Stochastic Modeling

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By Mark H. A. Davis (auth.)

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If J = {1,2, ... ,d} then X is just a random vector (an 1Rd-valued random variable). If J=£:+={0,1,2, ... } then X is a discrete-time process, while if J = [0, T] for some T > 0 or J = 1R +, then X is a continuous-time process. These are the only cases that will be considered, with the main accent on the continuous-time case. The finite-dimensional distributions of a process is the collection 13 STOCHASTIC PROCESSES 17 of joint distribution functions Fr,, ... ,rJa 1, ... , an) of the random variables (Xr,, ...

7) Px[X,EA] = p(t,x,A) (t,x,A)E~+ x Ex C. In a Markov family, only the measure P x depends on the initial point xEE; all the other ingredients are the same for every x. ] = IEz[f(X,)] lz=X,· Thus the behaviour of the process beyond time s is just that of another process started at X s· In applications Markov families are normally constructed on canonical spaces. The most common such space, certainly for the purposes of this book, is the space Q = DE[O, oo[ of right-continuous £-valued functions on~+ with left-hand limits.

Each production facility costs £p, and investment can be channelled into the current building project at any rate up to a maximum of £K per week. We simply assume that the project is complete and comes on stream when the cumulative investment in it reaches £p. Thus there is a minimum lead time of p/K to provide a new facility. 8(a) shows typical sample functions of demand d(t) and capacity c(t); the latter increases by K units each time a new facility is completed, and we assume that when this happens, further investment is chanelled immediately into the next project in the series.

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