Poster session 2
Tuesday, 7/15, 12:00-13:00, Congress Centre
P16: Pricing Mortgage-Backed Securities with the Hull-White PDE
Michał Wronka (Wroclaw University of Science and Technology)
stand: S2
This work explores the pricing of a selected Mortgage-Backed Security (MBS) using a Partial Differential Equation (PDE) grid approach. We begin by establishing the theoretical foundation, deriving the pricing PDE for a zero-coupon bond within the Hull-White framework, following a methodology similar to the Black-Scholes model in option pricing. Building on this, we introduce the unique structural components of MBS contracts and implement necessary modifications to the pricing PDE to account for key risk factors, including the prepayment factor (), current coupon, and Option-Adjusted Spread (OAS). Further, we contextualize MBS pricing by incorporating volatility dynamics calibrated under the Hull-White model. A detailed examination of prepayment modeling follows, where we analyze consumer prepayment behavior, emphasizing the significance of the S-shaped prepayment curve, burnout effects, and turnover rates. The study culminates in the development of a comprehensive pricing framework that integrates all these factors, ultimately yielding a prepayment function that directly feeds into the PDE-based valuation model. By synthesizing theoretical derivations with practical risk considerations, this work provides a structured approach to MBS pricing, offering insights into the interplay between prepayment dynamics and PDE-based valuation techniques.
P17: Uncovering Data Symmetries: Estimating Covariance Matrix in High-Dimensional Setting With ’gips’ R Package
Adam Przemysław Chojecki (Warsaw University of Technology, Faculty of Mathematics and Information Science)
stand: S3
Estimating high-dimensional covariance matrices is challenging when the sample has limited size. We present gips, an R package that automatically detects permutation symmetries among variables in a Gaussian model and uses them to constrain covariance estimation. By Bayesian inference over possible symmetry partitions, gips identifies groups of interchangeable dimensions and enforces these constraints to produce a reduced-variance estimator. In simulations, it accurately recovers true symmetries and matches popular shrinkage methods in accuracy while providing additional interpretation via symmetry structures.
This poster is based on Chojecki et al. \([2]\), which presents the package in greater detail. gips is available on CRAN, complete with documentation \([3]\), and examples, enabling straightforward application in data mining, computational biology, and other fields requiring structured covariance estimation.
Bibliography
\([1]\) Piotr Graczyk, Hideyuki Ishi, Bartosz Kołodziejek, Hélène Massam. Model selection in the space of Gaussian models invariant by symmetry. The Annals of Statistics, vol. 50, no. 3, pp. 1747–1774, June 2022. arXiv: 2004.03503; DOI: 10.1214/22-AOS2174.
\([2]\) Chojecki, A., Morgen, P., Kołodziejek, B. (2025). Learning Permutation Symmetry of a Gaussian Vector with gips in R. Journal of Statistical Software, vol. 112, no. 7, pp. 1–38. DOI: 10.18637/jss.v112.i07.
\([3]\) gips package documentation: przechoj.github.io/gips.
P18: An asymptotically probabilistic method for a class of partial integrodifferential equations
Alioune Coulibaly (Université Amadou Mahtar Mbow de Diamniadio - Dakar - Sénégal)
stand: S4
In this paper, we consider a nonlocal boundary condition and examine the
asymptotic behavior of the solution to a family of nonlocal partial
differential equations in the half-space. Our approach is fully
probabilistic and builds upon the works of Huang et al. (2022) and
Diakhaby et al. (2016). Reflected stochastic differential equations,
driven by multiplicative Lévy noise and with singular coefficients, play
an important role in our method.
