First-Passage Times and Optimal Control of Integrated Jump-Diffusion Processes

We consider a class of nonlinear stochastic systems driven by Wiener and Poisson processes. The Wiener process input enters into the equations additively to the dynamics while the Poisson process input enters into the equations multiplicatively to the control input. Examples of applied problems that may lead to system models of this kind are discussed. The optimal containment control problem is then formulated for these systems. It involves either maximizing the time of stay within an admissible set or a closely related performance measure. The optimal control and the optimal value function are characterized on the basis of Bellman's dynamic programming principle in the general case so that the optimal value function is a solution of a boundary value problem for a PDE. For a special case defined by more restrictive assumptions the method of successive approximations is used to show the existence of a solution to this boundary value problem and to set up an iterative solution procedure. An example is reported that illustrates the results.

The study of the First Passage Time (FPT) problem (also known as first passage problem, FPP) started more than a century ago, but its diverse applications in science and engineering mostly emerged in the last two to three decades. Assuming that X(t) is a one-dimensional stochastic process, the First Passage Time is defined as the time (T) when X(t) first crosses a threshold. Engineering reliability is obviously a suitable application domain, and indeed applications such as optimal dam design in hydrology and analysis of structural failure in civil and mechanical engineering are typical examples. Although we envision that the FPT problem has great potential in network and software reliability, it should be more useful for network security and survivability because the approaches developed for the FPT problem are mostly analytical. The assumption for this inference is that in reliability analysis, experimental or historical data are often more readily available, which makes statistical approaches such as survival analysis more convenient and likely more realistic. In contrast, data is generally more difficult to obtain in security and survivability analyses, and analytical approaches can be leveraged to play more important roles. Furthermore, security and survivability often have to deal with malicious actions that may be driven by sophisticated cognition and behavioral processes, which are highly variable over time and very difficult to detect with short term data. If the behavior of an intruder can be characterized with some stochastic process such as Brownian motion, then the FPT approach may be applied to find the closed-form solution of the probability density function (PDF) of the first passage time, which can be the time when the system breaks down or when the hacker is successful in compromising a network. In addition, the solutions to FPT depend on boundary and initial conditions of the corresponding partial differential equations, and they also describe the evolution of PDF over time. This may suggest that it is possible to model the behavior changes of an intruder over time and circumstances. Another advantage of FPT analysis is that it may help solve some non-Markov stochastic process problems in reliability analysis and survival analysis. In this article, we first briefly introduce the FPT problem with Brownian motion as an example, and then suggest its potential applications in software reliability and network security.

Brownian motion is an important stochastic process used in modelling the random evolution of stock prices. In their 1973 seminal paper—which led to the awarding of the 1997 Nobel prize in Economic Sciences—Fischer Black and Myron Scholes assumed that the random stock price process is described (i.e., generated) by Brownian motion. Despite its relative simplicity, the description of Brownian motion in advanced textbooks sometimes lacks an intuitive basis. The present exposition attempts to provide a simplified construction of standard Brownian motion based on a gambling analogy. This is followed by a description and explicit solution of two stochastic differential equations (known as arithmetic and geometric Brownian motion processes) that are driven by the standard Brownian motion process. The paper also illustrates the use of the Maple computer algebra system to simulate the standard and geometric Brownian motion processes.

This study aimed to execute Monte Carlo simulation method with Wiener Process, Generalized Wiener Process, Mean Reversion Process and Mean Reversion Jump Diffusion Process and to compare them and then expended with the idea of how to include negative and positive news shocks in the gold market to the Monte Carlo simulation. By enhancing the determination of the 3 standard deviation shocks within the process of Classic Mean Jump Diffusion Process, an enchanted model for the 1,96 and 3 standard deviation shocks were being used and additionally positive and negative shocks were added to the system in a different way. This new Mean Reversion Jump Diffusion Process that have been developed by Sümer, executes Monte Carlo simulation regarding the gold market return with five random variables that are chosen from Poisson distribution and one random variable chosen from the normal distribution. Additionally, by accepting volatilities as outlies over the 1,96 and 3 standard deviations with the effect of the new and good news and the standard deviations on the traditional approximate return and the standard deviations (volatility) and the obtained new approximate return and the new standard deviation (volatility) and compares them with the Monte Carlo simulations.

