From the above equations it is clear that the probability structure of a Gaussian

stochastic process is described by the moments of first and second order. ■

This is a typical characteristic of the Gaussian stochastic processes that makes them particularly useful for applications. Direct consequence of the above remark is that the higher order moments of every Gaussian characteristic function are expressed through the moments of first and second order. The theorem presented below, due to Isserlis 1918, give the necessary formula for the calculation of higher order moments.

Theorem 2.52 : Let the Gaussian stochastic process X t β

(

;

)

, t I∈ . Then I. The central moments of odd order, j=2k+ , are identically zero, 1 II. The central moments of even order, j=2k, are given by the formula,

(

)

((

(

)

)

(

(

)

)

(

(

)

)

)

( )

(

)

(

)

1 2 1 2 1 2 1 2 3 4 2 1 2 ... 1, ,...,2 1; 2; ... ; , , , n n n k k j j j j j j XX X n t t n t XX i i XX i i XX i i C t t t E X t m X t m X t m R t t R t t R t t β _{β β β} − ⋅ ⋅⋅⋅ _{≡ − − − =} =

∑

…

where the summation is over all the possible partitions of the 2k indices 1 2 1,1, ,1; 2, 2, , 2; , , , , n j j j n n n

… … … … at pairs

(

i i₁, ₂

) (

, ,i i_{3 4}

)

, ,…

(

i_2k−₁,i_2k

)

. The number of terms is

( )

(

)

(

)

2 ! 1 3 5 2 1 2 !k k k k = ⋅ ⋅ ⋅ ⋅… − . ■

For the Gaussian stochastic processes the following two important theorems hold,

Theorem 2.53 : Let the functions μ i

( )

, I → and R i i

( )

, , I I× → with the last one, satisfies the conditions

1) R s t

( )

, =R t s

( )

, , for any ,t s I∈

2) For any t t₁, , ,₂ … t_n ∈I the matrix

(

_m, _n

)

n n

R t t

×

⎡ ⎤

⎣ ⎦ is non-negative definite.

Then, there always exists a Gaussian stochastic process X t β

(

;

)

, t I∈ , with mean value function m t

( )

=μ

( )

t , t I∈ and covariance function R i i

( )

, , I I× → , i.e. it satisfies the relations

( )

(

,

)

( )

E X tβ _{β =μ} t _{, t I∈ (65)}

( ) (

)

(

, ,

)

( )

, E X tβ _β X s _{β =}R t s _{, ,t s I∈ (66)}

If the functions μ i

( )

and R i i

( )

, are real then there is always a real stochastic process X t β

(

;

)

,

t I∈ that satisfies the above relations. In this case equation (66) becomes

( ) (

)

(

, ,

)

( )

,

E X tβ _β X s _{β =}R t s _{, ,t s I∈ ■ (67)}

Theorem 2.54 : Let the stochastic process X t β

(

;

)

, t I∈ with finite variance

( )

(

2

)

,

E X tβ _{β < ∞, t I∈ ■ (68)}

Then there is always exists a Gaussian stochastic process Y t β

(

;

)

, t I∈ such that

( )

( )

Y X m t =m t , t I∈ (69)

( )

,

( )

, YY XX R t s =R t s , ,t s I∈ (70)

If the the stochastic process X t β

(

;

)

, t I∈ is real then there is always a real stochastic process

(

;

)

Y t β , t I∈ that satisfies the above relations. ■

The above theorems are formulated and proved at the monograph of DOOB, J.L., (Stochastic

2.3. References

ASH, R.B., 1972, Measure, Integration and Functional Analysis. Academic Press.

ASH, R.B., 2000, Probability and Measure Theory. Academic Press.

ATHANASSOULIS, G.A., 2002, Stochastic Modeling and Forecasting of Ship Systems. Lecture

Notes NTUA.

BROWN, R.G. & HWANG, P.Y.C., 1997, Introduction to Random Signals and Applied Kalman

Filtering. John Wiley &Sons.

CRAMER H. , LEADBETTER M. R.,1953, Stationary and Related Stochastic Process, Wiley NY.

DOOB, J.L., 1953, Stochastic Processes. John Wiley &Sons.

FRISTEDT, B. & GRAY. L., 1997, A Modern Approach to Probability Theory. Birkhäuser

GIΚHMAN, I.I. & SKOROΚHOD, A.V., 1996, Theory of Random Processes. Dover Publications.

GIΚHMAN, I.I. & SKOROΚHOD, A.V., 1974, The Theory of Stochastic Processes I. Spinger.

