We introduce a class of Markov processes, called $m$-polynomial, for which the calculation of (mixed) moments up to order $m$ only requires the computation of matrix exponentials. This class contains affine processes, processes with quadratic diffusion coefficients, as well as Lévy-driven SDEs with affine vector fields. Thus, many popular models such as exponential Lévy models or affine models are covered by this setting. The applications range from statistical GMM estimation procedures to new techniques for option pricing and hedging. For instance, the efficient and easy computation of moments can be used for variance reduction techniques in Monte Carlo methods.
Using Dupire's notion of vertical derivative, we provide a functional (path-dependent) extension of the Itô's formula of Gozzi and Russo (2006) that applies to C^{0,1}-functions of continuous weak Dirichlet processes. It is motivated and illustrated by its applications to the hedging or superhedging problems of path-dependent options in mathematical finance, in particular in the case of model uncertainty
In this paper we provide an exhaustive survey of the current state of the mathematics of filtration enlargement and an interpretation of the key results of the literature from the viewpoint of mathematical finance. The emphasis is on providing a well-structured compendium of known mathematical results that can be used by researchers in mathematical finance. We mainly state the results and discuss their role and significance, with references provided for the omitted proofs. The discussion of mathematical results is accompanied by numerous examples from mathematical finance.
Stochastic integrals are defined with respect to a collection $P = (P_i; \, i \in I)$ of continuous semimartingales, imposing no assumptions on the index set $I$ and the subspace of $\mathbb{R}^I$ where $P$ takes values. The integrals are constructed though finite-dimensional approximation, identifying the appropriate local geometry that allows extension to infinite dimensions. For local martingale integrators, the resulting space $\mathsf{S} (P)$ of stochastic integrals has an operational characterisation via a corresponding set of integrands $\mathsf{R} (C)$, constructed with only reference …
Assume that an agent models a financial asset through a measure Q with the goal to price / hedge some derivative or optimize some expected utility. Even if the model Q is chosen in the most skilful and sophisticated way, she is left with the possibility that Q does not provide an "exact" description of reality. This leads us to the following question: will the hedge still be somewhat meaningful for models in the proximity of Q? If we measure proximity with the usual Wasserstein distance (say), the answer is NO. Models which are similar w.r.t. Wasserstein distance may provide dramatically diffe…
Robust models in mathematical finance replace the classical single probability measure by a sufficiently rich set of probability measures on the future states of the world to capture (Knightian) uncertainty about the "right" probabilities of future events. If this set of measures is nondominated, many results known from classical dominated frameworks cease to hold as probabilistic and analytic tools crucial for the handling of dominated models fail. We investigate the consequences for the robust model when prominent results from the mathematical finance literature are postulate. In this vein, …
The paper introduces a simple way of recording and manipulating general stochastic processes without explicit reference to a probability measure. In the new calculus, operations traditionally presented in a measure-specific way are instead captured by tracing the behaviour of jumps (also when no jumps are physically present). The calculus is fail-safe in that, under minimal assumptions, all informal calculations yield mathematically well-defined stochastic processes. The calculus is also intuitive as it allows the user to pretend all jumps are of compound Poisson type. The new calculus is very…
We introduce a class of measure-valued processes, which -- in analogy to their finite dimensional counterparts -- will be called measure-valued polynomial diffusions. We show the so-called moment formula, i.e.~a representation of the conditional marginal moments via a system of finite dimensional linear PDEs. Furthermore, we characterize the corresponding infinitesimal generators and obtain a representation analogous to polynomial diffusions on $\mathbb{R}^m_+$, in cases where their domain is large enough. In general the infinite dimensional setting allows for richer specifications strictly be…
