Uncertain calculus with renewal process

Stability in mean for uncertain differential equation with jumps

An uncertain process is essentially a spectrum of uncertain variables indexed by the time. The uncertain renewal process is an uncertain process which counts the number of renewals that an uncertain system incurs.

Uncertain differential equation with jumps is a type of differential equations driven by both Liu process and the uncertain renewal process. Stability of uncertain differential equation with jumps, playing an important role in uncertain differential equation with jumps, means insensitivity of the state of a system with a small changes in the initial state. This paper focuses on the stability in p -th moment for uncertain differential equation with jumps, including the concept of stability in p -th moment, and the sufficient condition for uncertain differential equation with jumps being stable in p -th moment. In addition, the relationship between stability in measure and stability in p -th moment for the uncertain differential equation with jumps is also discussed.

Uncertain differential equation with jumps is a type of differential equation driven by two classes of uncertain processes, namely canonical process and renewal process. Based on uncertain differential equation with jumps, this paper proposes a stock model with jumps for uncertain financial markets. Furthermore, the European call and put option pricing formulas for the stock model are formulated and some mathematical properties of them are studied. Finally, some generalized uncertain stock models with jumps are discussed.

The renewal reward process is used to record the cumulative rewards of a system, which is widely applied in the queuing problems and insurance pricing problems. This paper studies a type of renewal reward processes with random inter-arrival times and uncertain rewards from the point of view of first hitting time. The analytic expressions of the chance distribution and the expected value of the first hitting time are derived, and a numerical method for calculating the chance distribution is designed based on the Monte-Carlo simulation. Besides, the concept of first hitting time is applied to the insurance risk process and is employed to model the ruin index of an insurance company.

Over the past decades, financial markets have increasingly exhibited features of both randomness and uncertainty, creating challenges for interest rate models that rely solely on stochastic or uncertain processes. These models often fail to adequately capture the dual nature of indeterminacy, limiting their relevance in volatile and unpredictable market conditions. This study aims to design and assess an interest rate model for uncertain-stochastic financial markets and to derive a framework for zero-coupon bond pricing under this setting. The methodology applies uncertain stochastic differential equations, which integrate elements of both probability theory and uncertainty theory, thereby accommodating aleatory and epistemic forms of indeterminacy. The proposed model extends the classical short-rate frameworks by introducing two sources of indeterminacy and provides theoretical derivations for bond pricing. Numerical illustrations are included to demonstrate the application of the model to zero-coupon bond valuation and to highlight differences from conventional approaches. The findings indicate that interest rates and zero-coupon bond prices in uncertain stochastic financial markets can be effectively modeled through uncertain random processes, leading to improved pricing accuracy and risk management in environments characterised by incomplete information and unpredictable shocks. The key conclusion is that incorporating uncertain stochastic differential equations into the interest rate and zero-coupon bonds’ prices modelling offers a more robust and flexible framework for uncertain stochastic markets. This study contributes to the growing body of uncertain stochastic finance by underscoring the need for hybrid models capable of guiding policymakers, investors and financial institutions in ensuring stability and resilience under future market uncertainties.

No previous study has involved uncertain delay differential equations with jump. In this paper, we consider the uncertain delay differential equations with V-jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, we give the equivalent integral equation. Next, we establish an existence and uniqueness theorem of solution to the differential equations we proposed in the finite domain and the infinite domain, respectively. Once more, the concept of stability for uncertain delay differential equations with V-jump is proposed. In addition, the sufficient condition for stability theorem is derived. To judge existence, uniqueness, and stability briefly, we provide some examples in the end.

An canonical process is stationary independent increment uncertain process whose increments are normal uncertain variables. Uncertain differential equation is a type of differential equation driven...

Uncertain process is a sequence of uncertain variables indexed by time and space. First, this paper presents a kind of uncertain process, known as the uncertain alternating renewal process, whose alternating interarrival times are uncertain variables. Then, it proves an uncertain alternating renewal theorem on the limit value of average working rate. Finally, an application of the alternating renewal theorem is discussed.

The Liu process is a new tool to deal with the noise process based on uncertainty theory. In this paper, we view the foreign exchange rate as an uncertain processes, described by uncertain differential equations driven by the Liu process, and build an uncertain currency model. Then, the uncertain currency option problems are discussed. Moreover, European and American currency option pricing formulas are derived for the proposed uncertain currency model and some mathematical properties are studied. Finally, two numerical examples are documented.

