This talk addresses the efficient Monte Carlo estimation of rare-event probabilities associated with a broad class of McKean--Vlasov stochastic differential equations (MV-SDEs), which arise in the analysis of mean-field systems in statistical physics, mathematical finance, and collective behaviour models. Standard Monte Carlo methods become computationally infeasible in this setting due to the rapid growth of the estimator's relative variance (coefficient of variation) in the rare-event regime. Using stochastic optimal control, an optimal importance sampling measure change is constructed to minimise the variance of the resulting estimator. The resulting double-loop Monte Carlo (DLMC) estimator with importance sampling significantly mitigates this growth in the coefficient of variation. The framework is further extended to the multilevel Monte Carlo setting to reduce computational complexity, leveraging propagation-of-chaos and strong antithetic coupling to ensure that the level differences vanish in the mean-field limit. To address the discontinuity of the probability observable, a numerical smoothing technique is introduced to recover optimal variance convergence rates. Numerical experiments on linear mean-field, Kuramoto, and Cucker--Smale models demonstrate computational savings of several orders of magnitude compared with standard Monte Carlo.
NOTES: online.
In a recent paper written jointly with Boaz Klartag, we prove that the variance of the Euclidean norm of any isotropic log-concave random vector is bounded above by a universal constant, not depending on the dimension. Thus, most of the mass of the random vector is concentrated in a thin spherical shell, whose width is order 1, while its radius is order root of the dimension. This confirms the thin-shell conjecture in high dimensional convex geometry. Our method relies on the construction of a certain coupling between log-affine perturbations of a given log-concave measure related to Eldan's stochastic localization and to the theory of non-linear filtering. Another ingredient is a recent breakthrough technique by Guan that was previously used in our proof of Bourgain's slicing conjecture, which is known to be implied by the thin-shell conjecture. In this talk, I'll first review the context and the history of the problem, before laying out the main steps of our proof.
NOTES: unusual time, online.
Abstract: Efficiently solving nonlinear stochastic optimal control problems remains a challenge with many applications. Existing optimality conditions typically rely on forward-backward stochastic differential equations (FBSDEs), which can be difficult to use in algorithms. In this talk, I will present new optimality conditions for stochastic optimal control, derived using rough path theory. This Pontryagin Maximum Principle uses the same Hamiltonian as in the deterministic setting, while avoiding FBSDEs. It unlocks the first indirect shooting method for stochastic optimal control, which only searches over the initial adjoint state and converges significantly faster than a direct method in numerical experiments. I will present applications to driving at the limits of handling and future directions towards robust and efficient uncertainty-aware control.
We consider coalescing random walks in 1+1 dimensional space-time, with a jump kernel that has finite moments up to order alpha. We view this system as a random set of coalescing paths, with one path starting from each point of space-time. When alpha>3, the diffusive scaling limit is known to be the Brownian web. We study the regime in which alpha is between 2 and 3. In this regime tightness fails (in the sense of continuous paths) due to erratic behaviour near the start times of some of the paths. We show that a surprising transition in behaviour occurs at alpha=9/4; when alpha>9/4 a diffusive scaling limit exists in which paths are essentially Brownian but some paths possess jumps at their initial times, whilst when alpha<9/4 tightness "truly" fails.
We study a family of balls-in-bins models with a power-law feedback and a local
interaction determined by an underlying graph on the bins. Specifically, for a fixed
graph on $d$ bins, and fixed positive real numbers $\beta_1, \dots, \beta_d$, at each time
step the model allocates a new ball to bin $i$ with probability proportional to
$U_i^{\beta_i}$, where $U_i$ is the total number of balls currently allocated to all bins
in the graph neighbourhood of bin $i$ (including bin $i$ itself).
In this talk, we focus attention on the case of a path graph on 3 bins, studying the
asymptotic behaviour of $X_n$, the vector of the number of balls allocated to each bin
after $n$ steps. Despite its apparent simplicity, the model exhibits a variety of
behaviours, depending on the parameters $\beta_1, \beta_2, \beta_3$. We analyse both the
symmetric ($\beta_i$ equal) and asymmetric ($\beta_i$ distinct) cases, presenting a
complete classification of the growth rates of the coordinates of $X_n$ when $\beta_i > 1$
for all $i$. In each case, we identify when the asymptotic behaviour is
deterministic, and when it is random.
