Ito Formula Research Articles

Analysis of a stochastic Leslie-Gower three-species food chain system with Holling-II functional response and Ornstein-Uhlenbeck process

<abstract><p>This paper studies a stochastic Leslie-Gower model with a Holling-II functional response that is driven by the Ornstein-Uhlenbeck process. Some asymptotic properties of the solution of the stochastic Leslie-Gower model are introduced: The existence and uniqueness of the global solution of the model are given; the ultimate boundedness of the model is proven; by constructing the Lyapunov function and applying Ito's formula, the existence of the stationary distribution of the model is demonstrated; and the conditions for system extinction are discussed. Finally, numerical simulations are used to validate our conclusion.</p></abstract>

Tool Degradation Prediction Based on Semimartingale Approximation of Linear Fractional Alpha-Stable Motion and Multi-Feature Fusion

Tool wear will reduce workpieces’ quality and accuracy. In this paper, the vibration signals of the milling process were analyzed, and it was found that historical fluctuations still have an impact on the existing state. First of all, the linear fractional alpha-stable motion (LFSM) was investigated, along with a differential iterative model with it as the noise term is constructed according to the fractional-order Ito formula; the general solution of this model is derived by semimartingale approximation. After that, for the chaotic features of the vibration signal, the time-frequency domain characteristics were extracted using principal component analysis (PCA), and the relationship between the variation of the generalized Hurst exponent and tool wear was established using multifractal detrended fluctuation analysis (MDFA). Then, the maximum prediction length was obtained by the maximum Lyapunov exponent (MLE), which allows for analysis of the vibration signal. Finally, tool condition diagnosis was carried out by the evolving connectionist system (ECoS). The results show that the LFSM iterative model with semimartingale approximation combined with PCA and MDFA are effective for the prediction of vibration trends and tool condition. Further, the monitoring of tool condition using ECoS is also effective.

Conditional expectation strategy under the long memory Heston stochastic volatility model

This article deals with an European option pricing via proportional transaction costs in the incomplete environment with and without arbitrage opportunities under two long memory versions of the Heston model. Observing and introducing a traded proxy for the volatility in the modern market, we use the conditional expectation and the delta hedging strategies and present the generalized fractional Ito formula to obtain the option price partial differential equations (PDEs). To solve these PDEs, we apply the finite difference method and employ the K-antithetic variates algorithm based on the Monte-Carlo simulation as a benchmark for this method. Finally, we provide numerical results to illustrate the effectiveness of the proposed model.

A New Proof of the Expansion of Iterated Ito Stochastic Integrals with Respect to the Components of a Multidimensional Wiener Process Based on Generalized Multiple Fourier Series and Hermite Polynomials

The article is devoted to a new proof of the expansion for iterated Ito stochastic integrals with respect to the components of a multidimensional Wiener process. The above expansion is based on Hermite polynomials and generalized multiple Fourier series in arbitrary complete orthonormal systems of functions in a Hilbert space. In 2006, the author obtained a similar expansion, but with a lesser degree of generality, namely, for the case of continuous or piecewise continuous complete orthonornal systems of functions in a Hilbert space. In this article, the author generalizes the expansion of iterated Ito stochastic integrals obtained by him in 2006 to the case of an arbitrary complete orthonormal systems of functions in a Hilbert space using a new approach based on the Ito formula. The obtained expansion of iterated Ito stochastic integrals is useful for constructing of high-order strong numerical methods for systems of Ito stochastic differential equations with multidimensional non-commutative noise.

Analisis Model Stokastik Penularan Virus Hepatitis B

The spread of hepatitis B virus (HBV) infection has been widely studied using the deterministic SIR model, in which individuals who recover from acute infection have temporary immunity to the virus. However, this deterministic model uses a constant rate of viral infection over time. This is not in accordance with the fact that the infection rate is a random parameter that depends on time. This study discusses the analysis of the stochastic model of hepatitis B virus transmission. The purpose of this study is to construct the SIR stochastic model by dividing the infection rate into two, namely the rate of acute and chronic infection following the Wiener process. The model is then searched for an analytical solution referring to the Ito formula. The analytical solution and the Wiener process are described by substituting parameter values in the form of acute and chronic infection rates (β+α), cure rate (γ), and initial values (S(0) and I(0)) to obtain the mean value (μ). and the standard deviation (σ) of dS(t) and dI(t). The results of the simulation show that the number of infected individuals (I(t)) will decrease rapidly if (γ) is greater but will increase rapidly if (β+α) and (I(0)) are greater.

