arXiv subjects

Jan Palczewski

Publications and source records attributed to Jan Palczewski.

At least 19 recordsLinked to original sources

arXiv2026

An Euler scheme for McKean SDEs with Besov drift: convergence rate and implementation

We study a one-dimensional McKean-Vlasov stochastic differential equation (SDE) with a drift equal to a product of a distribution depending on the state of the process and a non-linear function depending pointwise on the law density of the solution. Building on recent well-posedness results, we propose the first implementable numerical scheme for this class of SDEs. Our approach combines mollification of the distributional drift with the Euler-Maruyama scheme and a PDE-based approximation of the law via the associated Fokker-Planck equation. We prove strong convergence of the scheme and derive an explicit rate, showing how to balance the smoothing parameter with the time discretisation. Numerical experiments confirm the applicability of our scheme and demonstrate the significant influence of the McKean interaction term on the law of the solution.

math.PR↗

arXiv2025

Martingale theory for Dynkin games with asymmetric information

This paper provides necessary and sufficient conditions for a pair of randomised stopping times to form a saddle point of a zero-sum Dynkin game with partial and/or asymmetric information across players. The framework is non-Markovian and covers essentially any information structure. Our methodology relies on the identification of suitable super and submartingales involving players' equilibrium payoffs. Saddle point strategies are characterised in terms of the dynamics of those equilibrium payoffs and are related to their Doob-Meyer decompositions.

math.PR↗

arXiv2025

A new discrimination measure for assessing predictive performance of non-linear survival models

Non-linear survival models are flexible models in which the proportional hazard assumption is not required. This poses difficulties in their evaluation. We introduce a new discrimination measure, time-dependent Uno's C-index, to assess the discrimination performance of non-linear survival models. This is an unbiased version of Antolini's time-dependent concordance. We prove convergence of both measures employing Nolan and Pollard's results on U-statistics. We explore the relationship between these measures and, in particular, the bias of Antolini's concordance in the presence of censoring using simulated data. We demonstrate the value of time-dependent Uno's C-index for the evaluation of models trained on censored real data and for model tuning.

stat.ME↗

arXiv2025

Exact Bayesian inference for Markov switching diffusions

We develop the first exact Bayesian methodology for the problem of inference in discretely observed regime switching diffusions. Switching diffusion models extend ordinary diffusions by allowing for jumps in instantaneous drift and volatility. The jumps are driven by a latent, continuous time Markov switching process. We address the problem through an MCMC and an MCEM algorithm that target the exact posterior of diffusion parameters and the latent regime process. The algorithms are exact in the sense that they target the correct posterior distribution of the continuous model, so that the errors are due to Monte Carlo only. We illustrate the method on numerical examples, including an empirical analysis of the method's scalability in the length of the time series, and find that it is comparable in computational cost with discrete approximations while avoiding their shortcomings.

stat.CO↗

arXiv2023

Stopper vs. singular-controller games with degenerate diffusions

We study zero-sum stochastic games between a singular controller and a stopper when the (state-dependent) diffusion matrix of the underlying controlled diffusion process is degenerate. In particular, we show the existence of a value for the game and determine an optimal strategy for the stopper. The degeneracy of the dynamics prevents the use of analytical methods based on solution in Sobolev spaces of suitable variational problems. Therefore we adopt a probabilistic approach based on a perturbation of the underlying diffusion modulated by a parameter $\gamma>0$. For each $\gamma>0$ the approximating game is non-degenerate and admits a value $u^\gamma$ and an optimal strategy $\tau^\gamma_*$ for the stopper. Letting $\gamma\to 0$ we prove convergence of $u^\gamma$ to a function $v$, which identifies the value of the original game. We also construct explicitly optimal stopping times $\theta^\gamma_*$ for $u^\gamma$, related but not equal to $\tau^\gamma_*$, which converge almost surely to an optimal stopping time $\theta_*$ for the game with degenerate dynamics.

math.OC↗

arXiv2023

Convergence rate of numerical scheme for SDEs with a distributional drift in Besov space

This paper is concerned with numerical solutions of one-dimensional SDEs with the drift being a generalised function, in particular belonging to the H\"older-Zygmund space $C^{-\gamma}$ of negative order $-\gamma<0$ in the spatial variable. We design an Euler-Maruyama numerical scheme and prove its convergence, obtaining an upper bound for the strong $L^1$ convergence rate. We finally implement the scheme and discuss the results obtained.

math.PR↗

arXiv2023

Zero-sum stopper vs. singular-controller games with constrained control directions

