On the optimal control of stationary diffusion processes with natural boundaries and discounted cost

1971 ◽  
Vol 8 (3) ◽  
pp. 561-572 ◽  
Author(s):  
R. Morton
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Cost Function ◽  
Diffusion Processes ◽  
Expected Cost ◽  
Discounted Cost ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Stationary Diffusion ◽  
Natural Boundaries

SummaryResults similar to those in [3] are obtained for one-dimensional diffusion processes with discounted cost. The stronger assumption that at least one of the inaccessible boundaries is natural enables us to identify the solution of a differential equation corresponding to the future expected cost function.

On the optimal control of stationary diffusion processes with natural boundaries and discounted cost

1971 ◽  
Vol 8 (03) ◽  
pp. 561-572
Author(s):  
R. Morton
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Cost Function ◽  
Diffusion Processes ◽  
Expected Cost ◽  
Discounted Cost ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Stationary Diffusion ◽  
Natural Boundaries

Summary Results similar to those in [3] are obtained for one-dimensional diffusion processes with discounted cost. The stronger assumption that at least one of the inaccessible boundaries is natural enables us to identify the solution of a differential equation corresponding to the future expected cost function.

First-passage-time densities for time-non-homogeneous diffusion processes

1997 ◽  
Vol 34 (3) ◽  
pp. 623-631 ◽  
Author(s):  
R. Gutiérrez ◽  
L. M. Ricciardi ◽  
P. Román ◽  
F. Torres
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Continuous Functions ◽  
Probability Density Functions ◽  
Density Functions ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Natural Boundaries

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes

1989 ◽  
Vol 26 (4) ◽  
pp. 707-721 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Constructive Approach ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities ◽  
Reflecting Boundaries ◽  
Natural Boundaries

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

On the optimal control of stationary diffusion processes with inaccessible boundaries and no discounting

1971 ◽  
Vol 8 (03) ◽  
pp. 551-560 ◽  
Author(s):  
R. Morton
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Diffusion Processes ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Multidimensional Case ◽  
Potential Cost ◽  
One Dimensional ◽  
Dynamic Programming Equation ◽  
Stationary Diffusion

Summary Because there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.

On the optimal control of stationary diffusion processes with inaccessible boundaries and no discounting

1971 ◽  
Vol 8 (3) ◽  
pp. 551-560 ◽  
Author(s):  
R. Morton
Keyword(s):  
Optimal Control ◽  
Dynamic Programming ◽  
Diffusion Processes ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Multidimensional Case ◽  
Potential Cost ◽  
One Dimensional ◽  
Dynamic Programming Equation ◽  
Stationary Diffusion

SummaryBecause there are no boundary conditions, extra properties are required in order to identify the correct potential cost function. A solution of the Dynamic Programming equation for one-dimensional processes leads to an optimal solution within a wide class of alternatives (Theorem 1), and is completely optimal if certain conditions are satisfied (Theorem 2). Necessary conditions are also given. Several examples are solved, and some extension to the multidimensional case is shown.

A symmetry-based constructive approach to probability densities for one-dimensional diffusion processes

1989 ◽  
Vol 26 (04) ◽  
pp. 707-721 ◽  
Author(s):  
V. Giorno ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Constructive Approach ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Probability Densities ◽  
Reflecting Boundaries ◽  
Natural Boundaries

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

First-passage-time densities for time-non-homogeneous diffusion processes

1997 ◽  
Vol 34 (03) ◽  
pp. 623-631 ◽  
Author(s):  
R. Gutiérrez ◽  
L. M. Ricciardi ◽  
P. Román ◽  
F. Torres
Keyword(s):  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Continuous Functions ◽  
Probability Density Functions ◽  
Density Functions ◽  
First Passage ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Natural Boundaries

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

scholarly journals General LQG Homing Problems in One Dimension

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Mario Lefebvre ◽  
Foued Zitouni
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Diffusion Processes ◽  
Approximate Solutions ◽  
One Dimension ◽  
Control Problems ◽  
One Dimensional ◽  
Controlled Processes ◽  
Dimensional Diffusion ◽  
The One

Optimal control problems for one-dimensional diffusion processes in the interval (d1,d2) are considered. The aim is either to maximize or to minimize the time spent by the controlled processes in (d1,d2). Exact solutions are obtained when the processes are symmetrical with respect to d∗∈(d1,d2). Approximate solutions are derived in the asymmetrical case. The one-barrier cases are also treated. Examples are presented.

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