FOMENTO DE LA COLONIZACIÓN ORIENTAL EN VALLE DEL RÍO UPANO POR PARTE DEL CREA

As we saw in Chapter 3 the characteristic functional gives a complete description of a stochastic process. In this chapter we shall see, how we can derive functional differential equations for the characteristic functional of a continuous time stochastic process whose sample functions satisfy a given Ordinary Differential Equation. This fact is of great importance, since we can formulate the full statistical problem into one equation. We must also note that, having this equation, all the statistical information of the problem can be recovered. Hence, form the functional differential equation we can, for example, derive the moments equations associated with the problem.

The above formulation, although it contains the whole statistical information, it’s not yet of practical importance. The difficulty comes from the functional differential equation itself, since up to the present time very little has been done to solve this kind of equations. On the other hand, very few characteristic functions, can be generalized in infinite dimensional Hilbert spaces, in order to give representations of the characteristic functional.

In Section 5.1 we introduce the notion of a stochastic differential equation and prove some general theorem concerning the existence and uniqness of solutions for equations of this kind. Both existence and uniqness is proved in the sense of probability measure. Moreover we study the necessary conditions for bounded of all moments of the probability measure that describes the system (ODE) response. Additionally we investigate the dependence of the solution of the system from a parameter inhered into the ODE. The analysis follows the steps of GIΚHMAN, I.I. & SKOROΚHOD, A.V., (The Theory of Stochastic Processes III).

Section 5.2 deals with the formulation of functional differential equations for the joint characteristic functional that describes the joint probability structure of the excitation and the response. The formulation concerns stochastic systems described by nonlinear ODE’s with polynomial nonlinearities. We distinguish between the case of a m.s. continuous stochastic excitation and an orthogonal increment excitation. Before the general results we present two specific systems, the Duffing Oscillator and the Van der Poll Oscillator.

In the next section 5.3 we prove how the general functional differential equation describing the probability structure of a system can be reduced to partial differential equations for the characteristic functions of various orders. Special cases of the above reduction are the general form of the Fokker-Planck equation as well as the Liouville equation. Of great importance are the partial differential equations proved at Section 5.3.4 that describes the probability structure of the system for the case of m.s. continuous excitation.

In the last Section 5.5 we introduce the notion of the kernel characteristic functional that generalizes the corresponding idea of kernel density functions in infinite dimensional spaces. We prove that the Gaussian characteristic functional has the kernel property, based on existent

theorems for extreme values of stochastic processes. In Section 5.4.4 we present how the kernel characteristic functionals can be used efficiently for the solution of functional differential equations. The basic idea is the localization in the physical phase space domain (probability measure) expressed in terms of the characteristic functional. We must note that the Gaussian measures are of great importance for these constructions since every analytical calculation concerning infinite dimensional integrals holds for Gaussian measures. In this way we derive a set of partial differential equations that governs the functional parameters of the kernel characteristic functionals. Finally in Section 5.4.5 we make special assumptions to derive a more simplified set of equations. Moreover a simple nonlinear ODE is studied and numerical results are presented.

5.1. Stochastic Differential Equations – General Theory

In the present section we introduce the notion of a stochastic differential equation and prove some general theorems concerning the existence and uniqness of solutions of these equations. For this purpose it is necessary to generalize the notion of a stochastic integral introduced above. The following analysis is based on the approach of GIΚHMAN, I.I. & SKOROΚHOD, A.V., (The Theory of Stochastic Processes III).

5.1.1. General Problems of the Theory of Stochastic Differential Equations.

Assume that we are dealing with a motion of a system S in the phase space N _{and let}

( )

t =

(

x t x t1

( ) ( )

, 2 , ,… x tN

( ))

x denote the location of this system in N at time t . Assume also

that a displacement of system Σ located at point x during the time interval

(

t t, +Δt

)

can be

represented in the form,

(

t+Δt

)

−

( )

t =A

(

,t+Δt

)

−A

( )

,t +δ

x x x x (1)

Here, A

( )

x,t is in general a random function; the difference A

(

x,t+Δt

)

−A

( )

x,t

characterizes the action of an “external field of forces” at point x on Σ during the time period

(

t t, +Δt

)

and δ is a quantity which is of higher order of smallness in a certain sense than the difference A

(

x,t+Δt

)

−A

( )

x,t . If A

( )

x,t as a function of t is absolutely continuous, then

relation (1) can be replaced by the ordinary differential equation

( )

₍

_{( )}

₎

, t d t A t t dt = ′ x x (2)

Equation (2) defines the motion of Σ in N _for

0

t> under the initial conditiont x

( )

t =₀ x₀,

while A_t′ x

( )

,t determines the “velocity field” in the phase space at time t.

