Order Stochastic Differential Equation Research Articles

The Maximal and Minimal Distributions of Wealth Processes in Black–Scholes Markets

The Black–Scholes formula is an important formula for pricing a contingent claim in complete financial markets. This formula can be obtained under the assumption that the investor’s strategy is carried out according to a self-financing criterion; hence, there arise a set of self-financing portfolios corresponding to different contingent claims. The natural questions are: If an investor invests according to self-financing portfolios in the financial market, what are the maximal and minimal distributions of the investor’s wealth on some specific interval at the terminal time? Furthermore, if such distributions exist, how can the corresponding optimal portfolios be constructed? The present study applies the theory of backward stochastic differential equations in order to obtain an affirmative answer to the above questions. That is, the explicit formulations for the maximal and minimal distributions of wealth when adopting self-financing strategies would be derived, and the corresponding optimal (self-financing) portfolios would be constructed. Furthermore, this would verify the benefits of diversified portfolios in financial markets: that is, do not put all your eggs in the same basket.

Approximate controllability for Hilfer fractional stochastic differential systems of order 1

In this paper, we analyse the approximate controllability of Hilfer fractional stochastic differential systems of order 1 < μ < 2 in Hilbert spaces. The primary findings are carried out by using fractional calculus, stochastic analysis theory, cosine family, and Schauder's fixed point theorem. In particular, we draw a new set of sufficient conditions for the approximate controllability of Hilfer fractional stochastic differential equations of order 1 < μ < 2 under the assumption that the corresponding linear system is approximately controllable. In the end, an example is provided to illustrate the derived theory.

European Option Pricing Based on FSDE Driven by Fractional Brownian Motion

In the actual financial market, the classical Black-Scholes (B-S) model cant perfectly describe the process of stock price. Besides, memory effect is an important phenomenon in financial systems. Thus, in this paper, we establish a fractional order stochastic differential equations (FSDE) which is driven by fractional Brownian motion (fBm) to describe the effect of noise memory and trend memory in financial pricing. Finally, we derive a European option pricing formula based on the established model. After conducting an empirical analysis based on the SSE 50ETF, we find that the established model performs better than the traditional one.

First-order linear Marcus SPDEs

In this paper, we solve a Lévy driven linear stochastic first-order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic characteristics. It has the same form as a solution of a deterministic PDE or a solution of a stochastic PDE driven by a Brownian motion studied by Kunita [First order stochastic partial differential equations, in Stochastic Analysis. Proc. Taniguchi Int. Symp. Stochastic Analysis, Katata and Kyoto, 1982, North-Holland Mathematical Library, Vol. 32, ed. K. Itô (North-Holland Publishing Company, 1984), pp. 249–269; Stochastic Flows and Stochastic Differential Equations, Vol. 24 (Cambridge University Press, 1997)].

Short-term photovoltaic power prediction based on fractional Levy stable motion

Accurate prediction of photovoltaic (PV) power generation is the key to daily dispatch management and safe and stable grid operation. In order to improve the accuracy of the prediction, a finite iterative PV power prediction model with long range dependence (LRD) characteristics was developed using fractional Lévy stable motion (fLsm) and applied to a real dataset collected in the DKASC photovoltaic system in Alice Springs, Australia. The LRD prediction model considers the influence of current and past trends in the stochastic series on the future trends. Firstly, the calculation of the maximum steps prediction was introduced based on the maximum Lyapunov. The maximum prediction steps could provide the prediction steps for subsequent prediction models. Secondly, the order stochastic differential equation (FSDE) which describes the fLsm can be obtained. The parameters of the FSDE were estimated by using a novel characteristic function method. The PV power forecasting model with the LRD characteristics was obtained by discretization of FSDE. By comparing statistical performance indicators such as root max error, mean square error with Conv-LSTM, BiLSTM, and GA-LSTM models, the performance of the proposed fLsm model has been demonstrated. The proposed methods can provide better theoretical support for the stable and safe operation of PV grid connection. They have high reference value for grid dispatching department.

Strong convergence of a fractional exponential integrator scheme for finite element discretization of time-fractional SPDE driven by fractional and standard Brownian motions

The aim of this work is to provide the first strong convergence result of a numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order α∈(12,1) and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter H∈(12,1), more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results, and perform the spatial discretization using the standard finite element and the temporal discretization based on a generalized exponential time differencing method (GETD). We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter H. Numerical results are provided to illustrate our theoretical results.

The Moment for Some quotient Stochastic Differential Equation with Application

In this article we use the product stochastic differential equations in order to study the solution for some quotient stochastic differential equation by using it ’s formula, Then we find their moments (mean, variance and the k.th moments). Also we gave some examples to explain the method.