Let
\(\left( b,c,\sigma\right) :\mathrm{\textbf{D}^{3}\times\mathbb{R}^{d}} \longrightarrow\mathbb{R}^{d}\),
\(\left( \beta,\gamma,\varrho\right) :\mathrm{\partial\textbf{D}^{3}\times\mathbb{R}^{d-1}} \longrightarrow\mathbb{R}^{d}\),
and \(\varepsilon\) be small positive. Our principal focus is the
limit-solution, when \(\varepsilon\) goes to zero, of the following
nonlocal partial differential equation (PDE) with rapidly oscillating
coefficients : \[\label{eq1}
\left\lbrace
\begin{aligned}
&\frac{\partial u^{\varepsilon}}{\partial t}(t,x)=\mathcal{L}^{\sigma,b,c}_{\varepsilon}u^{\varepsilon}(t,x)+\frac{\scriptstyle 1}{\scriptstyle \varepsilon}g\big(x_{\varepsilon},u^{\varepsilon}(t,x) \big).u^{\varepsilon}(t,x) , &x\in\mathrm{\textbf{D}},\\
& \mathcal{L}^{\varrho,\beta,\gamma}_{\varepsilon} u^{\varepsilon}(t,x)+\frac{\scriptstyle 1}{\scriptstyle \varepsilon}h\big( x_{\varepsilon},u^{\varepsilon}(t,x)\big).u^{\varepsilon}(t,x) =0, &x\in\partial\mathrm{\textbf{D}},\\
&u^{\varepsilon}(0,x)=u_{0}(x), &x\in\overline{\mathrm{\textbf{D}}},
\end{aligned}
\right.\] where
\(\mathrm{\textbf{D}}=\big\{\left(x^{1},\ldots,x^{d} \right) \in\mathbb{R}^{d} : x^{1}>0\big\}\).
Definitively, the boundary \(\partial\mathrm{\textbf{D}}\) is supposed to
be homeomorphic to \(\mathbb{R}^{d-1}\). Letting set
\(x_{\varepsilon}:=\left(\frac{\scriptstyle x}{\scriptstyle \delta_{\varepsilon}}\right)_{\varepsilon>0},\ \delta_{\varepsilon}>0\),
the family of linear integro-differential operators
\(\mathcal{L}^{\varpi_{i},\tau_{i},\rho_{i}}_{\varepsilon}\) (\(i:=1,2\))
are given by \[\begin{split}
\mathcal{L}^{\varpi_{i},\tau_{i},\lambda_{i}}_{\varepsilon}f(x)
:=&
\int_{\mathbb{R}^{d+1-i}_{*}}\bigg[f\left( x+\varepsilon\varpi_{i}\left(x_{\varepsilon},y\right)\right) - f(x)
-\varepsilon\sum_{j=1}^{d}\varpi_{i}^{j}\left(x_{\varepsilon},y\right)\partial_{j}f(x)\bigg] \nu_{i}^{\varepsilon}(dy)\\%\boldsymbol{1}_{B} (y)
&+\left(\frac{\scriptstyle \varepsilon}{\scriptstyle \delta_{\varepsilon}}\right)^{\alpha-1}\sum_{j=1}^{d}\tau_{i}^{j}\left(x_{\varepsilon}\right)\partial_{j}f(x)+\sum_{j=1}^{d}\lambda_{i}^{j}\left(x_{\varepsilon}\right)\partial_{j}f(x),\ x\in\mathbb{R}^{d+1-i},
\end{split}\] with \(\varpi_{i}\) in
\(\bigl\{\sigma,\varrho\bigr\}\), \(\tau_{i}\) in \(\bigl\{b,\beta\bigr\}\)
and \(\lambda_{i}\) in \(\bigl\{c,\gamma\bigr\}\) respectively.
Theorem. \([\)Coulibaly (2025)\(]\)
Suppose assumptions (A\(_1\))–(A\(_6\)) hold true. Then, for every
\((t,x) \in\mathbb{R}^{*}_{+}\times\overline{\textrm{\textbf{D}}}\),
\[\lim_{\varepsilon\downarrow 0}\varepsilon\log u^{\varepsilon}(t,x)=V^{*}(t,x)=\inf_{\tau\in\Theta}\sup_{\left\{\phi\in \mathcal{D}\left(\left[0,t\right],\overline{\textrm{\textbf{D}}}\right),\phi(0)=x,\phi(t)\in U_{0}\right\}}\Big\{\overline{\Pi}\tau-S_{0,\tau}\left(\phi\right)\Big\}.\]
Corollary. \(\quad\) \[\lim_{\varepsilon\downarrow 0}u^{\varepsilon}(t,x)=\left\{\begin{array}{l} 0 \textrm{\textit{ from any compact}} \textrm{ of the set }\Big\{(t,x)\in\mathbb{R}^{*}_{+}\times\overline{\textbf{D}};\ V^{*}(t,x)< 0\Big\},\\ 1 \textrm{\textit{ from any compact}}\textrm{ of the set } \Big\{(t,x)\in\mathbb{R}^{*}_{+}\times\overline{\textbf{D}};\ V^{*}(t,x)=0\Big\}. \end{array}\right.\]
Bibliography
\([1]\) Diakhaby A., Ouknine Y. (2016). Generalized BSDEs, weak
convergence, and homogenization of semilinear PDEs with the
Wentzell-type boundary condition. Stoch. Anal. Appl., 34,
496–509.
