ac − b2 ) k= . 2µ2 Hence, the optimal decision and the optimal value of the objective function are given as follows, respectively (2kµ + b)x D=− , J(t, x) = kx2 e−βt . 2c
7
5
Portfolio Selection Problem
Let Xt be the wealth of an investor at time t. The investor allocates a fraction w of the wealth in a risky asset and remainder in a sure asset. The sure asset produces a rate of return b. The risky asset yields a mean rate of return µ along with a variance of σ 2 per unit time. That is to say, the risky asset earns a return drt in time interval (t, t + dt), where drt = µdt + σdCt , and Ct is a standard Liu process. Thus Xt+dt = Xt + b(1 − w)dt + drt (wXt ) = Xt + b(1 − w)Xt dt + (µdt + σdCt )(wXt ) = Xt + [µw + b(1 − w)]Xt dt + σwXt dCt . If we consider the consumption rate by an amount p, we obtain dXt = [µwXt + b(1 − w)Xt − p]dt + σwXt dCt . A portfolio selection problem is provided by ·Z ∞ ¸ λ 2 λ −βt p + [(µ − σ )wXt ] J(t, x) ≡ max E e dt p,w λ 0 subject to dXt = [µwXt + b(1 − w)Xt − p]dt + σwXt dCt , where β > 0, 0 < λ < 1. By the equation of optimality (4.2), we have that ¾ ½ pλ + [(µ − σ 2 )xw]λ + (µ − b)xwJx + bxJx − pJx = max L(p, w), −Jt = max e−βt p,w p,w λ where L(p, w) represents the term in the braces. The optimal (p, w) satisfies ∂L(p, w) = e−βt pλ−1 − Jx , ∂p ∂L(p, w) = e−βt [(µ − σ 2 )xw]λ−1 [(µ − σ 2 )x] + Jx (µ − b)x = 0, ∂w or
·
−βt
p = (Jx e
)
1 λ−1
(b − µ)Jx eβt w= µ − σ2
,
1 ¸ λ−1
1 . (µ − σ 2 )x
Hence 1 −Jt = e−βt λ
(
·
−βt
(Jx e
)
λ λ−1
(b − µ)Jx eβt + µ − σ2
λ ) ¸ λ−1
· ¸ 1 µ−b (b − µ)Jx eβt λ−1 + Jx µ − σ2 µ − σ2
1
+bxJx − Jx (Jx e−βt ) λ−1 , or
µ −Jt eβt =
1 −1 λ
¶"
µ 1+
b−µ µ − σ2
λ # ¶ λ−1
λ
(Jx eβt ) λ−1 + bxJx eβt .
We conjecture that J(t, x) = kxλ e−βt . Then Jt = −kβxλ e−βt , Jx = kλxλ−1 e−βt .
8
(5.1)
Substituting them into Equation (5.1) yields µ ¶" µ ¶ λ # λ 1 b − µ λ−1 λ kβx = −1 1+ (kλ) λ−1 xλ + kaλxλ , 2 λ µ−σ or 1
(kλ) λ−1 =
So we get
β − bλ · λ ¸. ³ ´ λ−1 b−µ (1 − λ) 1 + µ−σ 2
kλ =
λ−1
β − bλ · λ ¸ ³ ´ λ−1 (1 − λ) 1 + b−µ2 µ−σ
.
Therefore the optimal consumption rate and the optimal fraction of investment on risky asset is determined, respectively, by µ 1
p = x(kλ) λ−1 ,
w=
b−µ µ − σ2
1 ¶ λ−1
1
(kλ) λ−1 . µ − σ2
Remark 5.1. Note that the optimal consumption rate calls for the investor to consume a constant fraction of wealth at each moment, and optimal fraction of investment on risky asset is independent of total wealth. These conclusions are similar to that in the case of randomness [2].
6
Conclusion
Based on the concept of Liu process, we studied a fuzzy optimal control problem: optimizing the expected value of an objective function subject to a fuzzy differential equation. By using the Bellman’s principle of optimality in dynamic programming, we presented the principle of optimality and a fundamental result called equation of optimality for fuzzy optimal control. As application of the equation of optimality, we solved a portfolio selection problem.
References [1] Dixit, A.K. and Pindyck, R.S., Investment under Uncertainty, Princeton: Princeton University Press, 1994. [2] Kao, E.P.C., An Introduction to Stochastic Processed, Wadsworth Publishing Company, 1997. [3] Fleming, W.H. and Rishel, R.W., Deterministic and Stochastic Optimal Control, New York: SpringerVerlag, 1986. [4] Gao, J., Credibilistic option pricing: A new model, http://orsc.edu.cn/process/071124.pdf. [5] Harrison, J.M., Brownian Motion and Stochastic Flow Systems, New York: John Wiley & Sons, 1985. [6] Karatzas, I., Optimization problems in the theory of continuous trading, SIAM Journal on Control and Optimization, Vol.27, No.6, 1221-1259, 1989. [7] Merton, R.C., Optimal consumption and portfolio rules in a continuous time model, Journal of Economic Theory, Vol.3, 373-413, 1971. [8] Liu, B., Theory and Practice of Uncertain Programming, Heidelberg: Physica-Verlag, 2002. [9] Liu, B., Uncertainty Theory: An Introduction to its Axiomatic Foundations, Berlin: Springer-Verlag, 2004. [10] Liu, B., A survey of entropy of fuzzy variables, Journal of Uncertain Systems, Vol.1, No.1, 4-13, 2007.
