In this chapter, we define (stochastic) Ito-integrals $\int_0^t {HdM} $ H dM for local L 2 — martingales M and a fairly large class of adapted processes H. The integral is a random variable. It will be constructed as a suitable limit of Riemann-Stieltjes type approximations like 

$$\sum\limits_{i = 1}^n {{H_{{s_i}}} \cdot \left( {{M_{s{}_{i + 1}}} - {M_{{s_i}}}} \right)} $$

The Stochastic Integral

Optimal resource allocation using reinforcement learning for IoT content-centric services

A hybrid fuzzy goal programming approach with different goal priorities to aggregate production planning

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.

Uncertain Currency Model and Currency Option Pricing

Uncertain calculus is a branch of mathematics that deals with the integral and differential of functions of uncertain processes. This paper first introduces the Liu process as an uncertain process defined by the Liu integral. Some properties of Liu processes are investigated such as sample continuity property, finite variation property, and the fact that a continuously differentiable function of the Liu process is another Liu process, among others. Based on the Liu process, the uncertain integral is extended. Furthermore, some mathematical properties are proved, including the fundamental theorem, change of variable theorem, and integration by parts theorem. Finally, uncertain calculus with respect to multiple Liu processes is discussed.

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Liu process and uncertain calculus

Complex uncertain variables are measurable functions from an uncertainty space to the set of complex numbers and are used to model complex uncertain quantities. This paper introduces the convergence concepts of complex uncertain sequences: convergence almost surely (a.s.), convergence in measure, convergence in mean, convergence in distribution and convergence uniformly almost surely. In addition, relationships among them are discussed.

Convergence of complex uncertain sequences

Adams-Simpson method for solving uncertain differential equation

Based on uncertainty theory, multiproduct aggregate production planning model is presented, where the market demand, production cost, subcontracting cost, etc., are all characterized as uncertain variables. The objective is to maximize the belief degree of obtaining the profit more than the predetermined profit over the whole planning horizon. When these uncertain variables are linear, the objective function and constraints can be converted into crisp equivalents, the model is a nonlinear programming, then can be solved by traditional methods. An example is given to illustrate the model and the converting method.

Uncertain aggregate production planning

Considering the uncertain fluctuations in the financial market from time to time, we propose a currency model with floating interest rates within the framework of uncertainty theory. Different from the classical stochastic currency models, this paper is assumed that the domestic interest rate, the foreign interest rate and the exchange rate follow uncertain differential equations. After that, the pricing formulas of European and American currency options are derived. The simulation experiments presented in this paper illustrate the performance of the proposed model, and the relationship between the option pricing formulas and all relevant parameters is analyzed.

An uncertain currency model with floating interest rates

Entropy is used as a measure to characterize the uncertainty. So far, entropy for uncertain variables in the forms of logarithm function and triangular function has been proposed. This paper mainly studies the concept of triangular entropy, and verifies its properties such as translation invariance and positive linearity. As an application, this paper also considers a mean-variance portfolio selection problem with triangular entropy as a constraint.

Yufu Ning

Papers

Convergence of complex uncertain sequences

Adams-Simpson method for solving uncertain differential equation

Uncertain aggregate production planning

An uncertain currency model with floating interest rates

Triangular entropy of uncertain variables with application to portfolio selection