GIΚHMAN, I.I. & SKOROΚHOD, A.V., 1975, The Theory of Stochastic Processes II. Spinger.

GIΚHMAN, I.I. & SKOROΚHOD, A.V., 1975, The Theory of Stochastic Processes III. Spinger.

KLIMOV, G., 1986, Probability Theory and Mathematical Statistics. MIR Publishers.

KOVALENKO, I.N. & KUZNETSOV, N.YU. & SHURENKOV, V.M., 1996, Models of Random

Processes. CRC Press.

KOLMOGOROV, A.N., 1956, Foundation of the Theory of Probability.

LOÈVE, M., 1977, Probability Theory II, Springer.

PERCIVAL, D.B. & WALDEN, A.T., 1993, Spectral Analysis for Physical Applications. Cambridge

University Press.

PROHOROV, YU.V. & ROZANOV, YU.A., 1969, Probability Theory. Springer.

PUGACHEV, V.S. & SINITSYN , 1987, Stochastic Differential Systems. John Wiley &Sons.

ROGERS, L.C.G. & WILLIAMS, D., 1993, Diffusions, Markov Processes and Martingales.

Cambridge Mathematical Library.

SINAI, YA.G., 1976, Introduction to Ergodic Theory. Princeton University Press.

SNYDER, D.L., 1975, Random Point Processes. Wiley, New York.

SOBCZYK, K., 1991, Stochastic Differential Equations, Kluwer Academic Publishers.

SOIZE, C., 1993, Mathematical Methods in Signal Analysis. French, Masson, Paris.

SOIZE, C., 1994, The Fokker-Planck Equation and its Explicit Steady State Solutions. World

Scientific.

SOONG, T.T., 1973, Random Differential Equations. Academic Press.

SOONG, T.T. AND GRIGORIU, M., 1993, Random Vibration of Mechanical and Structural

Systems. Prentice-Hall.

SPILIOTIS, I., 2004, Stochastic Differential Equations. Simeon Press.

51

Probability Measures in Hilbert Spaces

As we have discussed at Section 1 of Chapter 2 the definition of the stochastic process as a measurable function which maps the space of elementary events

Ω

into the metric space

_[

lead us to the concept of a probability measure over an infinite dimensional space. In this chapter we will consider the case of probability measures defined on a Hilbert space. Every result presented above can be generalized to the case of a Banach space by replacing the inner product with the duality pair and making suitable changes and restrictions.

In the first section we will give the definition of a stochastic process using probability measures. Moreover we will give some special examples of probability measures to get a clearer picture of the issue. Based on these examples we will introduce the finite dimensional relative measures, i.e. induced probability measures for finite dimensional subspaces of the Hilbert space. The next step will be to give the necessary and sufficient conditions for a family of probability measures, defined on every subspace of a Hilbert space, to define a probability measure.

Section 2 deals with the special case of cylinder functions and connects their integral over an infinite dimensional Hilbert space, using induced probability measures for finite dimensional subspaces of the Hilbert space.

In Section 3 we define the notion of the characteristic functional for probability measures over Hilbert spaces. Again, using finite dimensional relatives we make exact calculations of the characteristic functional. Finally, using nuclear operators we prove Minlos-Sazonov Theorem which is the generalization of Bochner’s Theorem for the infinite dimensional case and gives the necessary and sufficient conditions for a functional to be characteristic functional of some probability measure.

In Section 4 we generalize the concepts of mean value and covariance of a random variable for probability measures defined over infinite dimensional spaces. As before, we can calculate this quantities using induced probability measures over finite subspaces.

Section 5 and 6 present exact formulas for the calculation of classical moments and characteristic functions. The interesting part of this section is that we can calculate, using generalized functions the above quantities for the (time) derivatives of the stochastic process. Section 7 deals with special case of Gaussian measures and their properties. We prove that for this particular case of measure an exact formula for the characteristic functional. We also calculate the mean value and covariance operators and study their properties.

In Section 8 a special case of characteristic functional is studied named the Tatarskii charactersitc functional. These functionals have the important property to represent a wide class of stochastic processes.

Finally in section 9 we study the reduction of the Tatarskii characteristic functional to some very important characteristic functionals, such as the characteristic functionals due to Abel, Cauchy, and Feynman.

3.1. Probability Measures in Polish Spaces and their finite-dimensional relatives

3.1.1. Stochastic Processes as Random variables defined on Polish Spaces

In what follows we will consider the case of probability measures defined on a Polish space

[

, i.e. a complete metric space that has a countable dense subset. We also denote with

_[

∗

the dual space of linear continuous functionals on

_[

. We recall the definition of a stochastic process with values in

_[

.

