Examples

The models depend on obtaining the joint transition density fu'w (x,y} of the process starting from Xo = u and Yo = v. The examples give some instances of the way in which they can be derived.

7.1 Maximum Process

An illustration of the bivariate process is obtained by taking a basic process Xt and constructing the bivariate process (Xt,Yt} with the maximum process Yt = sup{Xs | 0 < s < t}. A non-monotonic process with continuous sample paths is defined by the Wiener process Xt = aBt+pt with drift p and variance parameter a and where Bt is a standard Brownian motion. The state space is Q = R and is divided into intervals at points -to < S1 < ...,sj,...,sn so that Yt > sn or Xt > sn indicate entry into the critical set. The required densities and distributions can be deduced from results in [RW94]; the joint density of the process and its maximum with Xo = u is fu(x y) = 2(2y — x - u) [ (x — u — Vrf] [ 2(y — u)(y — x) Jt (x, y)= 2a2T jeW{ a2r

The marginal distribution of YT is

which gives an inverse Gaussian distribution [CF89] as the hitting time distribution.

7.2 The Integrated Process

When the underlying process Xt is a Wiener the two dimensional Kolmogorov diffusion (Xt,Yt) arises on setting Yt = /0<s<t Xsds. For a Brownian motion Bt the transition density of (Bt, j Bsds) starting from (u,w) is [McK63]

fr (x, y, t) = 4 exp (—6(v — y — tu)2 + 6(x — v)(y 2— v — tu) — 2(u — y)2

The linearity of the integral shows that Yt is also Gaussian and its moments are easily obtained. When the basic process Xt has drift p and volatility a the derived moments are vecE [Yt] = ipt2 vecV [Yt] = ia2t3 allowing the joint density to be written f0,0(x y) = ^Lexp ( 6(y— 1 pt2)2 16(x — pt)(y— 2pt2) 2(x — pt)2

na2t2 \ a2t3 a212 a2t

7.3 The Absolute Value

For the one dimensional Wiener process, Xt, the distribution of the absolute value of the process, Yt = |Xt|, is relatively easy and mimics the arguments for the maximum value. The distribution is clearly

The moments are vecE [ Yt, ] = + pt ^2$ ( fa^ — 1 vecE [ Yt2] = p2t2 + a2t .

7.4 Bessel Processes

Betensky [Bet98] uses Bessel processes in the comparison of treatment regimes. The Bessel process can be used directly, but in many cases the squared Bessel process allows simpler decision making and produces equivalent decisions. The squared Bessel processes are handled effectively through the well known properties of chi-squared and Wishart distributions.

The simplest approach is illustrated by the norm functional Yt = H^H which generates a Bessel process from a multivariate Wiener process. The basic process is based on an W-dimensional Brownian motion, vecBt = [B1(t),B1(t),...,Bs(t)] .

The process Yt = ||vecBt|| or more explicitly

is a Bessel process. For simplicity, the squared Bessel process n

is also used. The excursions of the two processes produce equivalent decision rules, and the second is simpler to handle. The Bessel process is a diffusion [OksOO] and Zt, a squared Bessel process denoted BESQsX0, [RY99], is a solution of

Zt = zo + St + 2 VZldBs o with S > 0 and z0 > 0. If S > 2, the process never reaches 0 for t > 0.

The squared Bessel process, is the square of the Euclidean norm of a S-dimensional Brownian motion. Because the distributions of the Bi(t) are normal, the probability density function is a chi-squared distribution if z0 = 0 and S > 0 the density is the chi-square

with S degrees of freedom.

if zo = 0 and S > 0, the density is a non-central chisquare written in terms of modified Bessel functions of the first kind [AS72, result 9.6.7]

The model is more realistic if the Brownian motion is replaced by a Wiener process with drift vecW t = [Wl(t), W2(t),..., WS (t)], Wi(t) = pit + aiBt

The distribution of vecWt is N (vecMt, Y*t) where vecMt is the vector of means and Y*t is the covariance matrix; St = (vecWt-vecMt)T(vecWt-vecMt) thus has a Wishart distribution with parameters 5/2 and (\/2)Y>t. The distribution of the "non-standard" squared Bessel process

i=l can thus be obtained.

7.5 Models for Imperfect Inspection

Imperfect inspection provides another example of associated processes. In this case Yt is simply the observed level of degradation subject to error. The simplest model of imperfect inspection is when the system degradation Xt is observable, but with error. here a simple independent additive error is assumed

Yt = Xt + st where st represents the error.

The increments and the error are distributed

The distributions G and H have densities g and h, and assume that h is symmetrical about zero.

Since Yt - Xt = st it is clear that fYt\Xt (y\x) = h(y - x)

From the definition of the process fXt\X0 (x\u) = G(x - u) .

These results combine to give the joint distribution of the future observed and true values of the degradation fYt,Xt\Xo (y,xIu) = fYt\Xo,Xt (yIu,x)fXt\Xo (xIu) = fYt\Xt (yYx)fXt\Xo (xIu) = h(y — x)g(x — u) .

To be specific assume here that the et — N(0,a2) and that underlying process is the Gamma process with increments [Abd75]

Plugging in the densities yields the joint density of the observed and true level of degradation, conditional on the true initial level of degradation, u . 1 ( 1, 3aT (x — u)a T-1 exp {—¡3(x — u)} fU(y, x) = ^exp {—(y — x)2)-f^-.

It follows from the monotonicity of the gamma process that the distribution of the hitting time of the critical set from an initial degradation u is pF = P(t: <h) = nav3{c-uj .

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