4.1. Problem 4
Now we consider the general maximization problem :
Maximize V+(X) = Z ∞
0
w+(P {X > y}) dy
subject to
E[ ρX ] = x0, X ≥ 0, a.s.
(Problem 4)
The objective of Problem 4 is to find the optimal random variable X∗. Here we turn finding the optimal random variable X∗ into seeking the distribution function of X∗ by the following steps.
Lemma 4.1 (Ross [17], Chapter 5 theoretical exercises 28.). Let X be a continuous random variable having the distribution function F (·). Then the random variable 1−F (X) follows uniform distribution over the interval (0, 1), that is,
1 − F (X) ∼ U(0, 1). (4.1)
Recall that for the distribution function F (·) of X defined by F (x) = P{X ≤ x} and F (·) satisfies
1. F (·) is nondecreasing and right continuous.
2. lim
b→∞F (b) = 1 and lim
b→−∞F (b) = 0.
17
18 4. MAIN RESULTS
If the distribution function F (·) is strictly increasing and continuous, then F−1 : [0, 1] → R exists. Unfortunately, the distribution function does not have an inverse in general. Therefore, we defined the following general inverse function of the distribution function.
Definition 4.2. Let F (·) be the distribution function of X. We defined the inverse function of F (·), for y ∈ [0, 1],
F−1(y) = inf{x ∈ R : F (x) ≥ y} with inf φ = ∞. (4.2)
Some properties of the inverse of the distribution function are :
1. F−1(·) is nondecreasing and left continuous.
2. F−1(F (x)) ≤ x.
3. F (F−1(y)) ≥ y.
4. F−1(y) ≤ x if and only if y ≤ F (x).
5. If Y ∼ U(0, 1) then F−1(Y ) has the same distribution as X.
The property 5 tells us that the inverse of the distribution function can translate results the uniform distribution to the other distributions.
Proposition 4.3 (Jin and Zhou [11], Lemma C.1). If X∗ is the optimal solution for Problem 4 and G∗(·) is the distribution function of X∗, then (G∗)−1¡
1 − Fρ(ρ)¢
has the same distribution as X∗ and
X∗ = (G∗)−1¡
1 − Fρ(ρ)¢
, (4.3)
where (G∗)−1(·) is the inverse function of G∗(·).
4.1. PROBLEM 4 19
Now we turn to the objective functional of Problem 4, Z ∞ Here we need an important assumption.
Assumption 4.4. We assume that the limit
y→∞lim[y · w+(1 − G(y))] = 0 (4.6) This assumption is very rational because lim
y→∞w+(1 − G(y)) = 0 and usually the utility function u(·) is bounded.
Let s = G(y). Then we can get
20 4. MAIN RESULTS
4.2. Problem 5
Proposition 4.3 suggests that in order to solve Problem 4 we only needs to seek among random variables of the form G−1(U), where G(·) is the distribution function of a nonnegative random variable. Applying (4.5) and (4.7), we turn Problem 4 into the following problem.
Maximize v(G) :=
Z 1
0
G−1(s) · (w+)0(1 − s) ds
subject to
Z 1
0
G−1(s) · Fρ−1(1 − s) ds = x0,
G(·) is the distribution function of a nonnegative r.v.
(Problem 5)
The following result, which is straightforward in view of Lemma 4.1 and Proposition 4.3, means that Problem 4 is equivalent to Problem 5.
Proposition 4.5 (Jin and Zhou [11], Proposition C.1).
If G∗(·) is optimal for Problem 5, then
X∗ := (G∗)−1(U)
is optimal for Problem 4. Conversely, if X∗ is optimal for Problem 4, then its distribution function G∗(·) is optimal for Problem 5 and X∗ = (G∗)−1(U), a.s..
4.3. Problem 6
Denoting
g(·) := G−1(·). (4.8)
Since g(·) is the inverse of the distribution function, so g : [0, 1] 7→ [0, ∞] is nondecreasing and left continuous with g(0) = 0.
4.4. MAIN IDEA AND RESULTS 21
Then we can rewrite Problem 5 into Maximize
4.4. Main idea and results
Our main idea is to find an inequality, with which we can solve Problem 6. The inequality have relation that the objective is less than or equal to the first constrain.
