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This paper investigates the finite-time stability and optimal impulsive control for stochastic age-structured HIV model with time-varying delay. A stochastic noise is introduced by using the Levy process to characterize the phenomenon of discontinuous jumps in virus transmission, which cannot be described by a continuous stochastic process (e.g., Brownian motion). By employing the comparison theorem and the bounded impulsive interval method, we obtain the sufficient conditions of finite-time stability for a stochastic HIV system. The effects of impulse, delay and Levy noise on the finite-time stability are considered in our sufficient conditions. Furthermore, optimal impulsive control is studied to seek the optimal and cost-effective strategy for HIV treatments, which shows that control strategies play an important role in HIV virus transmissions. Numerical simulations are performed to illustrate the validity of our results.

The first-passage time τ S ( x ) of a one-dimensional continuous stochastic process X ( t ) , starting from x ≤ S ( 0 ) , through a smooth boundary S(t) is investigated; in particular, diffusions a...

In the simplest case, we obtain a general solution to a problem of minimizing an integral of a nondecreasing right continuous stochastic process from zero to some nonnegative random variable tau, under the constraints that for some nonnegative random variable T, tau is between zero and T a.s. and the expected value of tau is some alpha. The nondecreasing process and T are allowed to be dependent. In fact a more general setup involving sigma-finite measures, rather than just probability measures is considered and some consequences for families of stochastic processes are given as special cases. Various applications are provided., Comment: 22 pages

Simulation of second-order fluid models requires specific techniques due to the continuous randomness of the considered processes. Things become particularly difficult when considering several dimensions, where correlation occurs, and classical concepts like absorption and reflection require specific extensions. In this work, we will focus on three different types of behaviors, with two correlations structures: either independence or total correlation. For the considered scenario, we will describe how to produce suitable traces of the underlying continuous stochastic process.

International audience; In this paper, we propose a new methodology for solving stochastic inversion problems through computer experiments, the stochasticity being driven by a functional random variables. This study is motivated by an automotive application. In this context, the simulator code takes a double set of simulation inputs: deterministic control variables and functional uncertain variables. This framework is characterized by two features. The first one is the high computational cost of simulations. The second is that the probability distribution of the functional input is only known through a finite set of realizations. In our context, the inversion problem is formulated by considering the expectation over the functional random variable. We aim at solving this problem by evaluating the model on a design, whose adaptive construction combines the so-called stepwise uncertainty reduction methodology with a strategy for an efficient expectation estimation. Two greedy strategies are introduced to sequentially estimate the expectation over the functional uncertain variable by adaptively selecting curves from the initial set of realizations. Both of these strategies consider functional principal component analysis as a dimensionality reduction technique assuming that the realizations of the functional input are independent realizations of the same continuous stochastic process. The first strategy is based on a greedy approach for functional data-driven quantization, while the second one is linked to the notion of space-filling design. Functional PCA is used as an intermediate step. For each point of the design built in the reduced space, we select the corresponding curve from the sample of available curves, thus guaranteeing the robustness of the procedure to dimension reduction. The whole methodology is illustrated and calibrated on an analytical example. It is then applied on the automotive industrial test case where we aim at identifying the set of control parameters leading to meet the pollutant emission standards of a vehicle.

This paper investigates optimal search paths of online shoppers for experience versus search goods, as they engage in continuous sequential search for product information. An optimal stopping rule is designed, based on reservation utilities where the instantaneous utility at each search is modelled as a continuous stochastic process. Furthermore, an empirical model validates the theoretical finding using browsing and purchase data from a Finnish multi-product retailer. The main finding is that, experience goods are associated with three times lower search intensities as compared to search goods. A set of hypotheses investigating varying implications of information search online for the two product groups in question, are tested. A proxy for the agents’ prior information is calculated based on historic search data via novel methodology from the field of information retrieval, such as Text frequency-Inverse document frequency, which exhibits an estimated twelve percent increase in search for search goods, while having no effect on experience goods. Choice probabilities help identify the informativeness of search, which is shown to be inversely proportional to the intensity of search. Finally, the role of personalised recommendations is studied in the context of online search and choice, which has completely opposing effects on the two product types.

