Sample function behavior of increasing processes of class L

Computer simulation of landscapes and skylines has recently attracted a great deal of interest: see [6, 7]. Specification of a ‘landscape’ requires a function f: D → ℝ on a subset D of ℝ2, selected so that the apparent irregularity and randomness of the surface {(t,f(t)): t ∈ D} corresponds to what might be observed in nature. It is natural to look to random fields (that is, stochastic processes in two variables), and in particular to Gaussian fields, for functions with such properties. Even when an appropriate random field has been selected, determination of a typical sample function is far from easy [7].

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Some sample function properties of a process with stationary independent increments

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1963-0152000-9 ◽

1963 ◽

Vol 107 (3) ◽

pp. 545-545 ◽

Cited By ~ 1

Author(s):

Shashanka Shekhar Mitra

Keyword(s):

Sample Function ◽

Independent Increments

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Sample function behavior of increasing processes with stationary, independent increments

Pacific Journal of Mathematics ◽

10.2140/pjm.1967.21.21 ◽

1967 ◽

Vol 21 (1) ◽

pp. 21-33 ◽

Cited By ~ 18

Author(s):

Bert Fristedt

Keyword(s):

Sample Function ◽

Independent Increments

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A Method for the Estimation of the Risk Premiums in Stop Loss Reinsurance

Astin Bulletin ◽

10.1017/s0515036100001367 ◽

1959 ◽

Vol 1 (2) ◽

pp. 71-77

Author(s):

Carl Philipson

Keyword(s):

Stochastic Process ◽

Random Function ◽

Time Interval ◽

Sample Function ◽

Individual Event ◽

Risk Premiums ◽

Accounting Period ◽

Stop Loss

The reinsurance to be treated in this note shall cover the excess over a certain limit Q of the total amount of claims for each accounting period paid by the ceding company. Regardless of the rule for the determination of the limit Q such a reinsurance shall in this context be called stop loss reinsurance.Generally, such a reinsurance is called either stop loss or excess of loss reinsurance depending on the rule for the determination of Q. The use of the term stop loss regardless of this rule is preferred here in order to avoid confusion with an excess of loss reinsurance which refers to the excess of the amount of one or more claims caused by an individual event.The total amount of claims paid by the ceding company c during the period t shall be denoted cXt. This random function constitutes for a fixed value of c a stochastic process with the discontinuous parameter t > o.(It may be remarked that for a fixed pair of values of c and tcXt can be regarded as a particular value of a sample function pertaining to the process of the continuous paramater τ generally considered in the collective theory of risk to which the total amount of claims up to time τ is attached, in this case o < τ ≤ 1. cXt can also be considered as the increments during the time interval t of the total amount of the claims up to τ.)

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Square of Brownian motion

Nagoya Mathematical Journal ◽

10.1017/s0027763000016755 ◽

1975 ◽

Vol 59 ◽

pp. 1-8

Author(s):

Hisao Nomoto

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

Correlation Function ◽

Finite Time ◽

Present Note ◽

Point Of View ◽

Time Interval ◽

Sample Function ◽

Finite Time Interval ◽

Zero Crossings

Let Xt be a stochastic process and Yt be its square process. The present note is concerned with the solution of the equation assuming Yt is given. In [4], F. A. Grünbaum proved that certain statistics of Yt are enough to determine those of Xt when it is a centered, nonvanishing, Gaussian process with continuous correlation function. In connection with this result, we are interested in sample function-wise inference, though it is far from generalities. A glance of the equation shows that the difficulty is related how to pick up a sign of . Thus if we know that Xt has nice sample process such as the zero crossings are finite, no tangencies, in any finite time interval, then observations of these statistics will make it sure to find out sample functions of Xt from those of Yt (see [2]). The purpose of this note is to consider the above problem from this point of view.

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