Existence, uniqueness and stability of solutions to fractional backward stochastic differential equations

ABSTRACT Many types of fractional stochastic differential equation (FrSDE), such as Caputo, fractional Brown motion derivatives, and Mittag-Later functions, exist. In recent decades, FrSDE has been a hot topic and can be applied to many fields of research, such as disease transmission, option pricing, and quantitative finance. FrSDEs have various research and applications in financial markets. After comparing many internationally known articles, the fractional order stochastic differential equation proposed in 2016 is most suitable for European option pricing. Over the years, many scholars have studied fractional Brownian motion, fractional ordinary differential equations, and backward differential equations, and no one has studied the application of fractional backward equations in the financial field. Therefore, in this article, to adapt to the financial market more accurately, we construct a backward equation for this kind of FrSDE and construct a new mapping and use the norm method to prove the existence, uniqueness, and stability of the solution to the backward equation. Finally, considering European call options, the Euler Maruyama simulation example of FrSDE is investigated.

Some Results on Backward Stochastic Differential Equations of Fractional Order

In this article, we deal with fractional stochastic differential equations, so-called Caputo type fractional backward stochastic differential equations (Caputo fBSDEs, for short), and study the well-posedness of an adapted solution to Caputo fBSDEs of order \(\alpha \in (\frac{1}{2},1)\) whose coefficients satisfy a Lipschitz condition. A novelty of the article is that we introduce a new weighted norm in the square integrable measurable function space that is useful for proving a fundamental lemma and its well-posedness. For this class of systems, we then show the coincidence between the notion of stochastic Volterra integral equation and the mild solution.

On the behaviour of solutions to a kind of third order neutral stochastic differential equation with delay

This article demonstrates the behaviour of solutions to a kind of nonlinear third order neutral stochastic differential equations. Setting x^{prime }(t)=y(t), y^{prime }(t) =z(t) the third order differential equation is ablated to a system of first order differential equations together with its equivalent quadratic function to derive a suitable downright Lyapunov functional. This functional is utilised to obtain criteria which guarantee stochastic stability of the trivial solution and stochastic boundedness of the nontrivial solutions of the discussed equations. Furthermore, special cases are provided to verify the effectiveness and reliability of our hypotheses. The results of this paper complement the existing decisions on system of nonlinear neutral stochastic differential equations with delay and extend many results on third order neutral and stochastic differential equations with and without delay in the literature.

Obc fractional order stochastic differential equation driven by l'evy noise

In this article, we consider a class of stochastic fractional differential equations (SFDEs) driven by L'evy noise in the sense of a newly defined OBC-fractional derivative. This is a generalized Caputo type fractional derivative introduced recently by Zaid Odibat and Dumitru Baleanu. Under some suitable sufficient conditions, we have employed fixed point theorem to obtain existence and uniqueness results for the considered equation. We have also presented anexample which illustrates the applicability of our obtained results.

An efficient computational scheme to solve a class of fractional stochastic systems with mixed delays

Providing effective numerical methods to approximate the solution of fractional order stochastic differential equations is of great importance, since the exact solution of this type of equations is not available in many cases. In this paper, a stepwise collocation method for solving a system of nonlinear stochastic fractional differential equations (NSFDEs) with mixed delays is presented. First, an approximation of the white noise term is considered and the convergence of the solution of the problem with this approximated white noise term to the solution of the main problem is proved. Then, a combination of a stepwise scheme and a Legendre collocation technique is introduced to solve the stochastic system. In each step, the problem is studied in a subdomain and the proposed method transforms the NSFDE with delays into a system of nonlinear algebraic equations. Moreover, the convergence analysis of the proposed numerical method is described. Two numerical test examples are provided to verify the efficiency of the numerical technique. Finally, a practical epidemic model is examined to show the applicability of this scheme.

The long-time behavior of a stochastic SIR epidemic model with distributed delay and multidimensional Lévy jumps

This paper reports novel theoretical and analytical results for a perturbed version of a SIR model with Gamma-distributed delay. Notably, our epidemic model is represented by Itô–Lévy stochastic differential equations in order to simulate sudden and unexpected external phenomena. By using some new and ameliorated mathematical approaches, we study the long-run characteristics of the perturbed delayed model. Within this scope, we give sufficient conditions for two interesting asymptotic properties: extinction and persistence of the epidemic. One of the most interesting results is that the dynamics of the stochastic model are closely related to the intensities of white noises and Lévy jumps, which can give us a good insight into the evolution of the epidemic in some unexpected situations. Our work complements the results of some previous investigations and provides a new approach to predict and analyze the dynamic behavior of epidemics with distributed delay. For illustrative purposes, numerical examples are presented for checking the theoretical study.

Discrete fractional stochastic Grönwall inequalities arising in the numerical analysis of multi-term fractional order stochastic differential equations

This paper is devoted to the rigorous derivation of some discrete versions of stochastic Grönwall inequalities involving a martingale, which are commonly used in the numerical analysis of multi-term stochastic time-fractional diffusion equations. A Grönwall lemma is also established to deal with the numerical analysis of multi-term stochastic fractional diffusion equations with delay. The proofs of the established inequalities are based on a corresponding deterministic version of the discrete fractional Grönwall lemma in case of smooth solutions and an inequality bounding the supremum in terms of the infimum for discrete time martingales. A numerical application is introduced finally in which the constructed inequalities are handled to derive a priori estimates for a discrete fractional stochastic model.