In the classical stochastic resetting problem, a particle, moving according to some stochastic dynamics, undergoes random interruptions that bring it to a selected domain, and then, the process recommences. Hitherto, the resetting mechanism has been introduced as a symmetric reset about the preferred location. However, in nature, there are several instances where a system can only reset from certain directions, e.g., catastrophic events. Motivated by this, we consider a continuous stochastic process on the positive real line. The process is interrupted at random times occurring at a constant rate, and then, the former relocates to a value only if the current one exceeds a threshold; otherwise, it follows the trajectory defined by the underlying process without resetting. We present a general framework to obtain the exact non-equilibrium steady state of the system and the mean first passage time for the system to reach the origin. Employing this framework, we obtain the explicit solutions for two different model systems. Some of the classical results found in symmetric resetting such as the existence of an optimal resetting, are strongly modified. Finally, numerical simulations have been performed to verify the analytical findings, showing an excellent agreement., Comment: 10 pages including: main text with 6 figures and appendices

A methodology to model two types of random perturbation that affect the operation of electric power systems (EPS) are presented. The first uncertainty is wind power generation and is represented by a one-dimensional and by a multidimensional continuous stochastic process. The second one is power demand, and is modeled by using an hybrid structure based on harmonic regression and the Ornstein–Uhlenbeck (O–U) process. The stochastic models are applied to a real Chilean case, using real data for parametric estimation and validation models.

The power output of solar photovoltaic (PV) systems is affected by solar radiation and ambient temperature. The commonly used evaluation techniques usually overlook the four weather states which are clear, cloudy, foggy, and rainy. In this paper, an ovel analytical model of the four weather conditions based on the Markov chain is proposed. The Markov method is well suited to estimate the reliability and availability of systems based on a continuous stochastic process. The proposed method is generic enough to be applied to reliability evaluation of PV systems and even other applications. Further aspects investigated include the new degradation model for reliability predication of PV modules. The results indicate that the PV module degradation over years, failures, and solar radiation must be considered in choosing an efficient PV system with an optimal design to achieve the maximum benefit of the PV system. For each aspect, a method is proposed, and the complete focusing methodology is expounded and validated using simulated point targets. The results also demonstrate the feasibility and applicability of the proposed method for effective modeling of the chronological aspects and stochastic characteristics of solar cells as well as the optimal configuration and sizing of large PV plants in terms of cost and reliability.

We prove the existence and uniqueness of the Lp-variational solution, with p > 1, of the following multivalued backward stochastic differential equation with p-integrable data: {−dYt + ∂yΨ(t,Yt)dQt∋H(t,Yt,Zt)dQt−ZtdBt,0≤tYτ = η, where τ is a stopping time, Q is a progressively measurable increasing continuous stochastic process and ∂yΨ is the subdifferential of the convex lower semicontinuous function y↦Ψ(t, y). In the framework of [14] (the case p ≥ 2), the strong solution found it there is the unique variational solution, via the uniqueness property proved in the present article.

In this chapter we define notions of stochastic continuity and differentiability and describe Lindeberg’s condition for continuity of stochastic Markovian trajectories. We also show that the Fokker–Planck equation describes a continuous stochastic process. Finally, we derive the stationary solutions of the Fokker–Planck equation and define potential function of dynamics.

A linear-quadratic optimal control problem with an infinite time horizon for a scalar linear stochastic differential equation with additive Rosenblatt noise is formulated and solved. The Rosenblatt process is a non-Gaussian continuous stochastic process which exhibits self-similarity and long-range dependence. The feedback form of the optimal control and the optimal cost are given explicitly. The main tool used to find the optimal control is an Ito-type formula for a Rosenblatt process with drift.

International audience; The FitzHugh-Nagumo is a well-known neuronal model that describes the generation of spikes at the intracellular level. We study a stochastic version of the model from a probabilistic point of view. The hypoellipticity is proved, as well as the existence and uniqueness of the stationary distribution. The bi-dimensional stochastic process is $\beta$-mixing. The stationary density can be estimated with an adaptive non-parametric estimator. Then, we focus on the distribution of the length between successive spikes. Spikes are difficult to define directly from the continuous stochastic process. We study the distribution of the number of up-crossings. We link it to the stationary distribution and propose an estimator of its expectation. We finally prove mathematically that the mean length of inter-up-crossings interval in the FitzHugh-Nagumo model is equal to its up-crossings rate. We illustrate the proposed estimators on a simulation study. Different regimes are explored, with no, few or high generation of spikes.

