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3023 results in Probability theory and stochastic processes

Page 140 of 152

  • Chapter 15 - Applications

  • from PART B - MARTINGALE THEORY
  • David Williams, Statistical Laboratory, University of Cambridge
  • Book:
    Probability with Martingales
    Published online:
    05 June 2012
    Print publication:
    14 February 1991, pp 153-171

  • Summary

    Introduction – please read!

    The purpose of this chapter is to give some indication of some of the ways in which the theory which we have developed can be applied to real-world problems. We consider only very simple examples, but at a lively pace!

    In Sections 15.1-15.2, we discuss a trivial case of a celebrated result from mathematical economics, the Black-Scholes option-pricing formula. The formula was developed for a continuous-parameter (diffusion) model for stock prices; see, for example, Karatzas and Schreve (1988). We present an obvious discretization which also has many treatments in the literature. What needs to be emphasized is that in the discrete case, the result has nothing to do with probability, which is why the answer is completely independent of the underlying probability measure. The use of the ‘martingale measure’ ℙ in Section 15.2 is nothing other than a device for expressing some simple algebra/combinatorics. But in the diffusion setting, where the algebra and combinatorics are no longer meaningful, the martingale-representation theorem and Cameron-Martin-Girsanov change-of-measure theorem provide the essential language. I think that this justifies my giving a ‘martingale’ treatment of something which needs only junior-school algebra.

    Sections 15.3-15.5 indicate the further development of the martingale formulation of optimality in stochastic control, at which Exercise E10.2 gave a first look. We consider just one ‘fun’ example, the ‘Mabinogion sheep problem’; but it is an example which illustrates rather well several techniques which may be effectively utilized in other contexts.

  • Chapter 1 - Measure Spaces

  • from PART A - FOUNDATIONS
  • David Williams, Statistical Laboratory, University of Cambridge
  • Book:
    Probability with Martingales
    Published online:
    05 June 2012
    Print publication:
    14 February 1991, pp 14-22

  • Summary

    Introductory remarks

    Topology is about open sets. The characterizing property of a continuous function f is that the inverse image f−1(G) of an open set G is open.

    Measure theory is about measurable sets. The characterizing property of a measurable function f is that the inverse image f−1(A) of any measurable set is measurable.

    In topology, one axiomatizes the notion of ‘open set’, insisting in particular that the union of any collection of open sets is open, and that the intersection of a finite collection of open sets is open.

    In measure theory, one axiomatizes the notion of ‘measurable set’, insisting that the union of a countable collection of measurable sets is measurable, and that the intersection of a countable collection of measurable sets is also measurable. Also, the complement of a measurable set must be measurable, and the whole space must be measurable. Thus the measurable sets form a σ-algebra, a structure stable (or ‘closed’) under countably many set operations. Without the insistence that ‘only countably many operations are allowed’, measure theory would be self-contradictory – a point lost on certain philosophers of probability.

Page 140 of 152