Profile

In a nutshell:
I am interested in everything random.
It is fascinating to see how our everyday life is penetrated by uncertainty and randomness and what we can do to turn random facts into certain formulae.
One of many examples is professional football in the premier league.

In my research I am, in particular, interested in the temporal and spatial evolution of randomness. This can be captured by the notion of a random or stochastic process (X_t)_{t>0}. Usually, we divide processes according to

• the underlying random distribution   (What is the probability law of X_t(ω)? Are we normal? Do we have fat tails?)
• the structure of their sample paths t → X_t(ω)   (Is it continuous? Does it have jumps?)
• the dimension of the parameter t   (t > 0 is time, but t could also be a geo-coordinate)
• the time-evolution (does the process have memory?)

I am most interested in Markovian jump processes, i.e. processes with discontinuous paths (representing sudden changes or shocks) without memory; in physics such processes are known as super-diffusions (since they spread faster than diffusing particles) or Lévy flights, and their probability distributions are, typically, non-normal – they fit to N.N. Taleb’s ““black swan” events ”.  In real life situations you will meet such behaviour in modelling stock and Forex prices, preying behaviour of animals or the spreading of US-Dollar notes.  

Apart from the existence of such processes, typical questions are the long- and short-time behaviour of the paths (does the process return or does it wander out to infinity? How often does it come back? Does it form patterns?), the regularity of the path (how smooth or how fractal is it?) etc. which is important for predictions and qualitative assessment.

What I’ve just described is often called “Stochastic Analysis”. The tools and methods used here are often taken from analysis, e.g. (stochastic) differential equations, semigroups, PDEs and pseudo differential operators to mention but a few, but they do have a probabilistic touch:
 

Rather than looking at an operator semigroup T_t f(x) we can “follow the path” of the process and understand the semigroup as E^x f(X_t).  This provides additional flexibility since we can choose t depending on the randomness ω adding a further degree of freedom. This allows us to relax the conditions on f  or to find solutions to boundary value problems.

Do not hesitate to contact me (or come to any of my courses) if you are interested in such problems – or if you need some help with any kind or random questions.
As I said at the beginning: I am interested in everything random.

2020 Mathematics Subject Classification (MSC 2020)

Key Words and Phrases

60G, 60H, 60J, 46E, 47D, 35S, 31C, 28A, 26A

Stochastic Process Markov Process Feller Process Lévy Process Jump process Path properties Stochastic Differential Equation Pseudo differential operator Integro-differential operator Negative definite symbol Infinitesimal generator Semigroup Laplace Transform Fourier Transform Stieltjes Transform Function spaces Besov Space Triebel-Lizorkin Space Sobolev Space Bernstein Function Completely Monotone Function Risk Uncertainty