Electric power systems have been changing from the conventional and traditional electric units to the efficient, economical, less-polluting and reliable ones. Building cooling, heating and power (BCHP) systems can yield these goals and save energy as well as improve the reliability of the system. However, for significant integration and the use of large amount of BCHP generation in electric power systems, some approaches should be followed in order to study the reliability of the BCHP systems. In this study, we focused on the reliability aspects of power systems incorporating BCHP systems in the local distribution systems. The Markov method based on the state-space analysis is used to investigate the impacts of implementation of BCHP systems on the power systems’ reliability. The Markov method is well suited to analyze the reliability of systems based on a continuous stochastic process. The Roy Billinton test system and the IEEE reliability test system are used to illustrate the results. Case studies show the effects of BCHP systems on general adequacy of electric power systems. Various case results demonstrate the efficiency and effectiveness of the proposed method.

Using properties of the cdf of a random variable defined as a saddle-type point of a real valued continuous stochastic process, we derive first-order asymptotic properties of tests for stochastic spanning w.r.t. a stochastic dominance relation. First, we define the concept of Markowitz stochastic dominance spanning, and develop an analytical representation of the spanning property. Second, we construct a non-parametric test for spanning via the use of an empirical analogy. The method determines whether introducing new securities or relaxing investment constraints improves the investment opportunity set of investors driven by Markowitz stochastic dominance. In an application to standard data sets of historical stock market returns, we reject market portfolio Markowitz efficiency as well as two-fund separation. Hence there exists evidence that equity management through base assets can outperform the market, for investors with Markowitz type preferences.

This paper introduces test and estimation procedures for abrupt and gradual changes in the entire jump behaviour of a discretely observed Ito semimartingale. In contrast to existing work we analyse jumps of arbitrary size which are not restricted to a minimum height. Our methods are based on weak convergence of a truncated sequential empirical distribution function of the jump characteristic of the underlying Ito semimartingale. Critical values for the new tests are obtained by a multiplier bootstrap approach and we investigate the performance of the tests also under local alternatives. An extensive simulation study shows the finite-sample properties of the new procedures., Comment: 98 pages, 13 figures

We present a new formalism for option pricing that does not require an assumption on the stochastic process of the underlying asset price and yet produces remarkably accurate results versus the market. The new formalism applies for general Markovian stochastic behavior including continuous and discontinuous (jump) processes and in its broadest scheme contains all known models for Markovian option pricing and some new ones. The method is based on obtaining the risk neutral density function that satisfies a consistency condition, guaranteeing no arbitrage. For example, we show that when the underlying asset undergoes a continuous stochastic process with deterministic time dependent standard deviation the formalism produces the Black-Scholes-Merton formula without using a Wiener process. We show that in the general case the price of European options depends only on all the moments of the price return of the underlying asset. We offer a method to calculate the prices of European options when the volatility smile at maturity is independent of the term structure prior to the maturity, as observed in options markets. In the continuous case where only moments up to second order contribute to the price then any set of three option prices with the same maturity contains the information to determine the whole volatility smile for this maturity. In all the many examples we examined our method generates option prices that match the option markets prices very accurately in all asset classes. This confirms that the options market exhibits no-arbitrage. Moreover, using bootstrapping we demonstrate how to determine the conditional density function from inception to maturity, thus allowing the calculation of path dependent options. The new formalism also allows for the replication of ‘W-shape’ volatility smile that infrequently appears in some equity markets.

A mathematical model for a non-linear continuous stochastic process is presented. Analysis of the stability by uncoupling system is realized.

A fundamental issue in the theory of continuous stochastic process is the interpretation of multiplicative white noise, which is often referred to as the It\^o-Stratonovich dilemma. From a physical perspective, this reflects the need to introduce additional constraints in order to specify the nature of the noise, whereas from a mathematical perspective it reflects an ambiguity in the formulation of stochastic differential equations (SDEs). Recently, we have identified a mechanism for obtaining an It\^o SDE based on a form of temporal disorder. Motivated by switching processes in molecular biology, we considered a Brownian particle that randomly switches between two distinct conformational states with different diffusivities. In each state, the particle undergoes normal diffusion (additive noise) so there is no ambiguity in the interpretation of the noise. However, if the switching rates depend on position, then in the fast switching limit one obtains Brownian motion with a space-dependent diffusivity of the It\^o form. In this paper, we extend our theory to include colored additive noise. We show that the nature of the effective multiplicative noise process obtained by taking both the white-noise limit $(\ensuremath{\kappa}\ensuremath{\rightarrow}0)$ and fast switching limit $(\ensuremath{\epsilon}\ensuremath{\rightarrow}0)$ depends on the order the two limits are taken. If the white-noise limit is taken first, then we obtain It\^o, and if the fast switching limit is taken first, then we obtain Stratonovich. Moreover, the form of the effective diffusion coefficient differs in the two cases. The latter result holds even in the case of space-independent transition rates, where one obtains additive noise processes with different diffusion coefficients. Finally, we show that yet another form of multiplicative noise is obtained in the simultaneous limit $\ensuremath{\epsilon},\ensuremath{\kappa}\ensuremath{\rightarrow}0$ with $\ensuremath{\epsilon}/{\ensuremath{\kappa}}^{2}$ fixed.