We consider a class of zero-sum stopper vs. singular-controller games in which the controller can only act on a subset $d_0<d$ of the $d$ coordinates of a controlled diffusion. Due to the constraint on the control directions these games fall outside the framework of recently studied variational methods. In this paper we develop an approximation procedure, based on $L^1$-stability estimates for the controlled diffusion process and almost sure convergence of suitable stopping times. That allows us to prove existence of the game's value and to obtain an optimal strategy for the stopper, under continuity and growth conditions on the payoff functions. This class of games is a natural extension of (single-agent) singular control problems, studied in the literature, with similar constraints on the admissible controls.

math.OC↗

arXiv2022

A Bayesian Mixture Model Approach to Expected Possession Values in Rugby League

The aim of this study was to improve previous zonal approaches to expected possession value (EPV) models in low data availability sports by introducing a Bayesian Mixture Model approach to an EPV model in rugby league. 99,966 observations from the 2021 Super League season were used. A set of 33 centres (30 in the field of play, 3 in the try area) were located across the pitch. Each centre held the probability of five possession outcomes occurring (converted/unconverted try, penalty, drop goal and no points). Weights for the model were provided for each location on the pitch using linear and bilinear interpolation techniques. Probabilities at each centre were estimated using a Bayesian approach and extrapolated to all locations on the pitch. An EPV measure was derived from the possession outcome probabilities and their points value. The model produced a smooth pitch surface, which was able to provide different possession outcome probabilities and EPVs for every location on the pitch. Differences between team attacking and defensive plots were visualised and an actual vs expected player rating system was developed. The model provides significantly more flexibility than previous approaches and could be adapted to other sports where data is similarly sparse.

stat.AP↗

arXiv2022

Exit game with private information

The timing of strategic exit is one of the most important but difficult business decisions, especially under competition and uncertainty. Motivated by this problem, we examine a stochastic game of exit in which players are uncertain about their competitor's exit value. We construct an equilibrium for a large class of payoff flows driven by a general one-dimensional diffusion. In the equilibrium, the players employ sophisticated exit strategies involving both the state variable and the posterior belief process. These strategies are specified explicitly in terms of the problem data and a solution to an auxiliary optimal stopping problem. The equilibrium we obtain is further shown to be unique within a wide subclass of symmetric Bayesian equilibria.

math.OC↗

arXiv2022

Use of Kernel Density Estimation to understand the spatial trends of attacking possessions in rugby league

Despite having the potential to provide significant insights into tactical preparations for future matches, very few studies have considered the spatial trends of team attacking possessions in rugby league. Those which have considered these trends have used grid based aggregation methods, which provide a discrete understanding of rugby league match play but may fail to provide a complete understanding of the spatial trends of attacking possessions due to the dynamic nature of the sport. In this study, we use Kernel Density Estimation (KDE) to provide a continuous understanding of the spatial trends of attacking possessions in rugby league on a team by team basis. We use the Wasserstein distance to understand the differences between teams (i.e. using all of each team's data) and within teams (i.e. using a single team's data against different opponents). Our results show that KDEs are able to provide interesting tactical insights at the between team level. Furthermore, at the within team level, the results are able to show patterns of spatial trends for attacking teams, which are present against some opponents but not others. The results could help sports practitioners to understand opposition teams' previous performances and prepare tactical strategies for matches against them.

stat.AP↗

arXiv2021

On the continuity of optimal stopping surfaces for jump-diffusions

We show that optimal stopping surfaces $(t,y)\mapsto x_*(t,y)$ arising from time-inhomogeneous optimal stopping problems on two-dimensional jump-diffusions $(X,Y)$ are continuous (jointly in time and space) under mild monotonicity and regularity assumptions of local nature.

math.PR↗

arXiv2021

Automatic model training under restrictive time constraints

We develop a hyperparameter optimisation algorithm, Automated Budget Constrained Training (AutoBCT), which balances the quality of a model with the computational cost required to tune it. The relationship between hyperparameters, model quality and computational cost must be learnt and this learning is incorporated directly into the optimisation problem. At each training epoch, the algorithm decides whether to terminate or continue training, and, in the latter case, what values of hyperparameters to use. This decision weighs optimally potential improvements in the quality with the additional training time and the uncertainty about the learnt quantities. The performance of our algorithm is verified on a number of machine learning problems encompassing random forests and neural networks. Our approach is rooted in the theory of Markov decision processes with partial information and we develop a numerical method to compute the value function and an optimal strategy.

stat.ML↗

arXiv2021

The American put with finite-time maturity and stochastic interest rate

In this paper we study pricing of American put options on the Black and Scholes market with a stochastic interest rate and finite-time maturity. We prove that the option value is a $C^1$ function of the initial time, interest rate and stock price. By means of Ito calculus we rigorously derive the option value's early exercise premium formula and the associated hedging portfolio. We prove the existence of an optimal exercise boundary splitting the state space into continuation and stopping region. The boundary has a parametrisation as a jointly continuous function of time and stock price, and it is the unique solution to an integral equation which we compute numerically. Our results hold for a large class of interest rate models including CIR and Vasicek models. We show a numerical study of the option price and the optimal exercise boundary for Vasicek model.

