Introduction to Stopping Time in Stochastic Finance Theory. Part II

Summary We start proceeding with the stopping time theory in discrete time with the help of the Mizar system [1], [4]. We prove, that the expression for two stopping times k1 and k2 not always implies a stopping time (k1 + k2) (see Theorem 6 in this paper). If you want to get a stopping time, you have to cut the function e.g. (k1 + k2) ⋂ T (see [2, p. 283 Remark 6.14]). Next we introduce the stopping time in continuous time. We are focused on the intervals [0, r] where r ∈ ℝ. We prove, that for I = [0, r] or I = [0,+∞[ the set {A ⋂ I : A ∈ Borel-Sets} is a σ-algebra of I (see Definition 6 in this paper, and more general given in [3, p.12 1.8e]). The interval I can be considered as a timeline from now to some point in the future. This set is necessary to define our next lemma. We prove the existence of the σ-algebra of the τ -past, where τ is a stopping time (see Definition 11 in this paper and [6, p.187, Definition 9.19]). If τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we can prove, that the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past (see Theorem 9 in this paper and [6, p.187 Lemma 9.21]). Suppose, that you want to use Lemma 9.21 with some events, that never occur, see as a comparison the paper [5] and the example for ST(1)={+∞} in the Summary. We don’t have the element +1 in our above-mentioned time intervals [0, r[ and [0,+1[. This is only possible if we construct a new σ-algebra on ℝ {−∞,+∞}. This construction is similar to the Borel-Sets and we call this σ-algebra extended Borel sets (see Definition 13 in this paper and [3, p. 21]). It can be proved, that {+∞} is an Element of extended Borel sets (see Theorem 21 in this paper). Now we use the interval [0,+∞] as a basis. We construct a σ-algebra on [0,+∞] similar to the book ([3, p. 12 18e]), see Definition 18 in this paper, and call it extended Borel subsets. We prove for stopping times with this given σ-algebra, that for τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we have the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past, see Theorem 25 in this paper. It is obvious, that {+∞} 2 extended Borel subsets. In general, Lemma 9.21 is important for the proof of the Optional Sampling Theorem, see 10.11 Proof of (i) in [6, p. 203].

Stopping sets: Gamma-type results and hitting properties

Recently in the paper by Møller and Zuyev (1996), the following Gamma-type result was established. Given n points of a homogeneous Poisson process defining a random figure, its volume is Γ(n,λ) distributed, where λ is the intensity of the process. In this paper we give an alternative description of the class of random sets for which the Gamma-type results hold. We show that it corresponds to the class of stopping sets with respect to the natural filtration of the point process with certain scaling properties. The proof uses the martingale technique for directed processes, in particular, an analogue of Doob's optional sampling theorem proved in Kurtz (1980). As well as being compact, this approach provides a new insight into the nature of geometrical objects constructed with respect to a Poisson point process. We show, in particular, that in this framework the probability that a point is covered by a stopping set does not depend on whether it is a point of the process or not.