Since the integrations of Problem 6 can be express convolution or inner produce type, we found some useful weighted inequalities for monotone functions, which play a key role in solving Problem 6 :
Proposition 4.6 (Heinig and Maligranda [10], Theorem 2.1). Let 0 < p ≤ q < ∞, u(s), v(s) ≥ 0 and f (0) = 0. The inequality
Taking p = q = 1, we get the following corollary.
Corollary 4.7. Let u(s), v(s) ≥ 0 and f (0) = 0. The inequality
22 4. MAIN RESULTS
Moreover, M is the best constant satisfying (4.11). Equation (4.12) admits an optimal solution t∗, then ”=” holds when f (x) = λI(t∗,1](x) where λ is any nonnegative constant.
Since the probability weighting function w+(·) is strictly increasing and differentiable, we get the derivative (w+)0(s) ≥ 0 for all 0 ≤ s ≤ 1. And Fρ−1(·) is the inverse function of the distribution function, so Fρ−1(·) is nondecreasing, that is, Fρ−1(s) ≥ 0 for all 0 ≤ s ≤ 1.
We use the result of Corollary 4.7 for Problem 6. If M = sup
then the following inequality
³ Z 1 Problem 6 is M · x0 if the equality of (4.14) holds. The constant M given by (4.13) can be simplified by Equation (4.16) admits an optimal solution c∗, then the optimal function of Problem 6 is of the form
g∗(x) = (G∗)−1(x) = λI(1−c∗,1](x), x ∈ [0, 1] (4.17) where λ > 0 is the constant satisfying λ ·
Z c∗
4.4. MAIN IDEA AND RESULTS 23
Remark 4.8. A maximization problem is called well-posed if the supremum of its objective is finite; otherwise it is called ill-posed.
Theorem 4.9. The following statements are equivalent:
(1) Problem 6 is well-posed for any x0 ≥ 0.
Furthermore, when one of the above (1)-(3) holds, the optimal solution to Problem 6 is of the form
Fρ−1(s) ds are strictly increasing for c, then the
· infinity only when c = 0. Therfore,
(2) ⇐⇒ lim
We now summarize the main result in the following theorem.
24 4. MAIN RESULTS
Theorem 4.10. Assume that lim
c→0
£(w+)0(c)/Fρ−1(c)¤
< ∞. Then the maximal value at terminal time T under LCPT is given by
sup
Equation (4.19) admits an optimal solution c∗. Then the corresponding optimal terminal wealth to Problem 3 is
X∗ = x0 ratio that all the money put in the bank account. Therefore, It means that the payoff
³
E[ ρ I{ρ<Fρ−1(c∗)}]
´−1
· x0 is better than the payoff if we invest all money in the bond.
The optimal terminal wealth X∗ mentioned in (4.20) tells us two different stories in economical view at terminal time. In the cases of {ρ < Fρ−1(c∗)} the payoff we gain is more than that we get from the bond market, and this payoff is fixed due to the deterministic coefficient. Furthermore, in the rest of part {ρ ≥ Fρ−1(c∗)} all the assets turn out to be zero. Those cases of {ρ < Fρ−1(c∗)} are profitable to the investors. It might be that the noise Wt of the stock price does not fluctuate dramatically.
4.5. The optimal wealth process and the optimal strategies
In this section, we want to find a portfolio π replicating the optimal terminal wealth X∗ of (4.20). Recall that ρ = ρT with
Let N(·) and ψ(·) be the distribution function and probability density function of a standard normal random variable respectively. ρ(t, T ) := ρT/ρt conditional on Ft follows
4.5. THE OPTIMAL WEALTH PROCESS AND THE OPTIMAL STRATEGIES 25
a log-normal distribution with parameter (µt,σ2t), where µt= −( r + 1
By (3.11), the replicating wealth process is given by Xt = E£
It is well known that the replicating portfolio is πt= −³α − r
β2
´
fx(t, ρt) ρt; (4.22) see, e.g., Bielecki [3], Equation(7.6). Now we calculate
fx(t, ρt) = −λ ψ
Plugging it in (4.22), we get the following result.
Theorem 4.12. The wealth-portfolio pair replicating X∗ is given by Xt = λ
CHAPTER 5