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In this paper, we considered the problem of optimally controlling a twodimensional dynamical system until it reaches either of two boundaries. We consider a controlled dynamical system (X(t), Y(t)) which is a generalization of the classic twodimensional Kermack-McKendrick model for the spread of epidemics. Moreover, the system is subject to random jumps of fixed size according to a Poisson process. The system is controlled until the sum X(t) + Y(t) equal to either 0 or d (> 0) for the first time. Particular problems are solved explicitly.

Filtered Poisson processes are used as models in various applications, in particular in statistical hydrology. In this paper, controlled filtered Poisson processes are considered. The aim is to minimize the expected time that the process will spend in the continuation region. The dynamic programming equation satisfied by the value function is derived. To obtain the value function, and hence the optimal control, a non-linear integro-differential equation must be solved, subject to the appropriate boundary conditions. Various cases for the size of the jumps are treated and explicit results are obtained. [ABSTRACT FROM AUTHOR]

The state of a machine is modeled as a controlled continuous-time Markov chain. Moreover, the machine is being serviced at random times. The aim is to maximize the time until the machine must be repaired, while taking the maintenance costs into account. The dynamic programming equation satisfied by the value function is derived, enabling optimal decision-making regarding inspection rates, and special problems are solved explicitly. This approach minimizes direct maintenance costs along with potential failure expenses, establishing a robust methodology for determining inspection frequencies in reliability-centered maintenance. The results contribute to the advancement of maintenance strategies and provide explicit solutions for particular cases, offering ideas for application in reliability engineering. [ABSTRACT FROM AUTHOR]

Suppose that in an M / G / 1 waiting queue, the server can choose between two service time distributions. We look for the choice that enables us to minimize the expected value of a cost criterion that takes into account the cost incurred by working faster and the time needed to empty the waiting line. The random final time is the first time there is no customer waiting for service. When the service times are exponential random variables, we can appeal to dynamic programming to obtain the optimal solution. In the general case, conditional probability is used. Particular problems in which the capacity of the system is finite are solved explicitly. [ABSTRACT FROM AUTHOR]

We consider two companies that are competing for orders. Let X 1 (n) denote the number of orders processed by the first company at time n, and let τ (k) be the first time that X 1 (n) < j or X 1 (n) = r , given that X 1 (0) = k . We assume that { X 1 (n) , n = 0 , 1 , ... } is a controlled discrete-time queueing system. Each company is using some control to increase its share of orders. The aim of the first company is to maximize the expected value of τ (k) , while its competitor tries to minimize this expected value. The optimal solution is obtained by making use of dynamic programming. Particular problems are solved explicitly. [ABSTRACT FROM AUTHOR]

The problem of controlling a compound Poisson process until it leaves an interval is considered. This type of problem is known as a homing problem. To determine the value of the optimal control, we must solve a nonlinear integro-differential equation. Exact and explicit solutions are obtained for two possible jumps size distributions.

Let $ (X_n,Y_n) $ (X n , Y n) be a two-dimensional discrete-time controlled Markov chain. The problem of optimally controlling the chain until the sum $ X_n+Y_n $ X n + Y n is equal to either of two constants is considered. The aim is to leave the continuation region as soon as possible. The optimal control is found in two particular cases by solving the partial difference equation satisfied by the value function. [ABSTRACT FROM AUTHOR]

Let ${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$ , where Y (t) is a one-dimensional diffusion process, and let $\tau(x,y)$ be the first time the process (X (t), Y (t)), starting from (x , y), leaves a subset of the first quadrant. The problem of computing the probability $p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$ is considered. The Laplace transform of the function p (x , y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of $\mathbb{E}[\tau(x,y)]$ and of the moment-generating function of $\tau(x,y)$ can also be derived. [ABSTRACT FROM AUTHOR]

The streamflow of a river is modelled as a jump-diffusion process. The jump size is distributed as an exponential random variable. The various parameters of the model are estimated by using the method of moments. An application to the Mississippi River is presented. [ABSTRACT FROM AUTHOR]

