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Method of Iterated Integrals for Solution of Stochastic Integrals with Applications to Monte Carlo Simulation of Stochastic Differential Equations.

The theory of stochasic integrals and stochastic differential equations was established by K Ito [3, 4] (also see [2]). In past four decade years, Ito’s stochastic analysis has established for itself the central role in modern probability theory. Ito’s theory of stochastic differential equations has been one of the most important tools. However, Ito’s construction of stochastic integrals over Brownian motion possesses an essentially random characterization, and is meaningless for a single Brownian path. The Ito map obtained by solving Ito’s stochastic differential equations is nowhere continuous on the Wiener space.

Explicit Euler method, implicit Euler method, trapezoidal rule and midpoint rule are wildly-used and well-known numerical methods for ordinary differential equations. Methods with two parameters are constructed in this paper, whose error analysis is also given by Taylor expansion. The new constructed methods include and extend the 4 well-known methods, and the error analysis with parameters brings us convenience for the error analysis of numerical methods. At last, we give three original methods, whose orders can be gotten directly by the error analysis with parameters.

Variable-stepsize Runge–Kutta methods for stochastic Schrödinger equations

Stochastic differential equations (SDEs), which models uncertain phenomena as the time evolution of random variables, are exploited in various fields of natural and social sciences such as finance. Since SDEs rarely admit analytical solutions and must usually be solved numerically with huge classical-computational resources in practical applications, there is strong motivation to use quantum computation to accelerate the calculation. Here, we propose a quantum-classical hybrid algorithm that solves SDEs based on variational quantum simulation (VQS). We first approximate the target SDE by a trinomial tree structure with discretization and then formulate it as the time-evolution of a quantum state embedding the probability distributions of the SDE variables. We embed the probability distribution directly in the amplitudes of the quantum state while the previous studies did the square-root of the probability distribution in the amplitudes. Our embedding enables us to construct simple quantum circuits that simulate the time-evolution of the state for general SDEs. We also develop a scheme to compute the expectation values of the SDE variables and discuss whether our scheme can achieve quantum speed-up for the expectation-value evaluations of the SDE variables. Finally, we numerically validate our algorithm by simulating several types of stochastic processes. Our proposal provides a new direction for simulating SDEs on quantum computers.

We consider an enzyme-substrate reaction-diffusion problem. Unsteady and steady state models are recalled. For the unsteady state case, the model is in the form of a second order partial differential equation. We solve the unsteady state model using the explicit numerical finite difference method, which is forward difference in time and centered difference in space. For the general steady state case, the model is in the form of a second order ordinary differential equation. We solve the general steady state model using the explicit first order Euler's numerical method. For the particular steady state case of the unsaturated catalytic kinetics, we derive the exact analytical solution using the characteristic method of ordinary differential equations. For the particular steady state case of the saturated catalytic kinetics, we derive the exact analytical solution using the direct-integration method. The obtained exact analytical solutions are identical with the existing exact analytical solutions derived using the variational iteration method. With the aid of computer, the enzyme-substrate reaction-diffusion problem can be solved and simulated successfully for both unsteady and steady state cases.

Several engineering applications require numerical schemes for the simulation of stochastic differential equations (SDEs) on the Stiefel manifold of n × p real matrices with orthonormal columns (n ≥ p). The geometry of the Stiefel manifold causes classical schemes to fail, as they leave the manifold after one time step. Motivated by deterministic schemes on the Stiefel manifold, we develop a scheme for SDEs which always remains on the manifold. Our scheme can be easily applied to any SDE on the Stiefel manifold. The derivation is direct and made accessible to an engineering audience. Simulations with n = 1000, p = 10 are provided.