9
[11] Liu, B., Uncertainty Theory, 2nd ed., Berlin: Springer-Verlag, 2007. [12] Liu, B., Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems, Vol.2, No.1, 3-16, 2008. [13] Liu, B. and Liu, Y.-K., Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy Systems, Vol.10, No.4, 445-450, 2002. [14] Liu, Y.-K. and Liu, B., Expected value operator of random fuzzy variable and random fuzzy expected value models, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems, Vol.11, No.2, 195-215, 2003. [15] Peng, J., Credibilistic stopping problems for fuzzy stock market, http://orsc.edu.cn/process /071125.pdf. [16] Qin, Z. and Li, X., Option pricing formula for fuzzy financial market, Journal of Uncertain Systems, Vol.2, No.1, 17-21, 2008. [17] Sugeno, M., Theory of Fuzzy Integrals and its Applications, Ph.D. Dissertation, Tokyo Institute of Technology, 1974. [18] Zadeh, L.A., Fuzzy sets, Information and Control, Vol.8, 338-353, 1965. [19] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, Vol.1, 3-28, 1978.
Appendix Let us give a formula for computing the expected value of aξ + bξ 2 if ξ is a fuzzy variable. Theorem 6.1. Let ξ be a fuzzy variable with an even and integrable membership function µ(x) which is decreasing on [0, +∞) and µ(0) = 1. Then E[aξ + bξ 2 ] =
1 2
Z
+∞
µ(x)(a + 2bx)dx − 0
1 2
Z
x0
µ(x)(a − 2bx)dx
(6.1)
0
if a ≥ 0, b > 0, or a ≤ 0, b < 0; and Z Z 1 x0 1 +∞ E[aξ + bξ 2 ] = µ(x)(a + 2bx)dx − µ(x)(a − 2bx)dx 2 0 2 0
(6.2)
if a ≥ 0, b < 0, or a ≤ 0, b > 0, where x0 = |a|/(2|b|). 2
Proof: (1) If b > 0, then aξ + bξ 2 ≥ − a4b ≡ y0 . For any y ≥ y0 , let x1 =
−a +
p
a2 + 4by , 2b
x2 =
−a −
p
a2 + 4by . 2b
Then {aξ + bξ 2 = y} = {ξ = x1 } ∪ {ξ = x2 }. Thus, the membership function of fuzzy variable aξ + bξ 2 is that if y < y0 , µaξ+bξ2 (y) = 0, if y ≥ y0 , µaξ+bξ2 (y) = 2Cr{aξ + bξ 2 = y} ∧ 1 = 2(Cr{ξ = x1 } ∨ Cr{ξ = x2 }) ∧ 1 = (2Cr{ξ = x1 } ∧ 1) ∨ (2Cr{ξ = x2 } ∧ 1) = µ(x1 ) ∨ µ(x2 ).
10
If a ≥ 0, then µ(x2 ) ≤ µ(x1 ). So, µaξ+bξ2 (y) = µ(x1 ) for y ≥ y0 . Thus Z
+∞
2
E[aξ + bξ ] =
Z 2
0
Cr{aξ + bξ ≥ y}dy − 0
Z
+∞
Z
0
1 µ(x1 )dy (let x1 = x, then ax + bx2 = y) 0 y0 2 Z Z 1 +∞ 1 0 = µ(x)(a + 2bx)dx − µ(x)(a + 2bx)dx 2 0 2 −x0 Z Z 1 +∞ 1 x0 = µ(x)(a + 2bx)dx − µ(x)(a − 2bx)dx. 2 0 2 0
=
1 µ(x1 )dy − 2
Cr{aξ + bξ 2 ≤ y}dy
y0
where x0 = a/(2b). We obtain the formula (6.1). If a ≤ 0, then µ(x2 ) ≥ µ(x1 ). So, µaξ+bξ2 (y) = µ(x2 ) for y ≥ y0 . Thus Z
+∞
2
E[aξ + bξ ] =
Z 2
0
Cr{aξ + bξ ≥ y}dy − 0
Z
+∞
Z
Cr{aξ + bξ 2 ≤ y}dy
y0 0
1 µ(x2 )dy (let x2 = x, then ax + bx2 = y) y0 2 0 Z Z 1 −∞ 1 0 = µ(x)(a + 2bx)dx − µ(x)(a + 2bx)dx 2 0 2 x0 Z Z 1 +∞ 1 x0 µ(x)(a + 2bx)dx − µ(x)(a − 2bx)dx = 2 0 2 0 =
1 µ(x2 )dy − 2
where x0 = −a/(2b). The formula (6.2) is proved. 2 (2) If b < 0, then aξ + bξ 2 ≤ − a4b . The formula (6.1) and (6.2) can be proved same as in (1). Remark 6.1. When b = 0, define x0 = +∞. Letting a = 1, the formula (6.1) or (6.2) reduces the formula of computing the expected value E[ξ] provided by Definition 2.1. If a = 0, the formula (6.1) or (6.2) reduces the formula of computing the expected value E[bξ 2 ] as Z 2
+∞
E[bξ ] = b
xµ(x)dx. 0
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