## Methods of Mathematical FinanceThis book is intended for readers who are quite familiar with probability and stochastic processes but know little or nothing about ?nance. It is written in the de?nition/theorem/proof style of modern mathematics and attempts to explain as much of the ?nance motivation and terminology as possible. A mathematical monograph on ?nance can be written today only - cause 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 mathematicsor statistics. Theyhaveledinexorably,however,to anes- lation in the level of mathematics (including probability, statistics, partial di?erential equations and their numerical analysis) used in ?nance, to a point where genuine research problems in the former ?elds are now deeply intertwined with the theory and practice of the latter. The ?rst revolution in ?nance began with the 1952 publication of “Po- folio Selection,” an early version of the doctoral dissertation of Harry Markowitz. This publication began a shift away from the concept of t- ing to identify the “best” stock for an investor, and towards the concept of trying to understand and quantify the trade-o?s between risk and - turn inherent in an entire portfolio of stocks. The vehicle for this so-called mean–variance analysis of portfolios is linear regression; once this analysis is complete, one can then address the optimization problem of choosing the portfolio with the largest mean return, subject to keeping the risk (i. e. |

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### Contents

A Brownian Model of Financial Markets | 1 |

12 Portfolio and Gains Processes | 6 |

13 Income and Wealth Processes | 10 |

14 Arbitrage and Market Viability | 11 |

15 Standard Financial Markets | 16 |

16 Completeness of Financial Markets | 21 |

17 Financial Markets with an Infinite Planning Horizon | 27 |

18 Notes | 31 |

46 Existence and Uniqueness of Equilibrium | 178 |

47 Examples | 189 |

48 Notes | 196 |

Contingent Claims in Incomplete Markets | 199 |

52 The Model | 201 |

53 Upper Hedging Price | 204 |

54 Convex Sets and Support Functions | 205 |

55 A Family of Auxiliary Markets | 208 |

Contingent Claim Valuation in a Complete Market | 36 |

22 European Contingent Claims | 39 |

23 Forward and Futures Contracts | 43 |

24 European Options in a ConstantCoefficient Market | 47 |

25 American Contingent Claims | 54 |

26 The American Call Option | 60 |

27 The American Put Option | 67 |

28 Notes | 80 |

SingleAgent Consumption and Investment | 88 |

32 The Financial Market | 90 |

33 Consumption and Portfolio Processes | 91 |

34 Utility Functions | 94 |

35 The Optimization Problems | 97 |

36 Utility from Consumption and Terminal Wealth | 101 |

37 Utility from Consumption or Terminal Wealth | 111 |

38 Deterministic Coefficients | 118 |

39 Consumption and Investment on an Inﬁnite Horizon | 136 |

310 Maximization of the Growth Rate of Wealth | 150 |

311 Notes | 153 |

Equilibrium in a Complete Market | 159 |

42 Agents Endowments and Utility Functions | 161 |

Consumption and Portfolio Processes | 163 |

44 The Individual Optimization Problems | 167 |

45 Equilibrium and the Representative Agent | 170 |

56 The Main Hedging Result | 211 |

57 Upper Hedging with Constant Coefficients | 220 |

58 Optimal Dual Processes | 225 |

59 Lower Hedging Price | 238 |

510 Lower Hedging with Constant Coefficients | 254 |

511 Notes | 257 |

Constrained Consumption and Investment | 260 |

62 Utility Maximization with Constraints | 261 |

63 A Family of Unconstrained Problems | 266 |

64 Equivalent Optimality Conditions | 275 |

65 Duality and Existence | 284 |

66 Deterministic Coefficients Cone Constraints | 291 |

67 Incomplete Markets | 302 |

68 Higher Interest Rate for Borrowing Than for Investing | 310 |

69 Notes | 318 |

Essential Supremum of a Family of Random Variables | 323 |

On the Model of Section 11 | 326 |

On Theorem 641 | 335 |

Optimal Stopping for ContinuousParameter Processes | 349 |

The Clark Formula | 363 |

References | 370 |

403 | |

411 | |

### Common terms and phrases

agent American call option American put option arbitrage asset pricing assume assumption bounded Brownian motion chapter condition constant constraint contingent claim convex convex set corresponding wealth process deﬁne Deﬁnition denote derivative securities dual equilibrium market equivalent Example existence expected utility financial market ﬁnite ﬁrst ﬁxed follows formula forward contract given hedging portfolio holds almost surely implies incomplete markets inequality initial wealth interest rate investment investor Karatzas and Shreve Lebesgue-a.e. t G local martingale martingale measure martingale-generating Mathematical Finance maximization money market nondecreasing nonnegative obtain optimal portfolio portfolio process portfolio-proportion process positive progressively measurable progressively measurable process proof of Theorem Proposition random variable RCLL Remark right-continuous satisﬁes satisfying Section semimartingale stochastic stock prices supermartingale supremum terminal wealth Theorem 5.3 unconstrained unique utility function value function vector zero

### References to this book

Controlled Markov Processes and Viscosity Solutions Wendell H. Fleming,Halil Mete Soner No preview available - 2006 |

Stochastic Finance: An Introduction in Discrete Time Hans Föllmer,Alexander Schied Limited preview - 2004 |