By Thomas Mikosch
I discovered this textbook super teaching-oriented and a very good creation to a truly demanding topic, akin to stochastic calculus. i might certainly suggest it for a Master's point monetary engineering direction.
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Additional info for Elementary stochastic calculus with finance in view
Because of the discrete parameter n which indicates how many arguments Zi follow, the functions Pe are not quite pdfs in the standard sense. However, this is a technical detail which can be avoided by explaining the notation more clearly. For example, Po(n; Zl! znlv) is just shorthand for PO(Z1, ... znln, v)p(n), so that Po(n; Z1,'" zn)dz 1 ... dZ n is just the probability 2Recall from the footnote on page 7 that measurement lines have an orientation, so the "start" of the line has a well-defined meaning here.
1 will discuss background models in more detail, but it will be useful to introduce one special case now: the spatial Poisson process. If the background features (and, by Assumption 5, the interior features) arise from a Poisson process then the likelihood of the measurements takes on a particularly simple form. To see this, adopt the following assumption. • Assumption 7 The background features on a one-dimensional measurement line obey a Poisson law with density A. That is, bdn) = e->'L (A~)n .
Proof. The standard way to calculate the distribution of order statistics is to find their cumulative density functions (cdfs) and then differentiate (Ripley, 1987). In this case it is a rather routine calculation. Wh~n c = 0, all n features are. distributed uniformly on [O,L] so Prob(z' ~ y) = (1":' 2y/L)n. Thus the cdf C(y) = Prob(z'::; y) = 1-(1-2y/L)n, and differentiating gives the result. The case c = 1, n = 0 has zero probability since a target boundary is present and we have assumed a non-detection probability of zero.