Path-dependent SDEs in Hilbert spaces

We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove existence and uniqueness of mild solutions, continuity of mild solutions with respect to perturbations of all the data of the system, G\^ateaux differentiability of generic order n of mild solutions with respect to the starting point, continuity of the G\^ateaux derivatives with respect to all the data. The analysis is performed for generic spaces of paths that do not necessarily coincide with the space of continuous functions

Modeling interest rate dynamics: an infinite-dimensional approach

We present a family of models for the term structure of interest rates which describe the interest rate curve as a stochastic process in a Hilbert space. We start by decomposing the deformations of the term structure into the variations of the short rate, the long rate and the fluctuations of the curve around its average shape. This fluctuation is then described as a solution of a stochastic evolution equation in an infinite dimensional space. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates and the structure of principal components of term structure deformations. Finally, we discuss calibration issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.Comment: Keywords: interest rates, stochastic PDE, term structure models, stochastic processes in Hilbert space. Other related works may be retrieved on http://www.eleves.ens.fr:8080/home/cont/papers.htm

Model uncertainty and its impact on the pricing of derivative instruments

Model uncertainty, in the context of derivative pricing, can be defined as the uncertainty on the value of a contingent claim resulting from the lack of precise knowledge of the pricing model to be used for its valuation. We introduce here a quantitative framework for defining model uncertainty in option pricing models. After discussing some properties which a quantitative measure of model uncertainty should verify in order to be useful and relevant in the context of risk measurement and management, we propose a method for measuring model uncertainty which verifies these properties and yields numbers which are comparable to other risk measures and compatible with observations of market prices of a set of benchmark derivatives. We illustrate the difference between model uncertainty and the more common notion of "market risk" through examples. Finally, we illustrate the connection between our proposed measure of model uncertainty and the recent literature on coherent and convex risk measures.decision under ambiguity; uncertainty; option pricing; risk measures; mathematical finance

Pathwise integration with respect to paths of finite quadratic variation

We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands. We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation.Comment: To appear in: Journal de Mathematiques Pures et Appliquee

Convergent multiplicative processes repelled from zero: power laws and truncated power laws

Random multiplicative processes $w_t =\lambda_1 \lambda_2 ... \lambda_t$ (with 0 ) lead, in the presence of a boundary constraint, to a distribution $P(w_t)$ in the form of a power law $w_t^{-(1+\mu)}$. We provide a simple and physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic ($t \to \infty$) distribution of $w_t$ and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable ${1 \over \sqrt{t}}(log w_t -)$; 2) the necessary and sufficient conditions for $P(w_t)$ to be a power law are that < 0 (corresponding to a drift $w_t \to 0$) and that $w_t$ not be allowed to become too small. We discuss several models, previously unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable $\log w_t$ undergoes a random walk biased to the left but is bounded by a repulsive ''force''. We give an approximate treatment, which becomes exact for narrow or log-normal distributions of $\lambda$, in terms of the Fokker-Planck equation. 3) For all these models, the exponent $\mu$ is shown exactly to be the solution of $\langle \lambda^{\mu} \rangle = 1$ and is therefore non-universal and depends on the distribution of $\lambda$.Comment: 19 pages, Latex, 4 figures available on request from [email protected]