A stochastic difference game is considered in which a player wants to minimize the time spent by a controlled one-dimensional symmetric random walk { X n , n = 0 , 1 , ... } in the continuation region C : = { 1 , 2 , ... } , and the second player seeks to maximize the survival time in C. The process starts at X 0 = x > 0 and the game ends the first time X n ≤ 0 . An exact expression is derived for the value function, from which the optimal solution is obtained, and particular problems are solved explicitly. [ABSTRACT FROM AUTHOR]

The problem of optimally controlling a standard Brownian motion until a fixed final time is considered in the case when the final cost function is an even function. Two particular problems are solved explicitly. Moreover, the best constant control as well as the best linear control are also obtained in these two particular cases.

A boundary value problem for a non-linear difference equation of order three is considered. We show that this equation can be interpreted as the equation satisfied by the value function in a stochastic optimal control problem. We thus obtain an expression for the solution of the non-linear difference equation that can be used to find an explicit solution to this equation. An example is presented. [ABSTRACT FROM AUTHOR]

Various linear and non-linear first-order differential equations with different arguments are considered. Exact solutions to these equations are provided. Systems of two coupled linear first-order differential equations are also solved explicitly and exactly. [ABSTRACT FROM AUTHOR]

An analytical solution is found to the problem of maximising the time spent in the first quadrant by the two-dimensional diffusion process (X(t), Y(t)), where Y(t) is a controlled Brownian motion and X(t) is proportional to its integral. Moreover, we force the process to exit the first quadrant through the y-axis. This type of problem is known as LQG homing and is very difficult to solve explicitly, especially in two or more dimensions. Here the partial differential equation satisfied by a transformation of the value function is solved by making use of the method of separation of variables. The exact solution is expressed as an infinite sum of Airy functions.

Let Y (t) be a one-dimensional jump-diffusion process and X (t) be defined by d X (t) = ρ [ X (t) , Y (t) ] d t , where ρ (· , ·) is either a strictly positive or negative function. First-passage-time problems for the degenerate two-dimensional process (X (t) , Y (t)) are considered in the case when the process leaves the continuation region at the latest at the moment of the first jump, and the problem of optimally controlling the process is treated as well. A particular problem, in which ρ [ X (t) , Y (t) ] = Y (t) − X (t) and Y (t) is a standard Brownian motion with jumps, is solved explicitly. [ABSTRACT FROM AUTHOR]

Given that the flow of a river has just increased, the distribution of the flow at the time of the next increase is computed. Two models for the fluctuations of the flow are considered: a filtered Poisson process and a diffusion process with jumps. The results are applied to the Delaware River. Key words: filtered Poisson process, diffusion process with jumps, compound Poisson process, Wiener process, goodness- of-fit test. On s'interesse a une riviere dont le debit vient d'augmenter et l'on calcule la distribution du debit au moment oU celui-ci augmente a nouveau. Deux modeles de fluctuation du debit sont utilises : un processus de Poisson filtre et un processus de diffusion avec sauts. Les resultats obtenus sont appliques au fleuve Delaware. [Traduit par la Redaction] Mots-cles : processus de Poisson filtre, processus de diffusion avec sauts, processus de Poisson compose, processus de Wiener, test de validite de l'ajustement., 1. Introduction Let X(t) denote the flow of a river at time t. In Lefebvre and Bensalma (2015), the authors considered two stochastic models for the fluctuations of X(t) over [...]

A controlled discrete-time stochastic process is proposed as a model for the state of a machine. In the proposed model, normal and random wear of the machine are considered. The random wear of the machine during a time unit can be either a discrete or a continuous random variable. The objective is to find a control policy that maximises the profits generated by the machine over its useful lifetime. In this problem, the optimiser must decide whether to do or not to do the maintenance work on the machine at each time unit. The significance of the paper is that the final time in the optimal control problem is a random variable denoting the first time the machine must be replaced. To obtain the optimal control, one can try to solve the dynamic programming equation, which reduces to a difference or an integral equation, satisfied by the value function. Finally, particular cases are solved exactly, or approximately, and explicitly to demonstrate and validate the proposed model. [ABSTRACT FROM AUTHOR]