Summary By viewing animal movement paths as realizations of a continuous stochastic process, we introduce a rigorous likelihood method for estimating the statistical parameters of movement processes. This method makes no assumption of a hidden Markov property, places no special emphasis on the sampling rate, is insensitive to irregular sampling and data gaps, can produce reasonable estimates with limited sample sizes and can be used to assign AIC values to a vast array of qualitatively different models of animal movement at the individual and population levels. To develop our approach, we consider the likelihood of the first two cumulants of stochastic processes, the mean and autocorrelation functions. Together, these measures provide a considerable degree of information regarding searching, foraging, migration and other aspects of animal movement. As a specific example, we develop the likelihood analyses necessary to contrast performance of animal movement models based on Brownian motion, the Ornstein–Uhlenbeck process and a generalization of the Ornstein–Uhlenbeck process that includes ballistic bouts. We then show how our framework also provides a new and more accurate approach to home-range estimation when compared to estimators that neglect autocorrelation in the movement path. We apply our methods to a data set on Mongolian gazelles (Procapra gutturosa) to identify the movement behaviours and their associated time and length scales that characterize the movement of each individual. Additionally, we show that gazelle annual ranges are vastly larger than those of other non-migratory ungulates.

This paper shows that the mobilized shear strength of a two-dimensional (2-D) spatially variable saturated undrained clay is closely related to the extreme value of a one-dimensional (1-D) continuous stationary stochastic process. This 1-D stochastic process is the integration of the 2-D spatially variable shear strength along potential slip curves. Based on this finding, a probability distribution model for the mobilized shear strength of the 2-D clay is developed based on a probability distribution model for the extreme value of the 1-D stochastic process. The latter (the model for the 1-D extreme value) has analytical expressions. With the proposed probability distribution model, the mobilized shear strength of a 2-D clay can be simulated without the costly random field finite element analyses.

This paper discusses the valuation of the Credit Default Swap based on a jump market, in which the asset price of a firm follows a double exponential jump diffusion process, the value of the debt is driven by a geometric Brownian motion, and the default barrier follows a continuous stochastic process. Using the Gaver-Stehfest algorithm and the non-arbitrage asset pricing theory, we give the default probability of the first passage time, and more, derive the price of the Credit Default Swap.

In this short note we will provide a sufficient and necessary condition to have uniqueness of the location of the maximum of a stochastic process over an interval. The result will also express the mean value of the location in terms of the derivative of the expectation of the maximum of a linear perturbation of the underlying process. As an application, we will consider a Brownian motion with variable drift. The ideas behind the method of proof will also be useful to study the location of the maximum, over the real line, of a two-sided Brownian motion minus a parabola and of a stationary process minus a parabola., Comment: More general results, including a proof of Groeneboom-Janson formula for the variance of the location of maximum of a Brownian motion minus a parabola, and a proof of the uniqueness of the location of the maximum of an Airy process minus a parabola. Final version, accepted for publication in the Journal of Applied Probability

Market participants that have a task to acquire a certain position in a listed security at a predetermined price on behalf of a third party with no time urgency, i.e. to fill a perpetual limit order, can optimize the profitability of their trading strategy in order to accomplish this task. We study the statistical properties of the profit distribution of a particular market-making strategy: one which increments the inventory as the underlying price approaches the limit order price S0 and locks in profits by gradually liquidating the inventory as the market drifts away from S0. We do so by adopting a simple model of market microstructure in which an unobservable continuous stochastic process, the microprice, drives the dynamics of limit and market orders. In this model, the arrival of market orders and updates of the limit order book are determined by the microprice crossing a discrete set of n equidistant levels between the price ticks. Assuming normal dynamics for the microprice and adopting a standard mean-variance framework, we are able to derive a closed-form solution for the optimal inventory profile which is remarkably simple: the cumulative amount held when the market price is Si is inversely proportional to Si-S0, the distance in price terms from the limit order price. Finally, we show that n represents a sort of micro-volatility of the market that is entirely independent of the diffusive volatility of the microprice and is a measure of the intensity of the bid-ask bounce.