Asian rainbow option is option on the minimum or the maximum of several average prices. In modern financial market, Asian rainbow option is an effective instrument for asset allocation and risk management. The investor with Asian rainbow option enjoys an entitlement to select a max or min from multiple assets with an exercise price at maturity date. The investor has to defray fee to acquire this right, which raises the option pricing issue. This paper mainly explores the pricing of Asian rainbow option in the uncertain financial environment, in which the underlying assets prices are treated as uncertain processes. Here, the pricing formulas of Asian rainbow option are derived under the condition that stock prices obey uncertain differential equations driven by independent Liu processes. Furthermore, some numerical examples are designed to compute the prices of these options.

Uncertain differential equations (UDEs) with jumps are an essential tool to model the dynamic uncertain systems with dramatic changes. The interest rates, impacted heavily by human uncertainty, are assumed to follow UDEs with jumps in ideal markets. Based on this assumption, two derivatives, namely, interest-rate caps (IRCs) and interest-rate floors (IRFs), are investigated. Some formulas are presented to calculate their prices, which are of too complex forms for calculation in practice. For this reason, numerical algorithms are designed by using the formulas in order to compute the prices of these structured products. Numerical experiments are performed to illustrate the effectiveness and efficiency, which also show the prices of IRCs are strictly increasing with respect to the diffusion parameter while the prices of IRFs are strictly decreasing with respect to the diffusion parameter.

In this paper, we discuss a new kind of stability, that is, finite-time stability, for uncertain differential equations, by formalizing some properties. As a possible application, we define a new class of uncertain multi-agent systems, according to the Liu’s uncertainty theory, as a counterpart of stochastic multi-agent systems. We formalize the governing equations, driven by canonical process, which is a type of uncertain process with stationary and independent increments. The concept of finite-time consensus in the context of uncertainty theory is consequently derived. A numerical procedure to estimate the settling time is proposed. The case with proportional delay was also considered.

This paper describes a new method for generating stationary integer-valued time series from renewal processes. We prove that if the lifetime distribution of renewal processes is nonlattice and the probability generating function is rational, then the generated time series satisfy causal and invertible ARMA type stochastic difference equations. The result provides an easy method for generating integer-valued time series with ARMA type autocovariance functions. Examples of generating binomial ARMA(p,p-1) series from lifetime distributions with constant hazard rates after lag p are given as an illustration. An estimation method is developed for the AR(p) cases.

russell: the Journal of Bertrand Russell Studies n.s. 35 (winter 2015–16): 183 The Bertrand Russell Research Centre, McMaster U. issn 0036–01631; online 1913–8032 c:\users\ken\documents\type3502\red\rj 3502 053 red.docx 2015-11-12 7:16 PM oeviews RUSSELL’S PRINCIPLES OF MATHEMATICS G. E. Moore f the philosophical books published in the United Kingdom in 1903, the most important is Mr. Russell’s Principles of Mathematics.1 In this book, Mr. Russell tells us, he has two main objects. His first object is to establish the two very important propositions (1) “that all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts” and (2) “that all its propositions are deducible from a very small number of fundamental logical principles.” The examination of the principal branches of pure mathematics, which is necessary to establish these two propositions, occupies the last six Parts of the book, which are entitled respectively “Number”, “Quantity”, “Order”, “Infinity and Continuity ”, “Space”, and “Matter and Motion”. In these parts there is much which cannot be easily understood without a special knowledge of Mathematics, and much which has little bearing on philosophy, except so far as it helps to establish Mr. Russell’s two main propositions; but there is much also which is of considerable importance for philosophy, quite apart from its bearing on these two propositions: in particular, Mr. Russell examines very carefully the conceptions of Infinity and Continuity, and attempts to shew that they involve no antinomies. Part i, on the other hand, is devoted to Mr. Russell’s second object—“the explanation of the fundamental concepts which mathematics accepts as indefinable”, and is almost entirely philosophical in its nature. I shall endeavour to give some account (1) of the meaning and consequences of Mr. Russell’s two propositions concerning the relation of Logic and Mathematics (2) of some of the more important points dealt with in Part i and (3) of the theory of Infinity and Continuity. Mr. Russell is eminently qualified for his task by a thorough knowledge of Mathematics and by great philosophical acumen ; and it is certain that no philosopher ought in future to handle any of the subjects discussed in this book, without taking account of the arguments advanced in it.2 1 The Principles of Mathematics. 2 [The remainder of Moore’s very long, unpublished review may be read in the Russell Archives, Rec. Acq. 116.—Ed.] l= ...