The purpose of this chapter is to overview elements of the theory of stochastic differential equations, based on Wiener processes, for use in the subsequent chapters. This theory was founded by K. Itô in 1942 (Itô, Zenkoku Shijo Sugaku Danwakai 244:1352–1400, 1942; Itô, On stochastic differential equations. Memoirs of the American Mathematical Society, vol 4. AMS, New York City, 1951). His aim was to construct Markov processes, governed by Kolmogorov’s differential equations via Wiener processes, and to analyze their sample paths. After that, stochastic differential equations have been used to describe dynamical processes in random environments of various fields. Here we consider stochastic differential equations with random coefficients, because we aim at studying stochastic control problems.The chapter is organized as follows. Section 1.1 is preliminaries. The basic definitions and results on stochastic processes are collected for later use. Stochastic differential equations and stochastic analysis will be introduced in Sect. 1.2. Section 1.3 deals with asset pricing problems as an application of previous results.KeywordsBackward Stochastic Differential EquationsAsset Pricing ProblemsConstructing Markov ProcessesRandom CoefficientsBurkholder Davis Gundy InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper uses stochastic differential equation and risk theory to study the probability model of insurance company's bankruptcy function and the proportion model of financial insurance investment. It also studies the application of stochastic probability to the investment decision-making of institutional investors. Taking the four major insurance industries in China as an example, it gives the strategy of venture capital.

Continuous numerical methods for ODEs with defect control

Numerical Methods for Ordinary and Stochastic Differential Equations

This paper presents the cyber-physical system (CPS) of a numerical method (the widely-used Euler- Maruyama method) and establishes a foundational theory of the CPSs of numerical methods for stochastic differential equations (SDEs), which transforms the way we understand the relationship between the numerical method and the underlying dynamical system. Unlike in the literature where they are treated as separate systems linked by inequalities, the CPS is a seamless integration of the SDE and the numerical method and we construct a new and general class of stochastic impulsive differential equations (SiDEs) that can serve as a canonic form of the CPSs of numerical methods. By the CPS approach, we show the equivalence and intrinsic relationship between the stability of the SDE and the stability of the numerical method using the Lyapunov stability theory we develop for our class of SiDEs. Applying our established theory, we present the CPS Lyapunov inequality that is the necessary and sufficient condition for meansquare stability of the CPS of the Euler-Maruyama method for linear SDEs. The proposed CPS and theory initiate the study of systems numerics and provoke many open and interesting problems for future work.

This paper presents a depth control simulation of autonomous underwater vehicle (AUV) using a decentralized system approach called the Open Control Platform (OCP). We briefly discuss various delay time phenomena from the actuator model, network latency, and computational time processing, which are estimated using the estimated delay time reflected in the system design infrastructure. The proposed structure consists of the optimal regulator equation with the delay results obtained by bringing the original problem to the quadratic servo design (LQ) and the delay compensation method, which modifies input delays (generated by human operators) and outputs. The numerical method for delayed differential equations (DDE) is a direct analogy from the classical numerical method of ordinary differential equations (ODE), and will be compared with the proposed method. Simulation results are presented, showing the performance of traditional methods and new methods based on OCP.

Most neutral delay differential equations (NDDEs) that appear in real life phenomena cannot be solved analytically. Therefore, scientists and engineers resort for numerical solutions that can be made as accurately as possible. Numerical methods for NDDEs have been of great interests, since the existing methods for ordinary differential equations cannot directly solve the delay terms in NDDEs. Most numerical methods have been developed to approximate only one new value in a single integration step and using a fixed stepsize. Variable stepsize algorithm requires extra computation cost since the coefficients of the method need to be recalculated whenever the stepsize changes. Hence, this study aims to develop an implicit block method for solving systems of first order NDDEs using variable stepsize. Formulae of three-point one-block method using variable stepsize are derived. The performance of the three-point one-block method is compared with that of the two-point one-block method. The maximum and average errors of the two methods are comparable and the three-point one-block method requires less number of total steps taken. The numerical results indicates that the three-point one-block method can approximate the solution for NDDEs as accurate and efficient as possible when compared with two-point one-block method.

A Study on Uncertainty of Discharge in River Channel Using Stochastic Partial Differential Equation