A fast and computationally efficient valuation tool assists governments involved in Public–Private Partnership (P3) projects to examine many contractual configurations and design a guarantee that minimizes cost and reasonably mitigates the risk. This paper presents a continuous stochastic process derived from the risk factor forecast, thereby providing a more realistic and flexible model. A new valuation approach is developed by using a finite-difference method based on this continuous stochastic process. In a numerical example with one risk factor, it is shown that this new valuation tool is 100 times faster than the existing simulation-based approach. Its superior speed presents the opportunity to examine different contractual configurations, and as a result, design a more cost effective guarantee contract. Exercise strategies are derived for a multiple-exercise (Australian) guarantees structure. This new approach can be used by a government to reserve budget for the guarantees. Finally, the con...

In structural reliability the dependence structure between random variables is almost exclusively modeled by Gauss (normal or Gaussian) copula; however, this implicit assumption is typically not corroborated. This paper is focusing on time-variant reliability problems with continuous stochastic processes, which are collection of dependent random variables and to our knowledge are not modeled by other than Gauss copula in structural reliability. Therefore, the aim of this contribution is to qualitatively and quantitatively analyze the impact of this copula assumption on failure probability. Three illustrative examples are studied considering bivariate Gauss, t, rotated Clayton, Gumbel, and rotated Gumbel copulas. Time-variant actions are modeled as stationary, ergodic, continuous stochastic processes, and the PHI2 method is adopted for the analyses. The calculations show that the copula function has significant effect on failure probability. In the studied examples, application of Gauss copula can four times underestimate or even 10 times overestimate failure probabilities obtained by other copulas. For normal structures agreement on copula type is recommended, while for safety critical ones inference of copula type from observations is advocated. If data are scare, multiple copula functions and model averaging could be used to explore this uncertainty.

In structural reliability the dependence structure between random variables is almost exclusively modeled by Gauss (normal or Gaussian) copula; however, this implicit assumption is typically not corroborated. This paper is focusing on time-variant reliability problems with continuous stochastic processes, which are collection of dependent random variables and to our knowledge are not modeled by other than Gauss copula in structural reliability. Therefore, the aim of this contribution is to qualitatively and quantitatively analyze the impact of this copula assumption on failure probability. Three illustrative examples are studied considering bivariate Gauss, t, rotated Clayton, Gumbel, and rotated Gumbel copulas. Time-variant actions are modeled as stationary, ergodic, continuous stochastic processes, and the PHI2 method is adopted for the analyses. The calculations show that the copula function has significant effect on failure probability. In the studied examples, application of Gauss copula can four times underestimate or even 10 times overestimate failure probabilities obtained by other copulas. For normal structures agreement on copula type is recommended, while for safety critical ones inference of copula type from observations is advocated. If data are scare, multiple copula functions and model averaging could be used to explore this uncertainty.

Let X={X(s)}"s"@?"S be an almost sure continuous stochastic process (S compact subset of R^d) in the domain of attraction of some max-stable process, with index function constant over S. We study the tail distribution of @!"SX(s)ds, which turns out to be of Generalized Pareto type with an extra 'spatial' parameter (the areal coefficient from Coles and Tawn (1996) [3]). Moreover, we discuss how to estimate the tail probability P(@!"SX(s)ds>x) for some high value x, based on independent and identically distributed copies of X. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. X represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall. The paper has two main purposes: first to formalize and justify the results of Coles and Tawn (1996) [3]; further we treat the problem in a non-parametric way as opposed to their fully parametric methods.

Given a continuous stochastic process (X t ) t∈[0,T], this article provides, in the first part, a stochastic process that is the best mean square approximation of the form , with W t Brownian motion. The function coefficients a(t) and b(t) depend on the process X t and are calculated in the case of several classical examples. In the second part, we extend the method for mean square approximations of the form . We also present simulations for each example, and show that replacing by the martingale is a more natural framework for the problem.