Let X(t) be a controlled jump-di®usion process starting at x 2 [a; b] and whose infinitesimal parameters vary according to a continuous-time Markov chain. The aim is to minimize the expected value of a cost function with quadratic control costs until X(t) leaves the interval (a; b), and a termination cost that depends on the final value of X(t). Exact and explicit solutions are obtained for important processes. [ABSTRACT FROM AUTHOR]

This paper presents a method for generating groups of electrostatic images for the estimation of soil parameters in the case of multilayered horizontal soil. The method can be utilized for the interpretation of resistivity sounding measurements of stratified soil. The maximum errors of the calculated resistivity values can also be estimated and are used to validate the soil model. Errors and variations in apparent resistivity values used in the interpretation process can originate from slow convergence in calculations or variations during field measurements, such as local fluctuations in soil resistivity at points of measurements and instruments precision. An estimation of the effect of these variations on the calculated soil model parameters can be used to provide a confidence level in the results. This paper demonstrates the necessity of evaluating the sensitivity of the soil parameters and proposes methods of estimating a confidence level in the soil model. Confidence levels are also used to delimit boundaries during geophysical inversion with respect to the information available in the field measurements. Simulation results are presented for three-layer soil. Index Terms--Electrostatic images, geophysical inversion, grounding, Marquardt method, resistivity measurements, resistivity sounding, soil resistivity, Wenner method.

The number of daily new cases of an epidemic is assumed to evolve as the exponential of a Wiener process with Poissonian jumps that are exponentially distributed. The model parameters can be estimated by using the method of moments. In an application to the COVID-19 pandemic in the province of Qu´ebec, Canada, the proposed model is shown to be acceptable. General formulas for the probability that a given increase in the number of daily new cases is due to the normal variations of the continuous part of the process or rather to a jump of this process are given. Based on these formulas, the probability of observing the likely start of a new wave of infections is calculated for the application to the COVID-19 pandemic. [ABSTRACT FROM AUTHOR]

We consider the classic formula obtained by Merton for the value of a contingent claim that expires at a fixed time T. In our case, the final time is random and depends on both the time t and the value of the stock at t. The partial integro-differential equation (PIDE), subject to the appropriate boundary conditions, is solved explicitly for a certain jump size distribution. The PIDE is first transformed into an integro-differential equation, and this equation is then reduced to an ordinary differential equation. [ABSTRACT FROM AUTHOR]

Let X(t) be a time-homogeneous jump-diffusion process. We assume that the jump size depends on the value of X(t). We obtain analytical results for the moments of T(x) and of X (T (x)) , where T(x) is the first time that the process leaves the interval (a, b). We also compute P [ X (T (x)) ≤ a ]. These results have applications in financial mathematics. [ABSTRACT FROM AUTHOR]

The problem of maximizing or minimizing the time spent by a stochastic process in an interval is considered for autoregressive processes. The control applied to the system is equal to 0, |$b$| or |$-b$|⁠. Particular cases are considered, and the appropriate integral equations are solved explicitly, either exactly or approximately. [ABSTRACT FROM AUTHOR]

Let X (t) be a one-dimensional controlled diffusion process, and Y (t) be a deterministic process. Various models are considered for (X ˙ (t) , Y ˙ (t)). The optimizer tries to bring the ratio X (t) / Y (t) to a certain value d as soon as possible. There is also a second optimizer whose aim is to prevent the ratio X (t) / Y (t) from reaching d as long as possible. Explicit solutions are obtained. The particular case when the control variables are assumed to be constants and the general case are treated. [ABSTRACT FROM AUTHOR]

Let { X (t) , t ≥ 0 } be a Wiener process with infinitesimal mean μ and variance 1, starting at X (0) ∈ [ 0 , 1 ] . Assume that X(0) is a random variable and define τ to be the first time that X(t) = 0 or 1. We look for probability density functions of X(0) such that P [ X (τ) = 0 ] = q ∈ (0 , 1) . Various cases are considered both when μ = 0 and μ ≠ 0. [ABSTRACT FROM AUTHOR]