\([2]\) Coulibaly A., (2025). An asymptotically probabilistic method for
a class of partial integrodifferential equations. AIMS Mathematics,
10(6): 13512–13523.
\([3]\) Huang Q., Duan J., Song R., (2022). Homogenization of nonlocal
partial differential equations related to stochastic differential
equations with Lévy noise, Bernoulli, 28, 1648–1674.
P19: Accounting for reporting delays in real-time phylodynamic analyses with preferential sampling
Volodymyr Minin (University of California, Irvine)
stand: S5
The COVID-19 pandemic demonstrated that fast and accurate analysis of continually collected infectious disease surveillance data is crucial for situational awareness and policy making. Coalescent-based phylodynamic analysis can use genetic sequences of a pathogen to estimate changes in its effective population size, a measure of genetic diversity. These changes in effective population size can be connected to the changes in the number of infections in the population of interest under certain conditions. Phylodynamics is an important set of tools because its methods are often resilient to the ascertainment biases present in traditional surveillance data (e.g., preferentially testing symptomatic individuals). Unfortunately, it takes weeks or months to sequence and deposit the sampled pathogen genetic sequences into a database, making them available for such analyses. These reporting delays severely decrease precision of phylodynamic methods closer to present time, and for some models can lead to extreme biases. Here we present a method that affords reliable estimation of the effective population size trajectory closer to the time of data collection, allowing for policy decisions to be based on more recent data. Our work uses readily available historic times between sampling and reporting of sequenced samples for a population of interest, and incorporates this information into the sampling model to mitigate the effects of reporting delay in real-time analyses. We illustrate our methodology on simulated data and on SARS-CoV-2 sequences collected in the state of Washington in 2021 \([1]\).
Coalescent models are continuous-time Markov chains used to model a genealogy from a sample of sequences \([2]\). Rodrigo et al. (1999) extended coalescent theory for heterochronous sampling \([3]\): \[\begin{split} P(\boldsymbol{g} | \boldsymbol{s}, \boldsymbol{n}, N_e(t)) &= \prod_{k = 2}^{n}P(t_{k-1} | t_k, \boldsymbol{s}, N_e(t)) \\ &= \prod_{k = 2}^{n} \frac{A_{0, k}}{N_e(t_{k-1})}\exp \biggl\{ -\int_{I_{0, k}} \frac{A_{0, k}}{N_e(t)}dt - \sum\limits_{i \geq 1} \int_{I_{i,k}} \frac{A_{i, k}}{N_e(t)} dt \biggr\}. \end{split}\] In the preferential sampling model, we model sampling events as a Poisson Process with intensity \(\lambda(t)\): \[\begin{split} \log\lambda(t) =& \beta_0 + \beta_1 \log[N_e(t)] + \beta_2 f_2(t) + ... + \beta_p f_p(t). \end{split}\] Altogether, the posterior we are interested in is \[Pr(\boldsymbol{\gamma}, \kappa, \boldsymbol{\beta} | \boldsymbol{g}, \boldsymbol{s}) \propto Pr(\boldsymbol{g} | \boldsymbol{\gamma}, \boldsymbol{s}) Pr(\boldsymbol{s} | \boldsymbol{\gamma}, \boldsymbol{\beta}) Pr(\boldsymbol{\gamma} | \kappa) Pr(\kappa) Pr(\boldsymbol{\beta}).\]
Bibliography
\([1]\) Medina, CM, Palacios JA, Minin VM. "Accounting for reporting delays in real-time phylodynamic analyses with preferential sampling." PLoS Computational Biology, vol. 21, no. 5, 2025, pp. e1012970.
\([2]\) Kingman JFC. "The coalescent." Stochastics Processes and Applications, vol. 13, no. 3, 1982, pp. 235–248.
\([3]\) Rodrigo AG, Felsenstein J. "Coalescent approaches." The Evolution of HIV. 1999, pp. 233–72.
P20: Asymptotic results for dynamic contagion processes with different exciting functions and application to risk models.
Shamiksha Shamiksha (Indian Institute of Technology Delhi)
stand: S6
Hawkes process is a particular class of point process (introduced by Hawkes \([2]\)) which is characterized by the self-exciting and/or multi-exciting property, i.e., the occurrence of future events is influenced by the past events. Initially, Hawkes and Oakes \([3]\) constructed the cluster form representation of the Hawkes process from a homogeneous Poisson process and used the same exciting function for the production of subsequent offspring. These processes are widely used in several fields including seismology, finance, DNA modeling, neuroscience, queuing systems and many others.