In this paper we consider the problem of a firm that faces a stochastic (Poisson) demand and must replenish from a market in which prices fluctuate, such as a commodity market. We describe the price evolution as a continuous stochastic process and we focus on commonly used processes suggested by the financial literature, such as the geometric Brownian motion and the Ornstein-Uhlenbeck process. It is well known that under variable purchase price, a price-dependent base-stock policy is optimal. Using the single-unit decomposition approach, we explicitly characterize the optimal base-stock level using a series of threshold prices. We show that the base-stock level is first increasing and then decreasing in the current purchase price. We provide a procedure for calculating the thresholds, which yields closed-form solutions when price follows a geometric Brownian motion and implicit solutions under the Ornstein-Uhlenbeck price model. In addition, our numerical study shows that the optimal policy performs much better than inventory policies that ignore future price evolution, because it tends to place larger orders when prices are expected to increase.

This article analyses the space–time interdependency of marked point processes considering marked point random and deterministic motions. In this context, marked points can move randomly or/and deterministically as a result of their own growing status or/and the growing status of their neighbours. A continuous stochastic process for the generation of space–time patterns of moving marked points is formulated and illustrated. Three deterministic models of collision of moving and growing hard particles are also developed. We consider deterministic particle motion promoted by: (a) particle's own growth when touching other neighbours; (b) the growth of touching neighbours; and (c) a combination of these two collision dynamics. Moreover, we extend this combined deterministic model into its counterpart stochastic approach. Our analysis suggests that the generating mechanism of the space–time stochastic process may not become fully reflected in snap-shot analyses of purely spatial patterns when assuming random mo...

This paper aims to introduce jump tests to the actuarial community. In actuarial science, semimartingales are extensively used in the models for interest rates, options, variable annuities and equity-linked annuities. Those models usually assume without justification that the underlying asset process follows a continuous stochastic process such as a geometric Brownian motion, for the market data sometimes tell a different story. Choosing between a continuous model and a model with jumps is not only important for pricing of insurance products but also crucial for implementing other post-sales risk management measures such as dynamic liability hedging. A test for jumps allows actuaries to rigorously test whether the underlying asset process has jumps, which is the first critical step in model selection. The ability to conduct the test should thus belong to the repertoire of every expert and practitioner working in this field. In this paper, we review several major tests for jumps, describe their advantages and disadvantages, and offer suggestions for their implementation. We also implement several tests using real data, enabling practitioners to apply these tests in their work.Keywords: Asset price; Black–Scholes; equity-linked annuity; variable annuity

Fractional Brownian motion (fBm) is a nonstationary self-similar continuous stochastic process used to model many natural phenomena. A realization of the fBm can be numerically approximated by discrete paths which do not entirely preserve the self-similarity. We investigate the self-similarity at different time scales by decomposing the discrete paths of fBm into intrinsic components. The decomposition is realized by an automatic numerical algorithm based on successive smoothings stopped when the maximum monotonic variation of the averaged time series is reached. The spectral properties of the intrinsic components are analyzed through the monotony spectrum defined as the graph of the amplitudes of the monotonic segments with respect to their lengths (characteristic times). We show that, at intermediate time scales, the mean amplitude of the intrinsic components of discrete fBms scales with the mean characteristic time as a power law identical to that of the corresponding continuous fBm. As an application we consider hydrological time series of the transverse component of the transport process generated as a superposition of diffusive movements on advective transport in random velocity fields. We found that the transverse component has a rich structure of scales, which is not revealed by the analysis of the global variance, and that its intrinsic components may be self-similar only in particular cases.

This paper presents a software reliability growth model based on Ito type Stochastic Differential Equations (SDE). As the size of a software system becomes larger, the number of faults remaining in the system during the testing phase can be considered to be a continuous stochastic process. In practice, if the per-fault detection rate is subject to certain random effects, we may consider the use of a SDE to describe the average behavior of the software fault detection process during the testing phase. As a result, we derive several software reliability measures by utilizing the mean value function which is the expected value of the SDE. We also derive the maximum likelihood estimators of the unknown parameters for the model. Futhermore, we compare our model with other software reliability growth models in terms of several reliability measures and goodness-of-fit for the same data set.

Chapter 6 extends coverage of stochastic processes needed to develop the financial models, including those that derive from stochastic differential equations. The differential of a stochastic process and stochastic integration is defined. Relevant properties and essential theorems concerning these two operations are covered. The Radon–Nikodym derivative is covered in detail, with applications to both discrete binomial probability measures and continuous normal densities. The chapter presents essential tools for pricing securities following continuous stochastic process, including the Cameron–Martin–Girsanov theorem, the martingale representation theorem, and Ito’s lemma. Using Ito’s lemma, the solution to the primary stochastic differential equation for modeling securities (geometric Brownian motion) is derived.

