The prolonged boom in the US and European stock markets has led to increased interest in the mathematics of security markets, most notably in the theory of stochastic integration. This text gives a rigorous development of the theory of stochastic integration as it applies to the valuation of derivative securities. It includes all the tools necessary for readers to understand how the stochastic integral is constructed with respect to a general continuous martingale. The author develops the stochastic calculus from first principles, but at a relaxed pace that includes proofs that are detailed, but streamlined to applications to finance. The treatment requires minimal prerequisites-a basic knowledge of measure theoretic probability and Hilbert space theory-and devotes an entire chapter to application in finances, including the Black Scholes market, pricing contingent claims, the general market model, pricing of random payoffs, and interest rate derivatives. Continuous Stochastic Calculus with Application to Finance is your first opportunity to explore stochastic integration at a reasonable and practical mathematical level.
It offers a treatment well balanced between aesthetic appeal, degree of generality, depth, and ease of reading.
MARTINGALE THEORY Covergence of Random Variables Conditioning Submartingales Convergence Theorems Optional Sampling of Closed Submartingale Sequences Maximal Inequalities for Submartingale Sequences Continuous Time Martingales Local Martingales Quadratic Variation The Covariation Process Semimartingales BROWNIAN MOTION Gaussian Process One Dimensional Brownian Motion STOCHASTIC INTEGRATION Measurability Properties of Stochastic Processes Stochastic Integration with Respect to Continuous Semimartingales Ito's Formula Change of Measure Representation of Continuous Local Martingales Miscellaneous APPLICATION TO FINANCE The Simple Black Scholes Market Pricing of Contingent Claims The General Market Model Pricing of Random Payoffs at Fixed Future Dates Interest Rate Derivatives APPENDIX Separation of Convex Sets The Basic Extension Procedure Positive Semidefinite Matrices Kolmogoroff Existence Theorem