Stochastic Processes: From Physics to Finance
This book introduces the theory of stochastic processes with applications taken from physics and finance. Fundamental concepts like the random walk or Brownian motion but also Levy-stable distributions are discussed. Applications are selected to show the interdisciplinary character of the concepts and methods. In the second edition of the book a discussion of extreme events ranging from their mathematical definition to their importance for financial crashes was included. The exposition of basic notions of probability theory and the Brownian motion problem as well as the relation between conservative diffusion processes and quantum mechanics is expanded. The second edition also enlarges the treatment of financial markets. Beyond a presentation of geometric Brownian motion and the Black-Scholes approach to option pricing as well as the econophysics analysis of the stylized facts of financial markets, an introduction to agent based modeling approaches is given.
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A Brief Survey of the Mathematics of Probability Theory
Beyond the Central Limit Theorem LÚvy Distributions
Modeling the Financial Market
Stable Distributions Revisited
Hyperspherical Polar Coordinates
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asset price assume Black-Scholes theory Brownian particle call option Chapman-Kolmogorov equation chapter constant correlation crash deﬁne deﬁnition derive differential equation diffusion process discussion drift efﬁcient energy evolution example expiry exponent exponential ﬁeld ﬁnal ﬁnance ﬁnancial market ﬁnd ﬁnite ﬁrst passage ﬁxed ﬂuid Fokker-Planck equation function Furthermore Gaussian distribution geometric Brownian motion given increments inﬁnite Inserting integral interval jumps kurtosis lattice LÚvy distribution limiting distribution Markov process martingale master equation mathematical normal obtain option pricing parameters portfolio power law price ﬂuctuations probability density probability distribution probability theory problem proﬁt properties put option quantum mechanics random variables random walk reﬂects regime result risk risk-less scale Sect solution speciﬁc Springer stable distributions stationary statistical physics stochastic process tion trading transform truncated LÚvy ﬂight underlying velocity volatility walker Weierstrass random walk Wiener process zero