## Partial Differential Equations in Economics and FinanceThis book reviews the basic theory of partial differential equations of the first and second order and discusses their applications in economics and finance. It starts with well-known applications to consumer and producer theory, and to the theory of option pricing and then introduces new applications that emerge from current research (some of which is the author's own) in bounded rationality, game theory, and multi-dimensional screening. |

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### Contents

Partial Differential Equations | 5 |

111 Examples | 6 |

12 The Complete Integral the General Integral and the Singular Integral | 8 |

121 Compatible Systems of the First Order PDEs | 9 |

13 The Second Order Partial Differential Equations | 10 |

132 Boundary Value Problems for Elliptic Equations | 11 |

133 The Cauchy Problem for a Parabolic Equation | 13 |

134 Examples | 15 |

613 A Discussion of One Experiment | 56 |

614 Conclusions | 60 |

62 A Model of Social Learning | 61 |

63 Locally Improving Adjustment Rules and the Deduced Utility | 64 |

64 Estimating the Utility Function from the Data | 71 |

65 Evolutionary Determination of Adjustment Rules | 72 |

66 A Note of Caution | 73 |

68 The Optimal Monopoly Pricing Under Viscous Demand and Customers Turnover | 76 |

14 Exercises | 18 |

15 Bibliographic Notes | 19 |

Stochastic Processes | 21 |

22 The Generator of the Stochastic Process | 22 |

23 Exercises | 24 |

24 Bibliographic Notes | 25 |

Economic Applications | 27 |

Consumer Theory | 31 |

32 Finding the Indirect Utility from the Marshallian Demand | 33 |

33 An Example of Finding the Indirect Utility Function from the Demand | 34 |

34 Exercises | 36 |

35 Bibliographic Notes | 37 |

Producer Theory | 39 |

42 The Main Partial Differential Equation of the Production Theory | 40 |

43 Examples Solved | 41 |

44 Exercises | 44 |

Pricing of the Financial Derivatives | 45 |

52 The Fundamental Equation of Derivatives Pricing | 47 |

53 Pricing of European Options | 48 |

54 Financial Markets and Bounded Rationality | 49 |

A Theory of Boundedly Rational Behavior | 51 |

61 A Model of Noisy Individual Adjustment | 53 |

612 The Multidimensional Case | 55 |

681 The Model | 78 |

682 The Population Turnover Rate and the Monopolists Profits | 81 |

683 Conclusions | 82 |

610 Exercises | 83 |

611 Bibliographic Notes | 84 |

Game Theory | 85 |

72 An Evolutionary Model of Reciprocity | 86 |

721 The Model | 89 |

Replicator Dynamics with Migration | 93 |

731 The Model | 97 |

732 Some Examples | 106 |

733 The Behavioral Foundations of the Model | 110 |

734 Discussion and Conclusions | 113 |

74 Exercises | 114 |

The Multidimensional Screening Model | 117 |

81 Hamiltonian Approach and the First Order Conditions | 118 |

82 An Example | 120 |

83 Conclusions | 121 |

84 Exercises | 122 |

Conclusions | 123 |

125 | |

133 | |

### Common terms and phrases

adjustment rule applications arbitrage assume assumption asymptotically stable Basov Basov Bibliographic Notes boundary conditions boundary problem bounded rationality boundedly rational behavior Cauchy problem chapter Consider constraint consumer theory continuously differentiable function contract in period conventions coordination games customers defined denote Dirichlet problem economic eigenvalues equilibrium selection evolution example exists fi(x financial derivative firms Foster and Young game theory geometric Brownian motion given gradient dynamics Hence homogeneous of degree implies incentive contract indirect utility function initial integral LBYE long-run outcome LWYE Mailath Marshallian demand maximization Merlo and Schotter migration monopolist multidimensional screening Nash equilibrium obtains optimal order partial differential partial differential equations player population production function quasilinear random variable reciprocal agents reciprocal workers replicator dynamics risk-dominant equilibrium satisfies screening models share of reciprocal social learning solution solve spatial stochastic process switching wave Theorem trait trust contract unique vector zero