We generalize the description of the dynamics of the order book of financial markets in terms of a Brownian particle embedded in a fluid of incoming, exiting, and annihilating particles by presenting a model of the velocity on each side (buy and sell) independently. The improved model builds on the time-averaged number of particles in the inner layer and its change per unit time, where the inner layer is revealed by the correlations between price velocity and change in the number of particles (limit orders). This allows us to introduce the Knudsen number of the financial Brownian particle motion and its asymmetric version (on the buy and sell sides). Not being considered previously, the asymmetric Knudsen numbers are crucial in finance in order to detect asymmetric price changes. The Knudsen numbers allows us to characterize the conditions for the market dynamics to be correctly described by a continuous stochastic process. Not questioned until now for large liquid markets such as the USD-JPY and EUR-USD exchange rates, we show that there are regimes when the Knudsen numbers are so high that discrete particle effects dominate, such as during market stresses and crashes. We document the presence of imbalances of particles depletion rates on the buy and sell sides that are associated with high Knudsen numbers and violent directional price changes. This indicator can detect the direction of the price motion at the early stage while the usual volatility risk measure is blind to the price direction.

We derive the continuity properties of the CDF of a random variable defined as a saddle-type point of a real valued continuous stochastic process on a compact metric space. This result facilitates the derivation of first order asymptotic properties of tests for stochastic spanning w.r.t. some stochastic dominance relation based on subsampling. As an illustration we define the concept of Markowitz stochastic spanning, derive an analytic representation upon the empirical analog of which we construct a relevant statistical test. The aforementioned result enables derivation of asymptotic exactness for the relevant procedure based on subsampling, when the metric space has the form of a simplicial complex, the spanning set is a compact subset and the significance level is chosen according to the number of extreme points of the complex inside the spanning set. Consistency is also derived. Such tests are of interest in financial economics since they can provide reductions of portfolio sets.

In this paper, we discuss an initial-boundary value problem associated with the Euler equations with a random noise in a simply connected two-dimensional bounded domain. We present two different results according to the space regularity of the random noise. When the random noise is regular in the space variables, we prove the existence of a unique solution as a Banach space-valued continuous stochastic process. If the noise is less regular in the space variables, we establish the existence of solutions defined over a given probability space under the assumption that the noise is given in terms of a standard Brownian motion.

Inference methods are proposed for the bivariate mean function of a continuous stochastic process with a two-dimensional domain. Nonparametric bivariate estimation is facilitated by thresholded projection estimators. Estimators adapt to the sparsity of the bivariate function. Oracle inequality results are developed to describe the adaptive inference methods. The construction of nonparametric bivariate confidence bands is presented. Implementation results show the applicability of the methods in practice.

A wide variety of priors have been proposed for nonparametric Bayesian estimation of conditional distributions, and there is a clear need for theorems providing conditions on the prior for large support, as well as posterior consistency. Estimation of an uncountable collection of conditional distributions across different regions of the predictor space is a challenging problem, which differs in some important ways from density and mean regression estimation problems. Defining various topologies on the space of conditional distributions, we provide sufficient conditions for posterior consistency focusing on a broad class of priors formulated as predictor-dependent mixtures of Gaussian kernels. This theory is illustrated by showing that the conditions are satisfied for a class of generalized stick-breaking process mixtures in which the stick-breaking lengths are monotone, differentiable functions of a continuous stochastic process. We also provide a set of sufficient conditions for the case where stick-breaking lengths are predictor independent, such as those arising from a fixed Dirichlet process prior.

We introduce a domain-theoretic framework for continuous-time, continuous-state stochastic processes. The laws of stochastic processes are embedded into the space of maximal elements of the normalised probabilistic power domain on the space of continuous interval-valued functions endowed with the relative Scott topology. We use the resulting ω -continuous bounded complete dcpo to obtain partially defined stochas- tic processes and characterise their computability. For a given continuous stochastic process, we show how its domain-theoretic, i.e., finitary, approximations can be con- structed, whose least upper bound is the law of the stochastic process. As a main result, we apply our methodology to Brownian motion. We construct a partially de- fined Wiener measure and show that the Wiener measure is computable within the domain-theoretic framework.