Talks

Andrea Macrina (UCL): Informed Martingale Optimal Transport

 We introduce randomized arcade processes (RAPs), a class of stochastic processes that interpolate in a strong sense through a finite sequence of random variables on the whole probability space. The filtrations generated by such processes are utilized to construct so-called filtered arcade martingales (FAMs) that interpolate between the given target random variables. These martingales are almost-sure solutions to the martingale interpolation problem and reveal an underlying stochastic filtering structure. In the special case of conditionally Markov RAPs, the dynamics of FAMs are informed through Bayesian updating. FAMs can be connected to martingale optimal transport (MOT) by considering optimally coupled target random variables. FAMs allow for the formulation of an information-based martingale optimal transport problem (IB-MOT) that enables the introduction of noise in MOT, in a similar fashion to how Schrödinger's problem introduces noise in optimal transport. The information-based transport problem is concerned with selecting an optimal martingale coupling for the target random variables under the influence of the noise that is generated by an arcade process.

Youness Boutaib (Liverpool):  The separation capacity of linear reservoirs with random connectivity matrix (Slides)

Recurrent neural networks (RNNs) constitute the simplest machine learning paradigm that is able to handle variable-length data sequences while tracking long-term dependencies and taking into account the temporal order of the received information. Reservoir computing (i.e. randomly choosing the connectivity matrix of the RNN) is a paradigm based on the idea that universal approximation properties can be achieved for several dynamical systems without the need to optimise all parameters. This technique simplifies the training of RNNs and has shown exceptional performance in a variety of tasks. Despite this, there is a fundamental lack in the mathematical understanding of the success of such approach. In this work, we explain this success by the separation capacity of such random reservoirs and discuss how the parameters of the problem (dimension of the reservoir, geometry of the classes of time-series, the choice of the probability distribution, etc.) impact the performance of the architecture. This is based on an ongoing work.

Ayse Sevtap Kestel (Middle East Technical University): Stop-loss reinsurance pricing and exposure curves under jump influence

The pricing in the stop-loss contracts is an important consideration of insurer and reinsurer. Based on historical loss amounts a stochastic model with the time-varying parameters to capture the time-dependent structure is developed. The analytical derivations of costs associated with reinsurance contract for reinsurer and insurer with constraints on time, loss amount, retention, and both retention and cap levels are made by including the extreme event influences which cause jumps in the course of the claim payments. Along with these, the analytical forms of exposure curves are derived for determining the premium share between reinsurer and insurer under prescribed constraints. An illustrative case study is given at which the calibration of time-varying parameters is made using dynamic maximum likelihood estimator The findings depict that implementation of a stochastic model with jump parameters improves the prediction power and ascertains a fair risk share between insurer and reinsurer.

(Joint work with Dr. M.Ozenc Mert)

Jan Palczewski (Leeds): Cancellable American put option with negative discounting

Cancellable American options, also known as game options or Israeli options, are American-style derivatives which give the writer the right to terminate the contract for a fixed penalty. I will talk about perpetual cancellable American put options on an asset whose dynamics follow exponential spectrally negative Levy process. The price and optimal strategies of the buyer and of the writer can be deduced from the solution of a corresponding Dynkin game. The new feature of the model is the negative interest rate which brings in difficulties (the payoff grows exponentially fast in time) and interesting strategies. We employ fully probabilistic arguments to argue the existence of the value and of the optimal strategies and characterise explicitly their form. We also prove smooth fit at boundaries of stopping sets enabling their numerical identification.

Lewis Ramsden (York): Exit Times for a Markov Modulated Random Walk with Applications in Risk Theory (Slides)

In this talk, we will discuss exit problems for general upward skip-free Markov additive chains (MACs) or Markov-modulated random walks. In particular, we will construct and characterise the so-called fundamental matrices $\widetilde{\bold{G}}$, $\widetilde{\bold{W}}(\cdot)$ and $\widetilde{\bold{Z}}(\cdot)$, demonstrating how these quantities can be used to derive standard exit problems and other fluctuation identities. The theory developed in this discrete setup is chosen to echo those of the theory for continuous-time Markov additive processes (MAPs), which allow us to identify the probabilistic construction, generating function and simple recursion relations for these matrices, as well as their connection to the so-called occupation mass functions.

In the second part of the talk, we will use these tools to develop the Gerber-Shiu theory for the classic and dual discrete risk processes in a regime-switching environment, including the associated constant dividend barrier problems. 

Teodor Holland (Leeds): Well-posedness of the stochastic reaction-diffusion equation with distributional drift and multiplicative noise

We consider the stochastic reaction-diffusion equation in $1+1$ dimensions driven by multiplicative space-time white noise

with distributional drift $b$ belonging in the negative Besov-H\"older space $C^\alpha$ with $\alpha>-1$.

We assume that the diffusion coefficient $\sigma$ is sufficiently regular and nondegenerate.

By using a combination of stochastic sewing and Malliavin calculus, we show that the equation admits a unique strong solution. As a by-product we also obtain quantitative bounds for  the density  of the multiplicative stochastic heat equation and its derivatives.

Noah Beelders (Liverpool): Spectrally Negative Lévy Processes Refracted at Poissonian Arrival Times

In this paper we develop the  fluctuation theory  for a spectrally negative Lévy process refracted at Poissonian arrival times.  By deriving some new generalised scale functions we compute the resolvent measure and derive fluctuations identities for the two-sided exit problem. To show the applicability of our results, we derive an explicit expression for the ruin probability in a surplus process with delayed implementation in the dividend payments.

Shijie Xu (Liverpool): A sharp upper bound for the expected occupation density of Itô processes with bounded irregular drift and diffusion coefficients

We find explicit and optimal upper bounds for the expected occupation density for an Itô-process when its drift and diffusion coefficients are unknown under boundedness and ellipticity conditions on the coefficients. This is related to the optimal bound for the expected interval occupation found in Ankirchner and Wendt(2021). In contrast, our bound is for a single point and the resulting formula is less involved. Our findings allow us to find explicit upper bounds for mean path integrals.

Jacob Smith (Leeds): A Martingale Theory for Dynkin Games with Asymmetric Information

Dynkin games are a natural formulation of a two-player optimal stopping problem, players compete against each other with one maximising a payoff and the other minimising it. A large body of work exists for these games—showing the existence of a value and of a Nash equilibrium in stopping times—however, we study these games under the presence of information asymmetry, i.e. where players know differing amounts about the game. This discrepancy in information in our set-up comes from an exogenous random variable, whose outcome determines the payoff processes. Here we will formulate the game, introduce randomised strategies, and then present a general result providing us with both necessary and sufficient conditions for the value and Nash equilibrium.