q-fin.MF↗

arXiv2020

On the value of non-Markovian Dynkin games with partial and asymmetric information

We prove that zero-sum Dynkin games in continuous time with partial and asymmetric information admit a value in randomised stopping times when the stopping payoffs of the players are general \cadlag measurable processes. As a by-product of our method of proof we also obtain existence of optimal strategies for both players. The main novelties are that we do not assume a Markovian nature of the game nor a particular structure of the information available to the players. This allows us to go beyond the variational methods (based on PDEs) developed in the literature on Dynkin games in continuous time with partial/asymmetric information. Instead, we focus on a probabilistic and functional analytic approach based on the general theory of stochastic processes and Sion's min-max theorem (M. Sion, Pacific J. Math., 8, 1958, pp. 171-176). Our framework encompasses examples found in the literature on continuous time Dynkin games with asymmetric information and we provide counterexamples to show that our assumptions cannot be further relaxed.

math.PR↗

arXiv2020

Optimal hedging of a perpetual American put with a single trade

It is well-known that using delta hedging to hedge financial options is not feasible in practice. Traders often rely on discrete-time hedging strategies based on fixed trading times or fixed trading prices (i.e., trades only occur if the underlying asset's price reaches some predetermined values). Motivated by this insight and with the aim of obtaining explicit solutions, we consider the seller of a perpetual American put option who can hedge her portfolio once until the underlying stock price leaves a certain range of values $(a,b)$. We determine optimal trading boundaries as functions of the initial stock holding, and an optimal hedging strategy for a bond/stock portfolio. Optimality here refers to the variance of the hedging error at the (random) time when the stock leaves the interval $(a,b)$. Our study leads to analytical expressions for both the optimal boundaries and the optimal stock holding, which can be evaluated numerically with no effort.

q-fin.MF↗

arXiv2019

Statistical Learning for Probability-Constrained Stochastic Optimal Control

We investigate Monte Carlo based algorithms for solving stochastic control problems with probabilistic constraints. Our motivation comes from microgrid management, where the controller tries to optimally dispatch a diesel generator while maintaining low probability of blackouts. The key question we investigate are empirical simulation procedures for learning the admissible control set that is specified implicitly through a probability constraint on the system state. We propose a variety of relevant statistical tools including logistic regression, Gaussian process regression, quantile regression and support vector machines, which we then incorporate into an overall Regression Monte Carlo (RMC) framework for approximate dynamic programming. Our results indicate that using logistic or Gaussian process regression to estimate the admissibility probability outperforms the other options. Our algorithms offer an efficient and reliable extension of RMC to probability-constrained control. We illustrate our findings with two case studies for the microgrid problem.

math.OC↗

arXiv2018

On the solution uniqueness in portfolio optimization and risk analysis

We consider the issue of solution uniqueness for portfolio optimization problem and its inverse for asset returns with a finite number of possible scenarios. The risk is assessed by deviation measures introduced by [Rockafellar et al., Mathematical Programming, Ser. B, 108 (2006), pp. 515-540] instead of variance as in the Markowitz optimization problem. We prove that in general one can expect uniqueness neither in forward nor in inverse problems. We discuss consequences of that non-uniqueness for several problems in risk analysis and portfolio optimization, including capital allocation, risk sharing, cooperative investment, and the Black-Litterman methodology. In all cases, the issue with non-uniqueness is closely related to the fact that subgradient of a convex function is non-unique at the points of non-differentiability. We suggest methodology to resolve this issue by identifying a unique "special" subgradient satisfying some natural axioms. This "special" subgradient happens to be the Stainer point of the subdifferential set.

q-fin.PM↗

arXiv2017

Regress-Later Monte Carlo for optimal control of Markov processes

We develop two Regression Monte Carlo algorithms (value and performance iteration) to solve general problems of optimal stochastic control of discrete-time Markov processes. We formulate our method within an innovative framework that allow us to prove the speed of convergence of our numerical schemes. We rely on the Regress Later approach unlike other attempts which employ the Regress Now technique. We exploit error bounds obtained in our proofs, along with numerical experiments, to investigate differences between the value and performance iteration approaches. Introduced in Tsitsiklis and VanRoy [2001] and Longstaff and Schwartz [2001] respectively, their characteristics have gone largely unnoticed in the literature; we show however that their differences are paramount in practical solution of stochastic control problems. Finally, we provide some guidelines for the tuning of our algorithms.

math.OC↗