Jump Processes

We propose a flexible framework for modeling the joint dynamics of an index and a set of forward variance swap rates written on this index. Our model reproduces various empirically observed properties of variance swap dynamics and enables volatility derivatives and options on the underlying index to be priced consistently, while allowing for jumps in volatility and returns. An affine specification using Lévy processes as building blocks leads to analytically tractable pricing formulas for volatility derivatives, such as VIX options, as well as efficient numerical methods for pricing of European options on the underlying asset. The model has the convenient feature of decoupling the vanilla skews from spot/volatility correlations and allowing for different conditional correlations in large and small spot/volatility moves. We show that our model can simultaneously fit prices of European options on S&P 500 across strikes and maturities as well as options on the VIX volatility index.

We analyze a continuous-time optimal stopping problem in which rewards accrue via a homogeneous Poisson process ("Clock A") while a competing inhomogeneous Poisson process ("Clock B") causes complete forfeiture upon arrival. Although optimal stopping under default risk and jump processes has been extensively studied in the mathematical literature, these results typically rely on stationary hazards, diffusionbased dynamics, or discounting for simplicity. Here, we consider a non-stationary undiscounted setting with deterministic jump rewards and time-dependent hazard rates and show that the resulting Hamilton-Jacobi-Bellman equation admits an exact solution. Our main contribution is to identify and exploit an affine invariance property of the value function that holds under total-forfeiture risk with deterministic reward jumps. This structure reduces the stochastic control problem to a system of linear ordinary differential equations in time, yielding an explicit representation of both the value function and the optimal stopping rule. The optimal policy is shown to be a deterministic time-dependent threshold on accumulated reward which is fully characterized by the integrated hazard rate and the reward intensity. The solution provides a transparent probabilistic interpretation in terms of survival-weighted rewards and delivers closed-form comparative expressions that are unavailable in more general formulations. Our results constitute a tractable baseline for non-stationary stopping problems with catastrophic risk and clarify the precise role of time-varying hazard in optimal exit decisions.

Estes slides apresentam um resumo de modelos dinâmicos bayesianos. Eles foram elaborados a partir do artigo de Migon, Alves, Menezes e Pinheiro (2023). Neles, apresento a ideia por trás dos modelos dinâmicos e sua relação com os modelos lineares generalizados. Todo o conteúdo pode ser obtido no artigo dos autores, mas grande parte da fundamentação mostrada nos slides não está lá; para isso, o livro Bayesian Forecasting and Dynamic Models, de Mike West e Jeff Harrisson  é a principal referência.
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These slides present an overview of Bayesian dynamic models. They were developed based on the article by Migon, Alves, Menezes, and Pinheiro (2023). In them, I introduce the ideas behind dynamic models and their relationship with generalized linear models. All of the content can be found in the authors’ article, but much of the theoretical foundation shown in the slides is not included there; for that, the book Bayesian Forecasting and Dynamic Models by Mike West and Jeff Harrison is the main reference.

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping problems related to the case with model ambiguity are investigated.

We study the optimal stopping problem for dynamic risk measures represented by Backward Stochastic Differential Equations (BSDEs) with jumps and its relation with reflected BSDEs (RBSDEs). We first provide general existence, uniqueness and comparison theorems for RBSDEs with jumps in the case of a RCLL adapted obstacle. We then show that the value function of the optimal stopping problem is characterized as the solution of an RBSDE. The existence of an optimal stopping time is obtained when the obstacle is left-upper semi-continuous along stopping times. Finally, robust optimal stopping problems related to the case with model ambiguity are investigated.