Application of Stochastic Differential Equation in Insurance Portfolio Construction Involving Leverage Function and Elasticity of Debt and Equity

Leverage effect specifies the functional relationship between stock returns and volatility. As stock price declines, volatility tends to rise. Thus, the variability in market prices of a company's stock has pervasive effect when measuring the level of leverage in the capital structure. The determination of portfolio value of an investor using the continuous time second order stochastic differential equation has major consequence on leverage function. Usually, structural stochastic value of leveraged firms treats company's portfolio as equity whose underlying instrument is the company's asset. In this paper, the objectives are to theoretically: (i) measure the value of insurance company's portfolio by second order stochastic differential equation, (ii) apply Ito's rule to obtain a value on its leverage function and, (iii) obtain the analytical correspondence between equity and volatility in a leveraged company through infinitesimal calculus. The stochastic second order differential equation of portfolio value under two arguments results in equilibrium position which provides the traded price of the derivative, furthermore the linear combination of first order derivative of volatilities with respect to equity and debt is vanishingly zero based on the underlying elasticity of stock volatilities. The resultant effect is that elasticity = of debt and equity cancel out and -15E, SOSE, 51

A note on the continuity for Caputo fractional stochastic differential equations.

The first aim of this paper is to establish the well-posedness for a type of Caputo fractional stochastic differential equations, and we obtain the global existence and uniqueness of solutions under some conditions consistent with the classic (integer order) stochastic differential equations. The second aim is that we consider the continuity of solutions on the fractional order of those equations.

Parametric estimation for a parabolic linear SPDE model based on discrete observations

We consider parametric estimation for a parabolic linear second order stochastic partial differential equation (SPDE) from high frequency data which are observed in time and space. By using thinned data obtained from the high frequency data, adaptive estimators of the coefficient parameters including the volatility parameter are proposed. Moreover, we give some examples and simulation results of the adaptive estimators of the SPDE model based on the high frequency data.

Fractional Lévy stable motion: Finite difference iterative forecasting model

In this study we use the fractional Lévy stable motion (fLsm) to establish a finite iterative forecasting model with Long Range Dependent (LRD) characteristics. The LRD forecasting model considers the influence of current and past trends in stochastic sequences on future trends. We find that the discussed model can accurately forecast the trends of stochastic sequences. This fact enables us to introduce the fLsm as the fractional-order model of Lévy stable motion. Self-similarity and LRD characteristics of the flsm model is introduced by using the relationship between self-similar index and the characteristic index. Thus, the order Stochastic Differential Equation (FSDE) which describes the fLsm can be obtained. The parameters of the FSDE were estimated by using a novel characteristic function method. The forecasting model with the LRD characteristics was obtained by discretization of FSDE. The Monte Carlo method was applied to demonstrate the feasibility of the forecasting model. The power load forecasting history data demonstrates the advantages of our model.

Time Series Irradiance Synthesizer for Optical GEO Satellite Downlinks in 5G Networks

Next generation 5G networks generate a need for broadband, low latency and power efficient backhauling and data-relay services. In this paper, optical satellite communications links, as an integrated component of 5G networks, are studied. More specifically, the Geostationary (GEO) satellite-to-ground optical communication link is investigated. Long-term irradiance statistics based on experimental measurements from the ARTEMIS program are presented and a new time series generator related to the received irradiance/power fluctuations due to atmospheric turbulence is reported. The proposed synthesizer takes into consideration the turbulence-induced scintillation effects that deteriorate the laser beam propagation, on the assumption of the Kolmogorov spectrum. The modeling is based on Rytov theory regarding weak turbulence conditions with the incorporation of first order stochastic differential equations. Summing up, the time series synthesizer is validated in terms of first and second order statistics with experimental results from the European Space Agency‘s ARTEMIS experimental optical downlink and simulated received power statistics for various weather conditions are presented using the proposed validated methodology. Some important conclusions are drawn.

Approximate Controllability of Second-Order Stochastic Differential Systems Driven by a Lévy Process

This paper addresses approximate controllability for a class of control systems represented by second order stochastic differential equations driven by Teugels martingales associated with a Levy process. The main technique is the fundamental solution theory constructed through Laplace transformation. By using the so-called resolvent condition and cosine family of linear operators, stochastic analysis, and the technique of stochastic control theory, a new set of sufficient conditions for the approximate controllability of the considered second order stochastic differential system are formulated and proved. Due to the fundamental solution theory, the nonlinear terms are only required to be partly uniformly bounded. As an illustration of the applications of the obtained results, an example is also provided in the end.