LetY(t) be a one-dimensional diffusion process anddX(t)=Y(t)dt. The process (X(t), Y(t)) is considered in the second quadrant. First, the probability that (X(t), Y(t)) will hit thex-axis before they-axis is computed explicitly whenY(t) is a standard Brownian motion or a particular case of the Bessel process. Next, an exact expression is obtained for the average time it takes (X(t), Y(t)) to exit the regionC={(x, y) ∈ ℝ2 :x < 0,0 <d 1 <y <d 2 whenY(t) is a standard Brownian motion. The solution is expressed as a generalized Fourier series.

Lifetime prognostics for power module bonding wires under non-constant stress conditions: A compound stochastic process degradation modeling approach

Owing to operating condition changing, physical mutation, and sudden shocks, degradation trajectories usually exhibit multi-phase features, and the abrupt jump often appears at the changing time, which makes the traditional methods of lifetime estimation unavailable. In this paper, we mainly focus on how to estimate the lifetime of the multi-phase degradation process with abrupt jumps at the change points under the concept of the first passage time (FPT). Firstly, a multi-phase degradation model with jumps based on the Wiener process is formulated to describe the multi-phase degradation pattern. Then, we attain the lifetime’s closed-form expression for the two-phase model with fixed jump relying on the distribution of the degradation state at the change point. Furthermore, we continue to investigate the lifetime estimation of the degradation process with random effect caused by unit-to-unit variability and the multi-phase degradation process. We extend the results of the two-phase case with fixed parameters to these two cases. For better implementation, a model identification method with off-line and on-line parts based on Expectation Maximization (EM) algorithm and Bayesian rule is proposed. Finally, a numerical case study and a practical example of gyro are provided for illustration.

In this paper, we discuss almost sure asymptotic stability for deterministic systems, which are randomized by adding one-dimensional standard Wiener processes. First, we clarify the difference between the deterministic Lyapunov stability theory and the stochastic stability theory proposed by Hasminskii, and show that local asymptotic stability in probability causes trouble in randomization problems. Second, we describe the Lyapunov stability theory based on almost sure stability, as proposed by Bardi and Cesaroni, and show that the stability is “almost the same” as the deterministic Lyapunov stability. Third, we survey randomization problems briefly, and show that Stratonovich integrals are valid for such problems based on one-dimensional Wiener processes. Finally, we derive the necessary conditions for stochastic Lyapunov functions to ensure almost sure stabilities and discuss the difference between Hasminskii's stabilities and those of Bardi and Cesaroni via linear stochastic systems with numerical simulations.

We consider the stochastic linear quadratic optimal control problem where the state equation is given by a stochastic differential equation of the Ito–Skorokhod type with respect to fractional Brownian motion. The dynamics are driven by strongly continuous semigroups and the cost functional is quadratic. We use the fractional isometry mapping defined between the space of square integrable stochastic processes with respect to fractional Gaussian white noise measure and the space of integrable stochastic processes with respect to the classical Gaussian white noise measure. By this mapping we transform the fractional state equation to a state equation with Brownian motion. Applying the chaos expansion approach, we can solve the optimal control problem with respect to a state equation with the standard Brownian motion. We recover the solution of the original problem by the inverse of the fractional isometry mapping. Finally, we consider a general form of the state equation related to the Gaussian colored noise, we study the control problem, a system with an algebraic constraint and a particular example involving generalized operators from the Malliavin calculus.

Given a Gaussian random walk (or a Wiener process), possibly with drift, observed through noise, we consider the problem of estimating its first-passage time $\tau_{\ell}$ of a given level $\ell$ with a stopping time $\eta$ defined over the noisy observation process. Main results are upper and lower bounds on the minimum mean absolute deviation $\inf_{\eta}{\mathbb{E} }|\eta-\tau_{\ell}|$ which become tight as $\ell\to\infty$. Interestingly, in this regime the estimation error does not get smaller if we allow $\eta$ to be an arbitrary function of the entire observation process, not necessarily a stopping time. In the particular case where there is no drift, we show that it is impossible to track $\tau_{\ell}$: $\inf_{\eta}{\mathbb{E} }|\eta-\tau_{\ell}|^{p}=\infty$ for any $\ell>0$ and $p\geq1/2$.

