Error calculus for finance and physics. The language of Dirichlet forms.
This interesting book proposes to look at probabilistic error calculus from the angle of Dirichlet forms. The motivation for doing so can be seen at the following simple example, in which a measurement \(x\) is distorted by an error \(\varepsilon Y\), where \(\varepsilon\) is small and \(Y\) is a centered random variable. When a nonlinear function \(f\) is applied, the bias becomes \[ {\mathbb E}[f(x+\varepsilon Y)-f(x)]=\tfrac12\varepsilon^2 \operatorname{var}(Y)^2f''(x)+\varepsilon^3O(1), \] whereas the variance is given by \(\varepsilon^2\operatorname{var}(Y)^2(f'(x))^2+\varepsilon^3O(1). \) In a multi- or even infinite-dimensional situation, the leading term in the variance calculation will involve a general square-field operator \(\Gamma(f,f)\), and this establishes the connection with the theory of Dirichlet forms as proposed in this book. The book is structured as follows. Chapter I explains the intutitive notion of an error structure, starting off from the simple example given above. Chapter II contains basic facts about Dirichlet forms and strongly continuous semigroups. Chapters III to V develop the formal theory of error structures based on Dirichlet forms. Chapter VI studies particular error structures such as error structures for sequences of i.i.d. random variables, Wiener space, and Poisson space. Chapter VII considers applications to mathematical finance. Particular models include the Black-Scholes and local volatility models. Moreover, some of the Greeks of an option are interpreted in terms of error calculus. The final Chapter VIII discusses applications to physics such as the calculation of lengths using the Cauchy-Favard method, the temperature equilibrium of a homogeneous solid, or a nonlinear oscillator subject to thermal interaction.
scientific article; zbMATH DE number 2118875