Our analysis employs the method of stochastic approximation for the symmetric case,
semimartingale methods for the asymmetric case, and liberal use of L\'evy's extension of
the Borel--Cantelli lemma. This is joint work with Mikhail Menshikov and Vadim
Shcherbakov.
We revisit the well-studied superhedging problem under proportional transaction costs in continuous time using the recently developed tools of set-valued stochastic analysis. By relying on a simple Black-Scholes-type market model for mid-prices and using continuous trading schemes, we define a dynamic family of superhedging sets in continuous time and express them in terms of set-valued integrals. We show that these sets, defined as subsets of Lebesgue spaces at different times, form a dynamic set-valued risk measure with multi-portfolio time-consistency. Finally, we transfer the problem formulation to a path-space setting and introduce approximate versions of superhedging sets that will involve relaxing the superhedging inequality, the superhedging probability, and the solvency requirement for the superhedging strategy with a predetermined error level. In this more technical framework, we are able to relate the approximate superhedging sets at different times by means of a set-valued Bellman’s principle. We conjecture that this pathwise principle can be used to obtain a set-valued differential structure that characterizes the superhedging sets. Joint work with Atiqah Almuzaini and Jin Ma.
Consider the following method for shuffling a deck of cards: at each step we select a card uniformly at random, and then transpose this with a card chosen uniformly from beneath the selected card. We call this a "one-sided transposition shuffle", and ask the question: how many shuffles does it take to "randomise" the deck? It turns out that we can give a precise answer to this question using a combination of ideas from probability and representation theory of the symmetric group; in particular, we are able to deduce an explicit formula for the eigenvalues of the shuffle by relating them to Young tableaux.
Joint work with O. Matheau-Raven and M. Bate.
The randomised reproducing graph is a random graph model, with some
similarities to duplication graphs, which shows a number of interesting
phase transitions. This talk will describe work in progress constructing
a local limit for this model. (Joint with Simon Irons)
In this talk I will present a result on the construction of the unique invariant measure of the singular SDE with fractional Brownian noise (fBm), equipped with a linear damping. We build up on the theory of regularisation by noise, developed in recent years by Catellier, Gubinelli, Galeati and many others, and merge it with ergodic theory of fractional SDEs, studied by Hairer and his coauthors. We establish tightness in the usual regime of weak existence ($\alpha > 1/2-1/(2H)$, where $\alpha$ is Besov-Holder regularity of the drift and H is Hurst index of fBm), and uniqueness of invariant measure under usual condition on well-posedness for singular SDEs driven by fractional Brownian motion ($\alpha > 1 - 1/(2H)$). To this end we employ a modification of stochastic sewing, which also allows us to show Gaussian tails of the solution. Our approach does not require any assumption on the size of the drift with respect to the damping strength.
NOTES: unusual time.
We study a random tree model known as the Preferential Attachment tree with Vertex Death. Here, one can both add vertices to the tree as well as kill vertices. This model mimics the non-monotone growth of real-world networks, absent in classical preferential attachment models. One initialises the tree with a single root vertex labelled 1. At every step n, either a new vertex labelled $n+1$ is added to the tree and connected to an already present alive vertex, selected preferentially according to a function $b$, or an already present vertex is selected preferentially according to a function $d$ and killed. Killed vertices can make no new connections.
We are interested in the behaviour of the richest alive vertex $I_n$ (with the largest degree) and the oldest alive vertex $ O_n$ (with the smallest label) in the tree $T_n$ as n tends to infinity. When $ I_n$ converges almost surely, we say that a persistent hub exists. When $I_n$ does not converge but $I_n/O_n$is tight, we say that persistence occurs, and when $I_n/O_n$ diverges to infinity we say lack of persistence occurs. We uncover three distinct regimes in which behaviour is different: (1) The Infinite Lifetime regime, where we provide conditions under which a persistent hub exists almost surely. (2) The Rich Are Old regime, where we provide conditions under which either persistence or lack of persistence occurs. (3) The Rich Die Young regime, where lack of persistence always occurs. We shall discuss how the three regimes can be identified and what drives the behaviour observed in each regime. Partially joint work with Markus Heydenreich.