In particular, the self-exciting property, and multi-exciting property property of the Hawkes processes provide a consistent model for examining the clustering of defaults during financial crises. However, defaults are also subject to external influences, thus in this regards, a new point process has been introduced by Dassios and Zhao \([1]\) known as the dynamic contagion process, which combines the characteristics of both self-excited and externally excited jumps.
In Pandey et al. \([4]\), we introduced an extension of the cluster representation of dynamic contagion process in which we used different exciting functions for different generations of offspring and studied their asymptotic properties including Central Limit Theorem and Large Deviation Principle. This consideration is important in a number of fields, e.g., in seismology, where main shocks produce aftershocks with possibly different intensities. Additionally, in a financial setting, we investigated the asymptotic behavior of the ruin probability in both finite and infinite time horizons. Specifically, we considered the risk model under the assumption that the dynamics of the contagion claims arrivals have different exciting functions and examined how it affects the ruin probability.
Bibliography
\([1]\) A. Dassios and H. Zhao, A dynamic contagion process, Advances in Applied Probability 43 (3) (2011) 814–846.
\([2]\) A. G. Hawkes, Spectra of some self-exciting and mutually exciting point processes, Biometrika 58 (1) (1971) 83–90.
\([3]\) A. G. Hawkes and D. Oakes, A cluster process representation of a self-exciting process, Journal of Applied Probability 11 (3) (1974) 493–503.
\([4]\) S. Pandey, D. Selvamuthu and P. Tardell, Asymptotic results for dynamic contagion processes with different exciting functions and application to risk models. Journal of Mathematical Analysis and Applications 547 (2025) 129392.
P21: Convergence of the loop-erased percolation explorer on UIHPT
Yuyang Feng (University of Chicago)
stand: S7
We study critical site percolation on a uniform infinite half-planar triangulation with a white-black boundary condition. Previous studies have shown the convergence of percolation interface-decorated maps to an \(\mathrm{SLE}_6\)-decorated \(\sqrt{8/3}\)-LQG surface under the local GHPU topology. In this work, we prove that the loop-erasure of the percolation interface also has a scaling limit and conjecture it to be an \(\mathrm{SLE}_{8/3}\)-type curve.
P22: Diffusion Limit for Markovian Models of Evolution in Structured Populations with Migration
Celia Garcia Pareja (KTH Royal Institute of Technology)
stand: S8
The evolution of microbial subpopulations that migrate within spatial structures has gained interest in recent years. Questions of relevance include, for instance, the ability of a migrant mutant to take over the population (fixate). Estimating fixation probabilities is, however, usually hindered by the lack of analytical formulas and by computational complexity of simulation-based strategies when considering large populations. In this work, we study several population genetics models where the population is divided into \(D\) subpopulations (called demes) consisting of two types of individuals, mutants and wild-types, that evolve through discrete Markovian updates. We prove that under certain assumptions all the considered models converge to the same diffusion approximation, which we call universal. This diffusion approximation is amenable to simulation strategies that underly methods of statistical inference while significantly reducing computational costs. In all models, each Markovian update follows two phases: First, a local growth phase in each subpopulation, where the growth of each type of individual depends on its fitness, and then a sampling phase that implements migration between subpopulations. Our proof relies on existing diffusion approximation results for degenerate diffusions, see \([1]\), but requires further technicalities due to fact that sample sizes in each deem are not necessarily fixed but change randomly with each update.
This is joint work with Alia Abbara and Anne-Florence Bitbol at EPF Lausanne.
Bibliography
\([1]\) Ethier, S. N.. "A class of degenerate diffusion processes occurring in population genetics." Comm. Pure Appl. Math., vol. 29, no. 5, 1976, pp. 483–493.
P23: Asymptotically Distribution-free Goodness-of-Fit Testing for Point Processes
Justin Baars (Universiteit van Amsterdam)
stand: S9
Consider an observation of a multivariate temporal point process \(N\) with law \(\mathcal P\) on the time interval \([0,T]\). To test the null hypothesis that \(\mathcal P\) belongs to a given parametric family, we construct a convergent compensated counting process to which we apply an innovation martingale transformation. We prove that the resulting process converges weakly to a standard Wiener process. Consequently, taking a suitable functional of this process yields an asymptotically distribution-free goodness-of-fit test for point processes. For several standard tests based on the increments of this transformed process, we establish consistency under alternative hypotheses. Finally, we assess the performance of the proposed testing procedure through a Monte Carlo simulation study and illustrate its practical utility with two real-data examples.