Definition 1.1 : Let

(Ω,U

( )Ω ,c

)

be a probability space, X Ω →:

[

be a point function from the sample space

Ω

to the Polish space

_[

, and

_{U [}( )

be a Borel field over

_[

. The mapping X is said to be a stochastic process, iff is

(U

( )Ω ,U [( ))

- measurable. ■

Any stochastic process induces a probability measure on the space it takes values.

Definition 1.2 : Let

(Ω,U

( )Ω ,c

)

be a probability space,

([ U [

,

(

))

a measurable

space, and X Ω →:

[

a stochastic process. On the Borel field

U [(

)

of the measurable space

(_{[ U [}

,

(

))

we define the induced (by :X Ω →

[

) probability measure on

(

)

U [

is defined by

( )

(

1

( ))

_{, for every}

( )

B = X− B B∈ [

c

c

U [

(1)

The triplet

(_{[ U [ c}

,

(

)

, _[

)

is called the induced probability space. ■

We shall now present some simple, specific examples of probability measures on Polish spaces. These examples are given by means of specific constructions permitting appropriate extensions of a finite-dimensional probability measure on the whole Polish space.

Example 1.3 : [The 1-D induced measure]: Let

c

be a probability measure in , and ˆe be a

unit vector in

_[

. Let, also, ζ: e

[ ]

ˆ→ IR be a bijection between the span of ˆe, (denoted by

[ ]

ˆe )

and the real axis , given by ζ

( )

aeˆ =a. Then, for any E ∈

U [(

)

we define

[ ]eˆ

( )

E =

(

ζ

(

E

∩[ ]

eˆ

))

c

c

(2)

[ ]ˆe

c

is a probability measure in

_[

since,

a)

c

_{[ ]eˆ}

( )

E =

c

(

ζ

(

E

∩[ ]

eˆ

))

≥0, E ∈

U [(

)

b)

c

_{[ ]eˆ}

( )

[ =

c

(

ζ

(

[

∩[ ]

eˆ

))

=

c

(

ζ

( )[ ]

eˆ

)

=

c

( )

=1

c) for any countable collection of mutually disjoint sets E E₁, ₂,..., in

U [( )

[ ]eˆ n n

[ ]

ˆ

(

n

[ ]

ˆ

)

[ ]eˆ

( )

n n n n n E ζ E e ζ E e E ⎛ ⎛ ⎞⎞ ⎛ ⎞_{⎟ ⎜ ⎜}⎛ ⎞_{⎟ ⎟⎟} ⎛ ⎞_⎟ ⎜ ⎟ = ⎜ ⎜⎜ ⎟ ⎟⎟ = ⎜ ⎟= ⎜ ⎟ ⎜ ⎜⎜ ⎟ ⎟⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜⎜ ⎟ ⎟⎟⎟ ⎜ ⎟ ⎝

∪

⎠ ⎝ ⎝⎝

∪ ∩

⎠ ⎠⎠ ⎝

∪

∩

⎠

∑

c

c

c

c

. ■

Example 1.4 : [The N-D induced measure]: Let

c

N be a probability measure in

N_{, and}

N

L be a linear subspace of

_[

, of dimensionN. Let

{

e e₁, , ...,₂ e_N

}

be a basis of L , and _N

[

1 2

]

: , ,..., N

N N

L e e e IR

ζ = → be the bijection ζ

(

a e_{1 1}+a e_{2 2}+ +... a e_{N N}

) (

= a a₁, ₂, ...,a_N

)

. Then,

for any E ∈

U [(

)

we define

( )

(

(

))

1 2, ,..., N N N . e e e E ζ E L ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ =

∩

c

c

■ (3)

The set function

1 2, ,..., N

e e e

⎡ ⎤

⎢ ⎥

⎣ ⎦

c

is a legitimate probability measure in

_[

obtained as an extension of the N-D probability measure

c

N over the whole Polish space. In what follows we will examine the existence of a probability measure which is not an induced measure from a finite dimensional measure, but it has its own infinite dimensional structure. Before we procced we recall some basic results from probability theory.

3.1.2. Cylindric Sets

To begin our discussion we first need to give some essential definitions, concerning a special class of sets. Thus we have,

In document NATIONAL TECHNICAL UNIVERSITY OF ATHENS SCHOOL OF NAVAL ARCHITECTURE & MARINE ENGINEERING Section of Naval & Marine Hydrodynamics (Page 59-65)