Karatzas I., Shreve S.E. Methods of mathematical finance

New York: Springer. - 1998. - 407 p. This book is intended for readers who are quite familiar with probability and stochastic processes but know little or nothing about finance. It is written in the definition/theorem/proof style of modern mathematics and attempts to explain as much of the finance motivation and terminology as possible. A mathematical monograph on finance can be written today only because of two revolutions that have taken place on Wall Street in the latter half of the twentieth century. Both these revolutions began at universities, albeit in economics departments and business schools, not in departments of mathematics or statistics. They have led inexorably, however, to an esca-lation in the level of mathematics (including probability, statistics, partial differential equations and their numerical analysis) used in finance, to a point where genuine research problems in the former fields are now deeply intertwined with the theory and practice of the latter. The first revolution in finance began with the 1952 publication of Portfolio Selection, an early version of the doctoral dissertation of Harry Markowitz. The second revolution in finance is connected with the explosion in the market for derivative securities.A Brownian Model of Financial Markets.
Contingent Claim Valuation in a Complete Market.
Single-Agent Consumption and Investment.
Equilibrium in a Complete Market.
Contingent Claims in Incomplete Markets.
Constrained Consumption and Investment.
Essential Supremum of a Family of Random Variables.
On the Model of Section 1.1.
On Theorem 6.4.1.
Optimal Stopping for Continuous-Parameter Processes.
The Clark Formula.