Analisis Model Stokastik Penularan Virus Hepatitis B

The spread of hepatitis B virus (HBV) infection has been widely studied using the deterministic SIR model, in which individuals who recover from acute infection have temporary immunity to the virus. However, this deterministic model uses a constant rate of viral infection over time. This is not in accordance with the fact that the infection rate is a random parameter that depends on time. This study discusses the analysis of the stochastic model of hepatitis B virus transmission. The purpose of this study is to construct the SIR stochastic model by dividing the infection rate into two, namely the rate of acute and chronic infection following the Wiener process. The model is then searched for an analytical solution referring to the Ito formula. The analytical solution and the Wiener process are described by substituting parameter values in the form of acute and chronic infection rates (β+α), cure rate (γ), and initial values (S(0) and I(0)) to obtain the mean value (μ). and the standard deviation (σ) of dS(t) and dI(t). The results of the simulation show that the number of infected individuals (I(t)) will decrease rapidly if (γ) is greater but will increase rapidly if (β+α) and (I(0)) are greater.

Time-Slicing Path-integral in Curved Space

Path integrals constitute powerful representations for both quantum and stochastic dynamics. Yet despite many decades of intensive studies, there is no consensus on how to formulate them for dynamics in curved space, or how to make them covariant with respect to nonlinear transform of variables (NTV). In this work, we construct a rigorous and covariant formulation of time-slicing path integrals for dynamics in curved space. We first establish a rigorous criterion for equivalence oftime-slice Green's function(TSGF) in the continuum limit (Lemma 1). This implies the existence of infinitely many equivalent representations for time-slicing path integral. We then show that, for any dynamics with second order generator, all time-slice actions are equivalent to a Gaussian (Lemma 2). We further construct a continuous family of equivalent path-integral actions parameterized by an interpolation parameterα∈[0,1](Lemma 3). The action generically contains term linear inΔx, whose concrete form depends onα. Finally we also establish the covariance of our path-integral formalism, by demonstrating how the action transforms under NTV. Theα=0representation of time-slice action is particularly convenient because it is Gaussian and transforms as a scalar, as long asΔxtransforms according toIto's formula.

A new stochastic diffusion process based on the Rayleigh density function

In this work we present a model of a stochastic diffusion process based on the Rayleigh distribution function. (its trend is proportional to the rayleigh probability density function). In the first part we start by obtaining the probabilistic characteristics of this model as the explicit expression of the process by applying the Ito formula. The conditional mean and mean function is analysed to obtain fits and predictions using simulated data. Then we developed the statistical inference of the model by estimating these parameters through the maximum likelihood methodology. Finally, in order to highlight the usefulness of this methodology, we include the results obtained from several simulation examples.

A Clark-Ocone Type Formula via Itô Calculus and its Application to Finance

An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which are not Malliavin differentiable. To this end, we make use of Ito's formula, instead of Malliavin calculus. As an application to mathematical finance, we shall give an explicit representation of locally risk-minimizing strategy of digital options for exponential Levy models. Since the payoff of digital options is described by an indicator function, we also discuss the Malliavin differentiability of indicator functions with respect to Levy processes.

Alòs Type Decomposition Formula for Barndorff-Nielsen and Shephard Model

The objective is to provide an Al\`os type decomposition formula of call option prices for the Barndorff-Nielsen and Shephard model: an Ornstein-Uhlenbeck type stochastic volatility model driven by a subordinator without drift. Al\`os (2012) introduced a decomposition expression for the Heston model by using Ito's formula. In this paper, we extend it to the Barndorff-Nielsen and Shephard model. As far as we know, this is the first result on the Al\`os type decomposition formula for models with infinite active jumps.

Infinite Horizon Stochastic Maximum Principle for Stochastic Delay Evolution Equations in Hilbert Spaces

In the present work, we investigate infinite horizon optimal control problems driven by a class of stochastic delay evolution equations in Hilbert spaces and the corresponding adjoint equation is given by an anticipated backward stochastic evolution equation (ABSEE). We first establish a priori estimate for the solution to ABSEEs by imposing restriction on unbounded operator $$A^*$$ , that is, the operator $$A^*$$ is maximal dissipative. In this way, the Ito inequality is applicable in our study, and we can also avoid the problem that neither Ito formula nor energy equation is available. Next, we obtain the existence and uniqueness results of solutions of linear backward stochastic evolution equations on infinite horizon by using approximating methods. Then, the existence and uniqueness results of ABSEEs on infinite horizon is obtained via the fixed-point theory. That is the highlight of innovation in this paper. Eventually, we establish necessary and sufficient conditions for optimality of the control problem on infinite horizon, in the form of Pontryagin’s maximum principle.

Existence and Uniqueness of Solutions for Some Basic Stochastic Differential Equations

Stochastic differential equation (sde) has plenty of applications on economic and physics. However, how to find the explicit solution usually remains a serious trouble. In this paper, we aim to study the uniqueness and existence of the stochastic differential equations. To this end, we used the Ito formula and the Lipschitz condition. As a result, we found that the Lipschitz condition is a sufficient condition for a stochastic differential equation to have a unique solution. If the Lipschitz condition can be proved by calculating the index, then it is possible to find the numerical solution.Research on the stochastic differential equation has gained much attention after the finding of the Ito formula. The Ito formula illustrates the chain rule of a semi-martingale. Due to the difficulty in obtaining the explicit solution, one primary problem is whether there exists a unique solution. One approach to solve this problem is the use of Lipschitz condition, which is used in the ordinary differential equations.

