Mario Lefebvre

Risk-sensitive optimal investment policy

A system of two ordinary differential equations representing the rate of change with respect to time of the liquid and the non-liquid assets of a company are considered. The control that maximizes a ( stochastic) risk-sensitive criterion is obtained. The corresponding deterministic problem is also solved

Moment generating function of a first hitting place for the integrated Ornstein-Uhlenbeck process

We consider the two-dimensional process (x(t),y(t)), where y(t)=dx(t)/dt is an Ornstein-Uhlenbeck process, Let T be the smallest t for which y(t) = [upsilon]. In this note we obtain an explicit expression, in terms of a parabolic cylinder function, for the moment generating function or the characteristic function of x(T) and we evaluate the expected value of x(T).

Analytical solutions to LQG homing problems in one dimension

The problem of optimally controlling one-dimensional diffusion processes until they leave a given interval is considered. By linearizing the Riccati differential equation satisfied by the derivative of the value function in the so-called linear quadratic Gaussian homing problem, we are able to obtain an exact expression for the solution to the general problem. Particular problems are solved explicitly.

Optimal control of an Ornstein-Uhlenbeck process

In this paper, we consider an Ornstein-Uhlenbeck process in both a finite and a semi-infinite interval. Depending on the form of the cost function, our aim is either to leave the interval as soon as possible or to maximize the time spent in the interval, taking into account the control costs in both cases. The model may represent the current in a simple electrical circuit. Since the exact solutions are in terms of special functions, approximate solutions are given. The deterministic cases are also solved.

Basic Probability Theory with Applications

This book presents elementary probability theory with interesting and well-chosen applications that illustrate the theory. An introductory chapter reviews the basic elements of differential calculus which are used in the material to follow. The theory is presented systematically, beginning with the main results in elementary probability theory. This is followed by material on random variables. Random vectors, including the all important central limit theorem, are treated next. The last three chapters concentrate on applications of this theory in the areas of reliability theory, basic queuing models, and time series. Examples are elegantly woven into the text and over 400 exercises reinforce the material and provide students with ample practice. This textbook can be used by undergraduate students in pure and applied sciences such as mathematics, engineering, computer science, finance and economics. A separate solutions manual is available to instructors who adopt the text for their course.

Multilayer resistivity interpretation and error estimation using electrostatic images

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

A homing problem for diffusion processes with control-dependent variance

Controlled one-dimensional diffusion processes, with infinitesimal variance (instead of the infinitesimal mean) depending on the control variable, are considered in an interval located on the positive half-line. The process is controlled until it reaches either end of the interval. The aim is to minimize the expected value of a cost criterion with quadratic control costs on the way and a final cost equal to zero (resp. a large constant) if the process exits the interval through its left (resp. right) end point. Explicit expressions are obtained both for the optimal value of the control variable and the value function when the infinitesimal parameters of the processes are proportional to a power of the state variable.

A bidimensional optimal landing problem

A bidimensional model for the rate of change of the height and the vertical velocity of an airplane with respect to time is considered. The cost criterion is risk-sensitive. By taking a parameter large enough, the final value of the height of the airplane will be near zero, so that the objective is to obtain an optimal landing of the airplane. The model proposed here is an improvement over the one-dimensional one considered by other authors.

General LQG Homing Problems in One Dimension

Optimal control problems for one-dimensional diffusion processes in the interval () are considered. The aim is either to maximize or to minimize the time spent by the controlled processes in (). Exact solutions are obtained when the processes are symmetrical with respect to . Approximate solutions are derived in the asymmetrical case. The one-barrier cases are also treated. Examples are presented.

First Hitting Place Probabilities for a Discrete Version of the Ornstein-Uhlenbeck Process

A Markov chain with state space and transition probabilities depending on the current state is studied. The chain can be considered as a discrete Ornstein-Uhlenbeck process. The probability that the process hits before 0 is computed explicitly. Similarly, the probability that the process hits before is computed in the case when the state space is and the transition probabilities are not necessarily the same when is positive and is negative.