Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- 1 Probability theory: basic notions
- 2 Maximum and addition of random variables
- 3 Continuous time limit, Ito calculus and path integrals
- 4 Analysis of empirical data
- 5 Financial products and financial markets
- 6 Statistics of real prices: basic results
- 7 Non-linear correlations and volatility fluctuations
- 8 Skewness and price-volatility correlations
- 9 Cross-correlations
- 10 Risk measures
- 11 Extreme correlations and variety
- 12 Optimal portfolios
- 13 Futures and options: fundamental concepts
- 14 Options: hedging and residual risk
- 15 Options: the role of drift and correlations
- 16 Options: the Black and Scholes model
- 17 Options: some more specific problems
- 18 Options: minimum variance Monte–Carlo
- 19 The yield curve
- 20 Simple mechanisms for anomalous price statistics
- Index of most important symbols
- Index
3 - Continuous time limit, Ito calculus and path integrals
Published online by Cambridge University Press: 06 July 2010
Book contents
- Frontmatter
- Contents
- Foreword
- Preface
- 1 Probability theory: basic notions
- 2 Maximum and addition of random variables
- 3 Continuous time limit, Ito calculus and path integrals
- 4 Analysis of empirical data
- 5 Financial products and financial markets
- 6 Statistics of real prices: basic results
- 7 Non-linear correlations and volatility fluctuations
- 8 Skewness and price-volatility correlations
- 9 Cross-correlations
- 10 Risk measures
- 11 Extreme correlations and variety
- 12 Optimal portfolios
- 13 Futures and options: fundamental concepts
- 14 Options: hedging and residual risk
- 15 Options: the role of drift and correlations
- 16 Options: the Black and Scholes model
- 17 Options: some more specific problems
- 18 Options: minimum variance Monte–Carlo
- 19 The yield curve
- 20 Simple mechanisms for anomalous price statistics
- Index of most important symbols
- Index
Summary
Comment oser parler des lois du hasard? Le hasard n'est-il pas l'antithèse de toute loi?
(Joseph Bertrand, Calcul des probabilités.)Divisibility and the continuous time limit
Divisibility
We have discussed in the previous chapter how the sum of two iid random variables can be computed if the distribution of these two variables is known. One can ask the opposite question: knowing the probability distribution of a variable X, is it possible to find two iid random variables such that X = δX1 + δX2, or more precisely such that the distribution of X can be written as the convolution of two identical distributions. If this is possible, the variable X is said to be divisible.
We already know cases where this is possible. For example, if X has a Gaussian distribution of variance σ2, one can choose X1 and X2 to be have Gaussian distribution of variance σ2/2. More generally, if X is a Lévy stable random variable of parameter aµ (see Eq. (1.20)), then X1 and X2 are also Lévy stable random variables with parameter aµ/2. However, all random variables are not divisible: as we show below, if X has a uniform distribution in the interval [a, b], it cannot be written as X = δX1 + δX2 with δX1, δX2 iid.
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- Theory of Financial Risk and Derivative PricingFrom Statistical Physics to Risk Management, pp. 43 - 54Publisher: Cambridge University PressPrint publication year: 2003
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