Qualitative behaviour of a stochastic hepatitis C epidemic model in cellular level.

In this paper, a mathematical model describing the dynamical of the spread of hepatitis C virus (HCV) at a cellular level with a stochastic noise in the transmission rate is developed from the deterministic model. The unique time-global solution for any positive initial value is served. The Ito's Formula, the suitable Lyapunov function, and other stochastic analysis techniques are used to analyze the model dynamics. The numerical simulations are carried out to describe the analytical results. These results highlight the impact of the noise intensity accelerating the extinction of the disease.

A model of hepatitis B virus with random interference infection rate.

According to the mechanism of drug inhibition of hepatitis B virus and the analysis of clinical data, it is found that random factors in long-term treatment produced uncertainty and resistance to hepatitis B virus infection rate, a model of hepatitis B virus with random interference infection rate is established. By constructing Lyapunov function and using Ito's formula, it is proved that the stochastic hepatitis B model has a unique global positive solution. The sufficient conditions for the asymptotic behavior of solution are given. The relationship between noise intensity and oscillation amplitude is obtained. The effects of noise intensity on the asymptotic behavior of the model and antiviral therapy are simulated, and the conclusion of the theorem is verified. An interesting phenomenon is also found that with the increase of noise intensity, the number of drug-resistant viruses will decrease, which will affect the accuracy of a single test of HBV DNA. Therefore, it is suggested to increase the frequency and interval of tests.

Pricing Options in a Delayed Market Driven by Le’vy Noise

In this paper we studied stochastic delayed differential equations driven by Le’vy noise. The analogue of Ito formula is considered. The Black-Scholes formula analogue for Vanilla call option price formula is derived.

Stochastic calculus on Fréchet spaces

We develop a stochastic calculus on Frechet spaces for a particular family of stochastic processes. The formulae obtained is applied to the case of tame Frechet spaces and tame maps. In particular, an Ito formula for tame maps is proved.

Modelling the fear effect in a two-species predator–prey system under the influence of toxic substances

Effect of anti-predator defense due to fear of predator felt by prey and effect of toxic substance released by external sources on prey–predator system is a serious mater of concern in mathematical biology. In the proposed model we have discussed a prey–predator system in which both the species are infected by environmental toxicant. In our consideration prey species is directly infected by environmental toxicant and predator gets infected by consuming infected prey. Prey’s growth rate is assumed to be affected by fear of predator. In this work the proposed predator–prey model is analyzed in presence of environmental fluctuation, i.e., stochastic analysis of this model is discussed. Using Ito formula: positivity, boundedness, uniform continuity criterion and global attractivity of solutions of this system have been established. Conditions for which the prey as well as the predator goes extinct have been derived. Conditions for persistence of the system have also been discussed. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different levels of toxicant and different levels of fear have been demonstrated by depicting figures in numerical simulation using MATLAB.

Alternatives to Black-76 Model for Options Valuation of Futures Contracts (Presentation Slides)

In these lectures’ notes I would like to introduce forwards, futures and options, and to review some results on Black-Scholes-73 and Black-76 models for positive prices, and also on alternatives models for negative prices for option valuation of futures contracts. I will focus on the first model introduced by Louis Bachelier in 1900, and on other ones, including Ornstein-Uhlenbeck model (1930) and Vasicek model (1977). For these models I will present some results related to the problem and project. In the end of this lecture notes there are three Appendices, A-C, which contain brief introductions to martingales, stochastic calculus (including Ito formula) and simulation. I’ll start with the References (they are also in the end of the lectures) to pay students’ attention to main literatures

Existence, uniqueness and continuous dependence of solutions to conformable stochastic differential equations

Abstract In this paper, we study conformable stochastic differential equations. Firstly, the Ito formula is established and used to discuss the explicit expression of solutions of linear differential equations. Secondly, the existence and uniqueness of solutions of nonlinear conformable stochastic differential equations are proved by the Picard iteration method, and the continuous dependence of solutions on initial values is proved by the Gronwall inequality, the exponential estimation of solutions is also given. Finally, some examples are given to illustrate the theoretically results and we compare the simulation results for the conformable stock model with different ρ.

Mixed-type stochastic differential equations driven by standard and fractional Brownian motions with Hurst indices greater than 1/3

In this paper we consider mixed-type stochastic differential equations driven by standard and fractional Brownian motions with Hurst indices greater than 1/3. There are proved theorems on the existence, uniqueness, and continuous dependence of solutions on the initial data. We provide an analog of the Ito formula to change variables. Asymptotic expansions of functionals on the solutions of mixed-type stochastic differential equations for small times are obtained. We receive analogs of the Kolmogorov equations for mathematical expectations and probability densities in the commutative case. Finally, we consider an application of mixed-type stochastic differential equations to solving the problem of macroeconomic variables extrapolation in credit risks models.