Integration with respect to Brownian motion and closely related processes (It?o processes) is introduced in Chapter 4. In Chapter 3, the two main stochastic processes used in Stochastic Calculus are given: Brownian motion (for calculus of continuous processes) and Poisson process (for calculus of processes with jumps). Some more technical results in these chapters may be skipped and referred to later when needed. These chapters have examples but no exercises. The first two chapters describe the basic results in Calculus and Probability needed for further development. A brief description of the contents follows (for more details see the Table of Contents). This text presumes less initial knowledge than most texts on the subject (M?etivier (1982), Dellacherie and Meyer (1982), Protter (1992), Liptser and Shiryayev (1989), Jacod and Shiryayev (1987), Karatzas and Shreve (1988), Stroock and Varadhan (1979), Revuz and Yor (1991), Rogers and Williams (1990)), however it still presents a fairly complete and mathematically rigorous treatment of Stochastic Calculus for both continuous processes and processes with jumps. For example, the problem of pricing an option and the problem of optimal filtering of a noisy signal, both rely on the martingale representation property of Brownian motion. It turns out that completely unrelated applied problems have their solutions rooted in the same mathematical result. For example, the change of measure technique is needed in options pricing calculations of conditional expectations with respect to a new filtration is needed in filtering.
These results are required in applications. This allows the reader to arrive at advanced results sooner. Simple proofs are presented, but more technical proofs are left out and replaced by heuristic arguments with references to other more complete texts. Every effort has been made to keep presentation as simple as possible, while mathematically rigorous. This is achieved by making use of many solved examples.
This text is aimed at gradually taking the reader from a fairly low technical level to a sophisticated one. This is a mathematical text, that builds on theory of functions and probability and develops the martingale theory, which is highly technical. This is where probability comes in and the result is a calculus for random functions or stochastic processes. The need for this calculus comes from the necessity to include unpredictable factors into modelling. From an applied perspective, Stochastic Calculus can be loosely described as a field of Mathematics, that is concerned with infinitesimal calculus on nondifferentiable functions. In Biology, Stochastic Calculus is used to model the effects of stochastic variability in reproduction and environment on populations. In Physics, Stochastic Calculus is used to study the effects of random excitations on various physical phenomena. In Engineering, Stochastic Calculus is used in filtering and control theory. One of the greatest demands has come from the growing area of Mathematical Finance, where Stochastic Calculus is used for pricing and hedging of financial derivatives, v vi PREFACE such as options. During the past twenty years, there has been an increasing demand for tools and methods of Stochastic Calculus in various disciplines. This book aims at providing a concise presentation of Stochastic Calculus with some of its applications in Finance, Engineering and Science. Text Introduction to stochastic calculus with applications Klebaner, Fima C - Nama Orang
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