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Preface to the Second EditionThe second edition is revised, expanded and enhanced. This is now a more complete text in Stochastic Calculus, from both a theoretical and an applications point of view. Changes came about, as a result of using this book for teaching courses in Stochastic Calculus and Financial Mathematics over a number of years. Many topics are expanded with more worked out examples and exercises. Solutions to selected exercises are included. A new chapter on bonds and interest rates contains derivations of the main pricing models, including currently used market models (BGM). The change of numeraire technique is demonstrated on interest rate, currency and exotic options. The presentation of Applications in Finance is now more comprehensive and selfcontained. The models in Biology introduced in the new edition include the age-dependent branching process and a stochastic model for competition of species. These Markov processes are treated by Stochastic Calculus techniques using some new representations, such as a relation between Poisson and Birth-Death processes. The mathematical theory of filtering is based on the methods of Stochastic Calculus. In the new edition, we derive stochastic equations for a non-linear filter first and obtain the Kalman-Bucy filter as a corollary. Models arising in applications are treated rigorously demonstrating how to apply theoretical results to particular models. This approach might not make certain places easy reading, however, by using this book, the reader will accomplish a working knowledge of Stochastic Calculus.

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We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.