A new explicit solution representation is provided for ARMA recursions with drift and either deterministically or stochastically varying coefficients. It is expressed in terms of the determinants of banded Hessenberg matrices and, as such, is an explicit function of the coefficients. In addition to computational efficiency, the proposed solution provides a more explicit analysis of the fundamental properties of such processes, including their Wold-Cramér decomposition, their covariance structure, and their asymptotic stability and efficiency. Explicit formulae for optimal linear forecasts based either on finite or infinite sequences of past observations are provided. The practical significance of the theoretical results in this work is illustrated with an application to U.S. inflation data. The main finding is that inflation persistence increased after 1976, whereas from 1986 onward, the persistence declines and stabilizes to even lower levels than the pre-1976 period. The first two author names are in alphabetical order. We thank the Editor, the Co-Editor, and two anonymous referees for their thoughtful comments and constructive suggestions. We gratefully acknowledge the helpful conversations we had with L. Giraitis, and G. Kapetanios in the preparation of the paper. We have also benefited from the comments given by participants (R. Baillie, M.

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Exploiting Stochasticity in Multi-agent Systems Alexandre Rodrigues Mesquita To control multi-agent systems one can exploit the recurrence properties of stochastic processes. We illustrate this principle through two applications. In both applications, systems are modeled as hybrid systems where Markov transitions on the discrete variables depend on the continuous variables.

This paper focuses on exploring the convergence properties of a generic player's trajectory and empirical measures in an N-player Linear-Quadratic-Gaussian Nash game, where Brownian motion serves as the common noise. The study establishes three distinct convergence rates concerning the representative player and empirical measure. To investigate the convergence, the methodology relies on a specific decomposition of the equilibrium path in the N-player game and utilizes the associated Mean Field Game framework.

We present a continuous time model of maturation and survival, obtained as the limit of a compartmental evolution model when the number of compartments tends to infinity. We establish in particular an explicit formula for the law of the system output under inhomogeneous killing and when the input follows a time-inhomogeneous Poisson process. This approach allows the discussion of identifiability issues which are of difficult access for finite compartmental models. The article ends up with an example of application for anticancer drug pharmacokinetics/pharmacodynamics.

We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases such as the TCP model or a model of switched vector fields, better results can be proved, especially as regards long time behaviour. We continue our review with an infinite dimensional example of neuronal activity. From the statistical point of view, these models provide specific challenges: we illustrate this point with the example of the estimation of the distribution of the inter-jumping times. We conclude with a short overview on numerical methods used for simulating PDMPs.

We study a class of one-dimensional particle systems with true (Bird type) binary interactions, which includes Kac's model of the Boltzmann equation and nonlinear equations for the evolution of wealth distribution arising in kinetic economic models. We obtain explicit rates of convergence for the Wasserstein distance between the law of the particles and their limiting law, which are linear in time and depend in a mild polynomial manner on the number of particles. The proof is based on a novel coupling between the particle system and a suitable system of nonindependent nonlinear processes, as well as on recent sharp estimates for empirical measures.

In this work, we study the problem of mean-variance hedging with a random horizon T ∧ τ , where T is a deterministic constant and τ is a jump time of the underlying asset price process. We first formulate this problem as a stochastic control problem and relate it to a system of BSDEs with a jump. We then provide a verification theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from filtration enlargement theory.

This paper addresses the problem of finding the optimal portfolio and consumption of a small agent in an economy. The novelty of this work is in considering that the financial market, in contrast to the celebrated Black-Scholes model, is composed of two sources of uncertainties: a Brownian motion and a continuous time Markov chain. While the Brownian motion intends to model the normal oscillations of the asset prices, the continuous time Markov chain aims at taking into account the abrupt variations that can occur in the parameters of the asset dynamics due to changes that take place in the state of the economy. The problem is formulated in terms of classical optimal stochastic control and the Hamilton-Jacobi-Bellman equation is solved to yield the solution.

We prove convex ordering results for random vectors admitting a predictable representation in terms of a Brownian motion and a non-necessarily independent jump component. Our method uses forward-backward stochastic calculus and extends the results proved in [4] in the one-dimensional case. We also study a geometric interpretation of convex ordering for discrete measures in connection with the conditions set on the jump heights and intensities of the considered processes.