Harald Oberhauser

Regularity Theory for Rough Partial Differential Equations and Parabolic Comparison Revisited

Partial differential equations driven by rough paths are studied. We return to the investigations of [Caruana, Friz and Oberhauser: A (rough) pathwise approach to a class of non- linear SPDEs, Annales de l’Institut Henri Poincare/Analyse Non Lineaire 2011, 28, pp. 27–46], motivated by the Lions–Souganidis theory of viscosity solutions for SPDEs. We continue and complement the previous (uniqueness) results with general existence and regularity statements. Much of this is transformed to questions for deterministic parabolic partial differential equations in viscosity sense. On a technical level, we establish a refined parabolic theorem of sums which may be useful in its own right.

A feature set for streams and an application to high-frequency financial tick data

We propose a set of features to study the effects of data streams on complex systems. This feature set is called the the signature representation of a stream. It has its origin in pure mathematics and relies on a relationship between non-commutative polynomials and paths. This representation had already signifcant impact on algebraic topology, control theory, numerics for PDEs, stochastic analysis and the theory of rough paths; more recently first steps have been taken to apply such methods to the study of big data streams. We show that the signature representation can provide an efficient summary of a stream and its effects. We then show that it can be combined with standard tools from machine learning. After introducing the signature for streams and some theoretical background, we apply this approach to a challenging real-world example: high-frequency financial data streams. In this context, the streams are tick-by-tick market data of a stock traded at the New York stock exchange NYSE and the effect of the stream is the profit and loss of complex investment strategies (i.e. a nonlinear functional of the stream). Our numerical results (applied to Thomson--Reuters tick data of several full trading days for IBM stocks) show that the signature of the price stream efficiently captures the necessary information to learn the return of an investment strategy. However, we emphasize that the underlying ideas are not limited to financial data streams and have the potential to be applied to many other areas in data mining where the non-commutative nature of streams is of importance, like text mining, bioinformatics or click history.

The functional Itō formula under the family of continuous semimartingale measures

Dupire [16] introduced a notion of smoothness for functionals of paths and arrived at a generalization of Itō’s formula that applies to functionals with a continuous dependence on the trajectories of the underlying process. In this paper, we study nonlinear functionals that do not have such continuity. By revisiting old work of Bichteler and Karandikar we show that one can construct pathwise versions of complex functionals like the quadratic variation, stochastic integrals or Itō processes that are still regular enough such that a functional Itō-formula applies.

Splitting Methods for SPDEs: From Robustness to Financial Engineering, Optimal Control, and Nonlinear Filtering

In this survey chapter we give an overview of recent applications of the splitting method to stochastic (partial) differential equations, that is, differential equations that evolve under the influence of noise. We discuss weak and strong approximations schemes. The applications range from the management of risk, financial engineering, optimal control, and nonlinear filtering to the viscosity theory of nonlinear SPDEs.

Robustness in Stochastic Filtering and Maximum Likelihood Estimation for SDEs

We consider complex stochastic systems in continuous time and space where the objects of interest are modelled via stochastic differential equations, in general high dimensional and with nonlinear coefficients. The extraction of quantifiable information from such systems has a long history and many aspects. We shall focus here on the perhaps most classical problems in this context: the filtering problem for nonlinear diffusions and the problem of parameter estimation, also for nonlinear and multidimensional diffusions. More specifically, we return to the question of robustness, first raised in the filtering community in the mid-1970s: will it be true that the conditional expectation of some observable of the signal process, given an observation (sample) path, depends continuously on the latter? Sadly, the answer here is no, as simple counterexamples show. Clearly, this is an unhappy state of affairs for users who effectively face an ill-posed situation: close observations may lead to vastly different predictions. A similar question can be asked in the context of (maximum likelihood) parameter estimation for diffusions. Some (apparently novel) counter examples show that, here again, the answer is no. Our contribution (Crisan et al., Ann Appl Probab 23(5):2139–2160, 2013); Diehl et al., A Levy-area between Brownian motion and rough paths with applications to robust non-linear filtering and RPDEs (2013, arXiv:1301.3799; Diehl et al., Pathwise stability of likelihood estimators for diffusions via rough paths (2013, arXiv:1311.1061) changed to yes, in other words: well-posedness is restored, provided one is willing or able to regard observations as rough paths in the sense of T. Lyons.