arXiv subjects

Tomasz Kosmala

Publications and source records attributed to Tomasz Kosmala.

7 recordsLinked to original sources

arXiv2025

Optimal stopping with nonlinear expectation: geometric and algorithmic solutions

We use the geometry of suitably generalised potentials to solve risk-sensitive Markovian optimal stopping problems. As in the linear case due to Dynkin and Yushkievich (1967), the value function is the pointwise infimum of those functions which dominate the gain function. An emphasis is placed on geometric and pathwise arguments, rather than exploiting convexity, positive homogeneity or related analytical properties. An algorithm is provided to construct the value function at the computational cost of a two-dimensional search.

math.OC↗

arXiv2025

Weak solutions of Navier-Stokes Equation with purely discontinuous Lévy Noise

In this paper we prove the existence of weak martingale solutions to the stochastic Navier-Stokes Equations driven by pure jump Lévy processes. Our proof consists of two parts. In the first one, mostly classical, we recall a priori estimates, from the paper by the third named author, for solutions to suitable constructed Galerkin approximations and we use the Jakubowski-Skorokhod Theorem to find a sequence of processes on a new probability space convergent point-wise to a limit process. In the second one, we show that the limit process is a weak martingale solution to the SNSEs by using an approach of Kallianpur and Xiong. The core of this method consists of a proof that a certain natural process on the new probability space is a purely discontinuous martingale and then to use a suitable representation theorem. In this way we propose a method of proving solutions to stochastic PDEs which is different from the method used recently by the first and fourth named author in their joint paper with E.\ Hausenblas, see \cite{Brz+Haus+Raza_2018_reaction_diffusion}.

math.PR↗

arXiv2022

Markov risk mappings and risk-sensitive optimal prediction

We formulate a probabilistic Markov property in discrete time under a dynamic risk framework with minimal assumptions. This is useful for recursive solutions to risk-sensitive versions of dynamic optimisation problems such as optimal prediction, where at each stage the recursion depends on the whole future. The property holds for standard measures of risk used in practice, and is formulated in several equivalent versions including a representation via acceptance sets, a strong version, and a dual representation.

math.OC↗

arXiv2022

RangL: A Reinforcement Learning Competition Platform

The RangL project hosted by The Alan Turing Institute aims to encourage the wider uptake of reinforcement learning by supporting competitions relating to real-world dynamic decision problems. This article describes the reusable code repository developed by the RangL team and deployed for the 2022 Pathways to Net Zero Challenge, supported by the UK Net Zero Technology Centre. The winning solutions to this particular Challenge seek to optimize the UK's energy transition policy to net zero carbon emissions by 2050. The RangL repository includes an OpenAI Gym reinforcement learning environment and code that supports both submission to, and evaluation in, a remote instance of the open source EvalAI platform as well as all winning learning agent strategies. The repository is an illustrative example of RangL's capability to provide a reusable structure for future challenges.

cs.LG↗

arXiv2021

Stochastic evolution equations driven by cylindrical stable noise

We prove existence and uniqueness of a mild solution of a stochastic evolution equation driven by a standard $α$-stable cylindrical Lévy process defined on a Hilbert space for $α\in (1,2)$. The coefficients are assumed to map between certain domains of fractional powers of the generator present in the equation. The solution is constructed as a weak limit of the Picard iteration using tightness arguments. Existence of strong solution is obtained by a general version of the Yamada--Watanabe theorem.

math.PR↗

arXiv2019

Variational solutions of stochastic partial differential equations with cylindrical Lévy noise

In this article, the existence of a unique solution in the variational approach of the stochastic evolution equation $$\dX(t) = F(X(t)) \dt + G(X(t)) \dL(t)$$ driven by a cylindrical Lévy process $L$ is established. The coefficients $F$ and $G$ are assumed to satisfy the usual monotonicity and coercivity conditions. The noise is modelled by a cylindrical Lévy processes which is assumed to belong to a certain subclass of cylindrical Lévy processes and may not have finite moments.

math.PR↗

arXiv2019

Stochastic integration with respect to cylindrical Lévy processes by p-summing operators

We introduce a stochastic integral with respect to cylindrical Lévy processes with finite $p$-th weak moment for $p\in [1,2]$. The space of integrands consists of $p$-summing operators between Banach spaces of martingale type $p$. We apply the developed integration theory to establish the existence of a solution for a stochastic evolution equation driven by a cylindrical Lévy process.

math.PR↗