This paper deals with the two-dimensional stochastic process (X(t), V(t)) where dX(t) = V(t)dt, V(t) = W(t) + ν for some constant ν and W(t) is a one-dimensional Wiener process with zero mean and variance parameter σ2= 1. We are interested in the first-passage time of (X(t), V(t)) to the plane X = 0 for a process starting from (X(0) = −x, V(0) = ν) with x &gt; 0. The partial differential equation for the Laplace transform of the first-passage time density is transformed into a Schrödinger-type equation and, using methods of global analysis, such as the method of dominant balance, an approximation to the first-passage density is obtained. In a series of simulations, the quality of this approximation is checked. Over a wide range of x and ν it is found to perform well, globally in t. Some applications are mentioned.

This article aims to investigate the optimal control of a class of fractional stochastic integrodifferential equations (SIDEs) driven by a Rosenblatt process and Poisson jumps in a Hilbert space. Fractional-order derivatives offer the advantage of more accurately modelling real-world scenarios compared to integer-order derivatives, particularly in problems involving memory or hereditary characteristics. Despite the widespread belief that the Rosenblatt process is insufficient and cannot be substituted by fractional Brownian motion (fBm), we show that the Rosenblatt process does, in fact, converge and can be used to real-world scenarios. We provide a detailed discussion of the Rosenblatt process, including its properties, and an explanation of how it can be approximated by the Wiener process. We study a novel set of sufficient conditions for the existence and uniqueness of mild solutions. Then the optimal control is demonstrated through the application of Balder’s theorem. Finally, we validate the theoretical findings with a practical example and a numerical simulation.

STOCHASTIC INTEGRATION

The nonlinear dynamic characteristics and optimal control of a giant magnetostrictive film (GMF)-shaped memory alloy (SMA) composite plate subjected to in-plane stochastic excitation are studied. GMF is prepared based on an SMA plate, and combined into a GMF—SMA composite plate. The Van der Pol item is improved to explain the hysteretic phenomena of GMF and SMA, and the nonlinear dynamics model of a GMF—SMA composite cantilever plate subjected to in-plane stochastic excitation is developed. The stochastic stability of the system is analyzed, and the steady-state probability density function of the dynamic response of the system is obtained. The condition of stochastic Hopf bifurcation is discussed, the reliability function of the system is provided, and then the probability density of the first-passage time is given. Finally, the stochastic optimal control strategy is proposed by the stochastic dynamic programming method. Numerical simulation shows that the stability of the trivial solution varies with bifurcation parameters, and stochastic Hopf bifurcation appears in the process; the system's reliability is improved through stochastic optimal control, and the first-passage time is delayed. A GMF—SMA composite plate combines the advantages of GMF and SMA, and can reduce vibration through passive control and active control effectively. The results are helpful for the engineering applications of GMF—SMA composite plates.

Stochastic bifurcation characteristics and optimal control of giant magnetostrictive film (GMF)-shape memory alloy (SMA) composite plate subjected to in-plane stochastic excitation were studied in this paper. Nonlinear difference item was introduced to interpret the hysteretic phenomena of both GMF and SMA, and then the nonlinear dynamic model of GMF-SMA composite plate subjected to in-plane stochastic excitation was developed. The stochastic stability of the system was analyzed, and the condition of stochastic Hopf bifurcation was discussed. The reliability function of the system was solved from backward Kolmogorov equation, and then the probability density of the first-passage time was obtained. Finally, the stochastic optimal control strategy was obtained in stochastic dynamic programming method. Numerical simulation shows that the stability of the trivial solution varies with bifurcation parameters, and stochastic Hopf bifurcation appears in the process; the reliability of the system is improved by stochastic optimal control, and the first-passage time is delayed. GMF-SMA composite plate combines the advantages of both GMF and SMA, and can reduce vibration through passive control and active control effectively. The results of this paper are helpful to application of GMF-SMA composite plate in engineering fields.