Oxford-Man Institute

Abstract:   Consider the square $[0,n]^2$ with points from a Poisson point process of intensity 1 distributed within it. In a seminal work, Baik, Deift and Johansson proved that the number of points $L_n$ (length) on a maximal increasing path (an increasing path that contains the most number of points), when properly centered and scaled, converges to the Tracy-Widom distribution. Later Johansson showed that all maximal paths lie within the strip of width $n^{\frac{2}{3} +\epsilon}$ around the diagonal with probability tending to 1 as $n \to \infty$. We shall discuss recent work on the Gaussian behaviour of the length $L_n^{(\gamma)}$ of a maximal increasing path restricted to lie within a strip of width $n^{\gamma}, \gamma< \frac{2}{3}$.

 

TBC

We consider a controlled system, in which an input $X: [0, T] \rightarrow E:= \mathbb{R}^{d}$ is a continuous but potentially highly oscillatory path and the corresponding output $Y$ is the line integral along $X$, for some unknown function $f: E \rightarrow E$. The rough paths theory provides a general framework to answer the question on which mild condition of $X$ and $f$, the integral $I(X)$ is well defined. It is robust enough to allow to treat stochastic integrals in a deterministic way. In this paper we are interested in identification of controlled systems of this type. The difficulty comes from the high dimensionality caused by the input of a function type. We propose novel adaptive and non-parametric algorithms to learn the functional relationship between the  input and the output from the data by carefully choosing the feature set of paths based on the rough paths theory and applying linear regression techniques. The algorithms is demonstrated on a financial application where the task is to predict the P$\&$L of the unknown trading strategy.

Abstract:In the first part of the talk, using classical hypergeometric identities, I will compute the harmonic measure of finite intervals and their complementaries for the Lévy stable process on the line. This gives a simple and unified proof of several results by Blumenthal-Getoor-Ray, Rogozin, and Kyprianou-Pardo-Watson. In the second part of the talk, I will consider the two-dimensional Markov process based on the stable Lévy process and its area process. I will give two explicit formulae for the harmonic measure of the split complex plane. These formulae allow to compute the persistence exponent of the stable area process, solving a problem raised by Zhan Shi. This is based on two joint works with Christophe Profeta.

Abstract: It has been recently shown that rough volatility models reproduce very well the statistical properties of low frequency financial data. In such models, the volatility process is driven by a fractional Brownian motion with Hurst parameter of order 0.1. The goal of this talk is to explain how such fractional dynamics can be obtained from the behaviour of market participants at the microstructural scales.

Using limit theorems for Hawkes processes, we show that a rough volatility naturally arises in the presence of high frequency trading combined with metaorders splitting. This is joint work with Thibault Jaisson.

We provide a result on prospect theory decision makers who are naïve about the time inconsistency induced by probability weighting. If a market offers a sufficiently rich set of investment strategies, investors postpone their trading decisions indefinitely due to a strong preference for skewness. We conclude that probability weighting in combination with naïveté leads to unrealistic predictions for a wide range of dynamic setups. Finally, I discuss recent work on the topic that invokes different assumptions on the dynamic modeling of prospect theory.

I prove that if markets are weak-form efficient, meaning current prices fully reflect all information available in past prices, then P = NP, meaning every computational problem whose solution can be verified in polynomial time can also be solved in polynomial time. I also prove the converse by showing how we can "program" the market to solveNP-complete problems. Since P probably does not equal NP, markets are probably not efficient. Specifically, markets become increasingly inefficient as the time series lengthens or becomes more frequent. An illustration by way of partitioning the excess returns to momentum strategies based on data availability confirms this prediction.

For more info please visit: http://philipmaymin.com/academic-papers#pnp

The model of random interlacements is a one-parameter family of random subsets of $\Z^d$, which locally describes the trace of a simple random walk on a $d$-dimensional torus running up to time $u$ times its volume. Here, $u$ serves as an intensity parameter.

Its complement, the so-called vacant set, has been show to undergo a non-trivial percolation phase transition in $u$, i.e., there is $u_*(d)\in (0,\infty)$ such that for all $u<u_*(d)$ the vacant set has a unique infinite connected component (supercritical phase), while for $u>u_*(d)$ all connected components are finite.

So far all results regarding geometric properties of this infinite connected component have been proven under the assumption that $u$ is close to zero. 

I will discuss a recent result, which states that throughout most of the supercritical phase simple random walk on the infinite connected component is transient, provided that the dimension is high enough.

This is joint work with Alexander Drewitz

Abstract: L\'evy processes are increasingly popular for modelling stochastic process data with jump behaviour. In practice statisticians only observe discretely sampled increments of the process, leading to a statistical inverse problem. To understand the jump behaviour of the process one needs to make inference on the infinite-dimensional parameter given by the L\'evy measure. We discuss recent developments in the analysis of this problem, including in particular functional limit theorems for commonly used estimators of the generalised distribution function of the L\'evy measure, and their application to statistical uncertainty quantification methodology (confidence bands and tests). 

We consider the problem of minimising the commute or shuttle time for a diffusion between the endpoints of an interval. The control is the scale function for the diffusion. We show that the dynamic version of the problem has the same solution as the static version if we start at an end point and consider the much harder case where the starting point is in the interior.