Application of martingale theory in enterprise investment decision making

From the point of view of the basic option model, enterprise investment decision making under uncertainty is studied based on the martingale method. The study shows that investment options and yields are increasing functions of time, and when the option equals the yield, the investment opportunity cost is the least, which is the appropriate time for the enterprise investment. Under the condition that the investment yield is an increasing function of time, the investment opportunity cost is also an increasing function of time after the time when the investment option equals the investment yield. So the investors should invest as soon as possible, otherwise they should stop investment forever in this project. It is impossible to acquire more investment yields by indefinitely delaying the investment. Meanwhile, the study also shows that the martingale method, used widely in financial investment theory, is a powerful tool for enterprise investment decision making.

Upper Bounds for the L_p-norms of the Maximal Functions of Martingales

Let 2≤p【∞ and let (f n) be a martingale. Using exponential bounds of the probabilities of the type P(|f n|】λ‖T(f n)‖ ∞) for some quasi-linear operators acting on martingales, we estimate upper bounds for the L p-norms of the maximal functions of martinglaes. Our result is the extension and improvements of the results obtained previously by HITCZENKO and ZENG .

数理模型下抵押贷款共同保险的Martingale评价

假设未偿付额可由风险信用评估得到,房产价格服从一般Ito过程,构建了抵押贷款共同保险的数理评价模型,利用Martingale评价方法,得到了房屋抵押贷款共同保险的精确定价公式。

LORENTZ MARTINGALE SPACES AND INTERPOLATION

In this article, the authors introduce some new Lorentz spaces for martingales, which are extensions of Hardy spaces of martingales. Then they discuss their basic properties, embedding relationships, and interpolation spaces between them, during which the use of rearrangement good-λ-inequality plays an important role.

ATOMIC DECOATOMIC DECOMPOSITIONS AND DUALS OF WEAK HARDY SPACES OF B-VALUED MARTINGALES

In this article, several weak Hardy spaces of Banach-space-valued martingales are introduced, some atomic decomposition theorems for them are established and their duals are investigated. The results closely depend on the geometrical properties of the Banach space in which the martingales take values.

TWO-WEIGHT WEAK-TYPE MAXIMAL INEQUALITIES FOR MARTINGALES

In this article, some necessary and sufficient conditions are shown in order that the inequality of the form Ф1（λ）Pu（f^＊〉λ）≤Ev （Ф2（C｜f∞｜）） holds with some constant C 〉 0 independent of martingale f = （fn）n≥0 and λ 〉 0, where Фl and Ф2 are a pair of Young functions, f^＊=sup n≥0｜fn｜ adn f∞=lim n→∞ fn a.e.

Two-Parameter Banach Space Valued Strong Martingales and Characterizations of p-Smoothable Spaces

In this paper we introduce the concept of two-parameterB-valued strong martingales and investigate some features of these strong martingales. We also characterizep-smoothable Banach spaces in terms of these strong martingales.

GENERALIZED ROSENTHAL'S INEQUALITY FOR BANACH-SPACE-VALUED MARTINGALES

A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.

Davis's Inequalities for Two-Parameter Banach Space Valued Strong Martingales and Geometrical Properties of Banach Spaces

In this paper we investigated theL 1 norm inequalities of theP square and the maximal functions of two-parameterB-valued strong martingales, which can be applied to characterizep-smoothness andq-convexity of Banach spaces.

INTERPOLATION OF LORENTZ-ORLICZ MARTINGALE SPACES

In this paper, we apply function parameters to real interpolation of Lorentz- Orlicz martingale spaces. Some new interpolation theorems are formulated which generalize some known results in Lorentz spaces An introduced by Sharpley.

Atomic Decompositions of Weak Hardy Spaces of B-Valued Martingales

The atomic decompositions of weak Hardy spaces of Banach-space-valued martingales are given. With the help of the atomic decompositions, some inequalities for B-valued martingales are established in the case 0〈r≤1. Here the results are connected closely with the p-uniform smoothness and q-uniform convexity of Banach spaces which the martingales take values in.

BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT

In this article, by extending classical Dellacherie＇s theorem on stochastic se- quences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis in- equality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier＇s inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

SEVERAL WEAK-TYPE WEIGHTED INEQUALITIES IN ORLICZ MARTINGALE CLASSES

The aim of this paper is to establish several necessary and sufficient conditions in order that the weighted inequality ρ（M f 〉 λ）Φ（λ） ≤ C ∫_Ω~Ψ （C｜f｜）σdμ,λ 〉 0 or ρ（Mf〉λ） ≤ C∫-Ω~Φ（Cλ^-1 ｜f｜）σdμ,λ 〉0 holds for every uniformly integral martingale f=（f_n）, where M is the Doob＇s maximal operator, Φ, Ψ are both Φ-functions, and e, σ are weights.

CONVERGENCE OF WEIGHTED AVERAGES OF MARTINGALES IN NONCOMMUTATIVE BANACH FUNCTION SPACES

Let x （xn）≥1 be a martingale on a noncommutative probability space n （M, r） and （wn）n≥1 a sequence of positive numbers such that Wn = ∑ k=1^n wk →∞ as n →∞ We prove that x = （x.）n≥1 converges in E（M） if and only if （σn（x）n≥1 converges in E（.hd）, where E（A//） is a noncommutative rearrangement invariant Banach function space with the Fatou property and σn（x） is given by σn（x） = 1/Wn ∑k=1^n wkxk, n=1, 2, .If in addition, E（Ad） has absolutely continuous norm, then, （an（x））≥1 converges in E（.M） if and only if x = （Xn）n≥1 is uniformly integrable and its limit in measure topology x∞∈ E（M）.

THE MINIMAL OPERATOR AND WEIGHTED INEQUALITIES FOR MARTINGALES

In this article the authors introduce the minimal operator on martingale spaces, discuss some one-weight and two-weight inequalities for the minimal operator and characterize the conditions which make the inequalities hold.

WEAK ATOMIC DECOMPOSITIONS OF MARTINGALE HARDY-LORENTZ SPACES AND APPLICATIONS

In this article, we establish some atomic decomposition theorems for martin- gale Hardy-Lorentz spaces. As applications, with the help of weak atomic decompositions, some interpolation theorems for martingale Hardy-Lorentz spaces are proved.

Banach Space Valued Martingales with Respect to Complex Measure and Their Inequalities

Let d μ=ψ d ν be a complex valued measure where ν is a non negative measure and ψ is a complex valued function which satisfies b + p or b + ∞∩a 1 condition. We prove some basic martingale inequalities such as B G inequalities, weak ( p,p) and strong (p,p) type inequalities for Banach space valued martingale with respect to complex measure μ .

Φ-INEQUALITIES AND LAWS OF LARGE NUMBERS OF HARDY MARTINGALETRANSFORMS

Using stopping time method we proved the Φ-inequalities, pointwise convergence, strong and weak laws of large numbers of Hardy martingale transforms with values in complex Banach spaces, and applying them to give several characterizations of AUMD spaces.

WEIGHTED INEQUALITIES FOR THE GEOMETRIC MAXIMAL OPERATOR ON MARTINGALE SPACES

In this article, the authors introduce two operators-geometrical maximal operator Mo and the closely related limiting operator M0^＊, then they give sufficient conditions under which the equality M0=MM0^＊ holds, and characterize the equivalent relations between the weak or strong type weighted inequality and the property of W∞-weight or W∞^＊-weight for the geometrical maximal operator in the case of two-weight condition. What should be mentioned is that the new operator-the geometrical minimal operator is equal to the limitation of the minimal operator sequence, and the results for the minimal operator proved in [12] makes the proof elegant and evident.

A FOUR-WEIGHT WEAK TYPE MAXIMAL INEQUALITY FOR MARTINGALES

In this article, some necessary and sufficient conditions are shown in order that weighted inequality of the form ■holds a.e. for uniformly integrable martingales f =(f_n)n≥0 with some constant C > 0,where Φ_1,Φ_2 are Young functions, w_i(i = 1,2,3, 4) are weights, f~* =sup n≥0 |f_n| and f_∞=lim n→∞ f_n a.e. As an application, two-weight weak type maximal inequalities of martingales are considered, and particularly a new equivalence condition is presented.