It is obvious that equation (2) cannot describe motions such as Brownian, i.e. motions which do not possess a finite velocity in the phase space or motions which possess discontinuities in the phase space. To obtain an equation which will describe a motion of systems of this kind it is expedient to replace relation (1) with an equation of an integral type. For this purpose we visualize that the time interval

[

t t₀,

]

is subdivided into subintervals by the subdividing points

1, , ,2 n

t t … t =t. It then follows from (1) that

( )

( )

0 1

(

( )

1

)

(

( )

)

1 0 0 , , n n i i i i i i i t t − t t₊ t t − = = − =

∑

− +

∑

x x A x A x δ

Since δ are of a small order, it is natural to assume that, _i 1

0 0 n i i − = →

∑

δ as n → ∞. In this case the last equality formally becomes,

( )

( )

(

( )

)

0 0 , t t t − t =

∫

s s ds x x A x (3)

and the expression

( )

(

)

0 , t t s s ds

∫

A x

can be called a stochastic integral in the random field A x

( )

,t along the random curve

( )

s , s∈

[

t t0,

]

x ; the integral should be interpreted as the limit, in a certain sence, to be defined

more precisely, of sums of the form

( )

(

)

(

( )

)

1 1 0 , , n i i i i i t t t t − + = −

∑

A x A x

Relation (3) is called a Stochastic Differential Equation and is written in the form

( )

(

( )

,

)

,

( )

0 0, 0

t

d x t =A x t t x t =x t≥t

Under sufficiently general assumptions, for example, if A x

( )

,t is a martingale for

each _∈ N

x , one can assume that

( )

,t =

( )

,t +

( )

,t

A x a x β x (4)

where β

( )

x,t as a function of t is a martingale and process a x

( )

,t is representable as the

difference of two monotonically nondecreasing natural processes. In this connection it makes sense to suppose that the right-hand side of equation (3) can be represented according to formula (4) and introduce further restrictions on functions a x

( )

,t and β

( )

x,t in various ways.

For example, we may assume that the function a x

( )

,t appearing in expression (4) is an

absolute continuous function of t while β

( )

x,t  as a function of t  is a local square

integrable martingale. (Some more general assumptions concerning β

( )

x,t are considered

below).

In what follows, equation (3) will be written in the form

( )

( )

(

( )

)

(

( )

)

0 0 0 , , t t t t t = t +

∫

s s ds+

∫

s ds x x a x β x (5) or,

( )

(

( )

,

)

(

( )

,

)

,

( )

0 0 t d x t =a x t t dt+β x t dt x t =x

In the case when β

(

x

( )

t dt =,

)

0, equation (5) is called an ordinary differential equation (with a random right-hand side).

Often fields β

( )

x,t =

(

β₁

( )

x, ,t β₂

( )

x, , ,t … β_N

( )

x,t

)

of the form

( )

(

)

( )

1 0 , , , 1, , t _M n nm m m t s d s n N β γ μ = =

_∫

∑

= … x x (6)

are considered. Here μ_m

( )

s m, = …1, ,M are local mutually orthogonal square integrable martingales and γ_nm

(

x,s

)

are random functions satisfying conditions which assure the

existence of corresponding integrals. In this case, the second integral in equation (5) may be defined as the vector-valued integral with components

( )

(

)

(

( )

)

( )

0 0 1 , , , 1, , t t _M n nm m m t t s ds s s d s n N β γ μ = =

∑

=

∫

x

∫

x …

and the theory of stochastic integrals described in Section 1 of Chapter 4 can be utilized. However, if we confine ourselves to functions β

( )

x,t of type (6) a substantial amount of

generality is lost. This can be seen from the fact that the joint characteristic of processes

( )

,

n t

In document Los proyectos de colonización semidirigida por el Centro de Reconversión Económica de Azuay, Cañar y Morona Santiago (CREA) en el oriente ecuatoriano como motor de desarrollo urbano-rural. estudio del proyecto Upano-Palora (1969-2016) (página 85-88)