P25: Evolution of a quantitative trait in a metapopulation setting: Propagation of chaos meets adaptive dynamics
Josué Tchouanti Fotso (LIFEWARE, Inria Center of Saclay)
stand: S11
This work is about studying the role of migration as a key driver of the evolution of spatially structured populations. To this end, we consider a metapopulation setting where evolutionary processes at the level of each patch is modelled by a Moran model, which describes the evolution of a quantitative trait in a population of fixed size by two main mechanisms : trait resampling and mutations. Migrations are added in order to take into account interactions between patches and the question we would like to answer is: how do these migrations influence the long term evolution of the population at the level of a single patch and at the level of the entire metapopulation ?
For this purpose, we study several scaling limits of the model. Assuming rare mutations and migrations, we adapt a technique from Champagnat & Lambert \([1]\) in order to get a mean-field network of Trait Substitution Sequence (TSS) describing long-term successive dominant traits in each patch. We derive the propagation of chaos as the metapopulation becomes large. Patches are therefore i.i.d copies of each other, with a TSS described by a McKean-Vlasov pure jump process. In the limit where mutations have small effects and migration is further slowed down accordingly, we obtain the convergence of the TSS, in the new migration timescale, to the solution of a stochastic differential equation which can be referred to as a new canonical equation of adaptive dynamics. This equation includes an advection term representing selection, a diffusive term due to the genetic drift, and a jump term – representing the effect of migration – to a state distributed according to the law of the solution.
See the details in Lambert et al. \([2]\).
Bibliography
\([1]\) N. Champagnat, A. Lambert. Evolution of discrete populations and the canonical diffusion of adaptative dynamics. The Annals of Applied Probability, 17(1):102–155, 2007.
\([2]\) A. Lambert, H. Leman, H. Morlon, J. Tchouanti. Evolution of a trait distributed over a large fragmented population : Propagation of chaos meets adaptive dynamics. Preprint arxiv:2503.13154, 2025.
P27: Bell shaped sequences and first passages locations for two-dimensional random walks
Jacek Wszoła (Wrocław University of Science and Technology)
stand: S13
Bell-shaped functions have been studied since the 1940s and are widely used in probability theory. Our goal is to define and describe their discrete counterparts – bell-shaped sequences.
We will also consider a two-dimensional random walk \((X_n, Y_n)\) on the lattice \(\mathbb{Z}^2\), with diagonal jumps \((\pm 1, \pm1)\) and jump probabilities depending only on the position of the second coordinate \(Y_n\). The first passage location is a random variable \(X_\tau\), where \(\tau = \min\{n \geq 0: Y_n=0\}\) denotes the first passage time to zero. We will discuss results concerning the distribution of the first passage locations and its connections to bell-shaped sequences.
Bibliography
\([1]\) M. Kwaśnicki, J. Wszoła. ‘Bell-shaped sequences.’ Studia Mathematica, 271 (2023), 151-185.
\([2]\) M. Kwaśnicki, J. Wszoła. ‘Two-sided bell-shaped sequences.’ Preprint, ArXiv:2404.11274
\([3]\) J. Wszoła. ‘First passage locations for two-dimensional lattice random walks and the bell-shape.’ Preprint, ArXiv:2501.14393
P28: The order-complete AL-space of quasimartingales
Timothy Minn Kang (Imperial College London)
stand: S14
Quasimartingales form a class of stochastic processes which, broadly speaking, can be considered as a stochastic analogue of functions of finite variation. Riesz spaces are vector spaces endowed with a partial order and where any two elements have a supremum and infimum.
In this work, we overview our study of the vector space of càdlàg quasimartingales through the framework of Riesz spaces. In particular, after presenting some basic definitions and known results associated with quasimartingales, we present a result and outline our original proof stating that this space is a Riesz space, when endowed with a particular order, and that it satisfies additional properties within this framework.
More specifically, this result states that the space of càdlàg quasimartingales is an order-complete Riesz space. Additionally, when endowed with a particular norm which we will define, this space is an AL-space.
We then present two additional results of ours. The first includes various characterisations of the notion of singularity for càdlàg quasimartingales which we have found. The second result states that the space of càdlàg quasimartingales of class(D) is, in some specific sense, the dual space of the space of adapted, bounded, continuous processes.
This is based on joint work with Pietro Siorpaes and Thomas Cass