scholarly journals Some Theorems for Inverse Uncertainty Distribution of Uncertain Processes

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 14
Author(s):  
Xiumei Chen ◽  
Yufu Ning ◽  
Lihui Wang ◽  
Shuai Wang ◽  
Hong Huang
Keyword(s):  
Uncertainty Theory ◽  
Independent Increment ◽  
Monotone Function ◽  
Real Life ◽  
Uncertain Process ◽  
Uncertainty Distribution ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
Independent Increment Process

In real life, indeterminacy and determinacy are symmetric, while indeterminacy is absolute. We are devoted to studying indeterminacy through uncertainty theory. Within the framework of uncertainty theory, uncertain processes are used to model the evolution of uncertain phenomena. The uncertainty distribution and inverse uncertainty distribution of uncertain processes are important tools to describe uncertain processes. An independent increment process is a special uncertain process with independent increments. An important conjecture about inverse uncertainty distribution of an independent increment process has not been solved yet. In this paper, the conjecture is proven, and therefore, a theorem is obtained. Based on this theorem, some other theorems for inverse uncertainty distribution of the monotone function of independent increment processes are investigated. Meanwhile, some examples are given to illustrate the results.

scholarly journals A New Formula for Calculating Uncertainty Distribution of Function of Uncertain Variables

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2429
Author(s):  
Yuxing Jia ◽  
Yuer Lv ◽  
Zhigang Wang
Keyword(s):  
Uncertainty Theory ◽  
Monotone Function ◽  
Uncertain Variable ◽  
Mathematical Tool ◽  
Human Beings ◽  
Uncertainty Distribution ◽  
Uncertain Variables ◽  
New Ideas ◽  
New Formula ◽  
Theory Uncertainty

As a mathematical tool to rationally handle degrees of belief in human beings, uncertainty theory has been widely applied in the research and development of various domains, including science and engineering. As a fundamental part of uncertainty theory, uncertainty distribution is the key approach in the characterization of an uncertain variable. This paper shows a new formula to calculate the uncertainty distribution of strictly monotone function of uncertain variables, which breaks the habitual thinking that only the former formula can be used. In particular, the new formula is symmetrical to the former formula, which shows that when it is too intricate to deal with a problem using the former formula, the problem can be observed from another perspective by using the new formula. New ideas may be obtained from the combination of uncertainty theory and symmetry.

scholarly journals Risk-Neutral Pricing Method of Options Based on Uncertainty Theory

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2285
Author(s):  
Hong Huang ◽  
Yufu Ning
Keyword(s):  
Stock Price ◽  
Uncertainty Theory ◽  
Independent Increment ◽  
Special Kind ◽  
Uncertain Process ◽  
Put Options ◽  
Liu Process ◽  
Stationary Independent Increment ◽  
Risk Neutral ◽  
Relationship Of

In order to rationally deal with the belief degree, Liu proposed uncertainty theory and refined into a branch of mathematics based on normality, self-duality, sub-additivity and product axioms. Subsequently, Liu defined the uncertainty process to describe the evolution of uncertainty phenomena over time. This paper proposes a risk-neutral option pricing method under the assumption that the stock price is driven by Liu process, which is a special kind of uncertain process with a stationary independent increment. Based on uncertainty theory, the stock price’s distribution and inverse distribution function under the risk-neutral measure are first derived. Then these two proposed functions are applied to price the European and American options, and verify the parity relationship of European call and put options.

Characterization of stable processes by identically distributed stochastic integrals

1980 ◽  
Vol 12 (3) ◽  
pp. 689-709 ◽  
Author(s):  
M. Riedel
Keyword(s):  
Stochastic Process ◽  
Stable Process ◽  
Convergence In Probability ◽  
Stochastic Integrals ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Subject

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

Uncertain Independent Increment Processes are Contour Processes

2021 ◽  
Vol 14 (03) ◽  
Author(s):  
Kai Yao
Keyword(s):  
Basic Property ◽  
Independent Increment ◽  
Extreme Values ◽  
Time Integral ◽  
Sample Paths ◽  
Independent Increment Process ◽  
Uncertainty Space

Uncertain processes are used to model dynamic indeterminate systems associated with human uncertainty, and uncertain independent increment processes are a type of uncertain processes with independent uncertain increments. This paper mainly verifies a basic property about the sample paths of uncertain independent increment processes, which states that uncertain independent increment processes defined on a continuous uncertainty space are contour processes, a type of uncertain processes with a spectrum of sample paths as the skeletons. Based on this property, the extreme values and the time integral of an uncertain independent increment process are investigated, and their inverse uncertainty distributions are obtained.

A characterization of stable processes

1969 ◽  
Vol 6 (02) ◽  
pp. 409-418 ◽  
Author(s):  
Eugene Lukacs
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Interior Point ◽  
Random Variable ◽  
Stable Processes ◽  
Process With Independent Increments ◽  
Independent Increments

Let X(t) be a stochastic process whose parameter t runs over a finite or infinite n terval T. Let t 1 , t 2 ɛ T, t 1 〈 t2; the random variable X(t 2) – X(t 1) is called the increment of the process X(t) over the interval [t 1, t 2]. A process X(t) is said to be homogeneous if the distribution function of the increment X(t + τ) — X(t) depends only on the length τ of the interval but is independent of the endpoint t. Two intervals are said to be non-overlapping if they have no interior point in common. A process X(t) is called a process with independent increments if the increments over non-overlapping intervals are stochastically independent. A process X(t) is said to be continuous at the point t if plimτ→0 [X(t + τ) — X(t)] = 0, that is if for any ε > 0, limτ→0 P(| X(t + τ) — X(t) | > ε) = 0. A process is continuous in an interval [A, B] if it is continuous in every point of [A, B].

Sign in / Sign up

Export Citation Format

Share Document