A forward--backward stochastic algorithm for quasi-linear PDEs

We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward--backward SDEs, which provides an efficient probabilistic representation of this type of equation. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE. In particular, our work provides an alternative to the method described in [Douglas, Ma and Protter (1996) Ann. Appl. Probab. 6 940--968] and weakens the regularity assumptions required in this reference.Comment: Published at http://dx.doi.org/10.1214/105051605000000674 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Stopped diffusion processes: boundary corrections and overshoot

For a stopped diffusion process in a multidimensional time-dependent domain \D, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $\Delta$ and stopping it at discrete times $(i\Delta)_{i\in\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of \D, and its amplitude is equal to $0.5826 (...) |n^*\sigma|(t,x)\sqrt \Delta$ where $\sigma$ stands for the diffusion coefficient of the process. The procedure is thus extremely easy to use. In addition, we prove that the rate of convergence w.r.t. $\Delta$ for the associated weak error is higher than without shifting, generalizin g previous results by \cite{broa:glas:kou:97} obtained for the one dimensional Brownian motion. For this, we establish in full generality the asymptotics of the triplet exit time/exit position/overshoot for the discretely stopped Euler scheme. Here, the overshoot means the distance to the boundary of the process when it exits the domain. Numerical experiments support these results.Comment: 39 page

Non Linear Singular Drifts and Fractional Operators: when Besov meets Morrey and Campanato

Within the global setting of singular drifts in Morrey-Campanato spaces presented in [6], we study now the H{\"o}lder regularity properties of the solutions of a transport-diffusion equation with nonlinear singular drifts that satisfy a Besov stability property. We will see how this Besov information is relevant and how it allows to improve previous results. Moreover, in some particular cases we show that as the nonlinear drift becomes more regular, in the sense of Morrey-Campanato spaces, the additional Besov stability property will be less useful

Heat kernel of supercritical SDEs with unbounded drifts

Let $\alpha\in(0,2)$ and $d\in\mathbb{N}$. Consider the following SDE in $\mathbb{R}^d$:$$\mathrm{d} X_t = b(t,X_t)\mathrm{d} t + a(t,X_{t-})\mathrm{d} L^{(\alpha)}_t,\ \ X_0 = x,$$where $L^{(\alpha)}$ is a $d$-dimensional rotationally invariant $\alpha$-stable process, $b:\mathbb{R}_ + \times\mathbb{R}^d \to\mathbb{R}^d$ and $a:\mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}^d \otimes \mathbb{R}^d$ are H\''older continuous functions in space, with respective order $\beta,\gamma\in (0,1)$ such that $(\beta\wedge \gamma)+\alpha>1$, uniformly in $t$. Here $b$ may be unbounded.When $a$ is bounded and uniformly elliptic, we show that the unique solution $X_t(x)$ of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. {In particular, we cover the whole \textit{supercritical} range $\alpha\in (0,1)$.} Our proof is based on \textit{ad hoc} parametrix expansions and probabilistic techniques

On Multidimensional stable-driven Stochastic Differential Equations with Besov drift

We establish well-posedness results for multidimensional non degenerate $\alpha$-stable driven SDEs with time inhomogeneous singular drifts in L^r-B^{--1+$\gamma$}_{p,q} with $\gamma$ < 1 and $\alpha$ in (1, 2], where L^r and B^{--1+$\gamma$}_{p,q} stand for Lebesgue and Besov spaces respectively. Precisely, we first prove the well-posedness of the corresponding martingale problem and then give a precise meaning to the dynamics of the SDE. Our results rely on the smoothing properties of the underlying PDE, which is investigated by combining a perturbative approach with duality results between Besov spaces

Trends in Cancer Incidence in Different Antiretroviral Treatment-Eras amongst People with HIV

Despite cancer being a leading comorbidity amongst individuals with HIV, there are limited data assessing cancer trends across different antiretroviral therapy (ART)-eras. We calculated age-standardised cancer incidence rates (IRs) from 2006–2021 in two international cohort collaborations (D:A:D and RESPOND). Poisson regression was used to assess temporal trends, adjusted for potential confounders. Amongst 64,937 individuals (31% ART-naïve at baseline) and 490,376 total person-years of follow-up (PYFU), there were 3763 incident cancers (IR 7.7/1000 PYFU [95% CI 7.4, 7.9]): 950 AIDS-defining cancers (ADCs), 2813 non-ADCs, 1677 infection-related cancers, 1372 smoking-related cancers, and 719 BMI-related cancers (groups were not mutually exclusive). Age-standardised IRs for overall cancer remained fairly constant over time (8.22/1000 PYFU [7.52, 8.97] in 2006–2007, 7.54 [6.59, 8.59] in 2020–2021). The incidence of ADCs (3.23 [2.79, 3.72], 0.99 [0.67, 1.42]) and infection-related cancers (4.83 [4.2, 5.41], 2.43 [1.90, 3.05]) decreased over time, whilst the incidence of non-ADCs (4.99 [4.44, 5.58], 6.55 [5.67, 7.53]), smoking-related cancers (2.38 [2.01, 2.79], 3.25 [2.63–3.96]), and BMI-related cancers (1.07 [0.83, 1.37], 1.88 [1.42, 2.44]) increased. Trends were similar after adjusting for demographics, comorbidities, HIV-related factors, and ART use. These results highlight the need for better prevention strategies to reduce the incidence of NADCs, smoking-, and BMI-related cancers

Discrétisations associées à un processus dans un domaine et Schémas numériques probabilistes pour les EDP paraboliques quasi-linéaires

PARIS-BIUSJ-Thèses (751052125) / SudocPARIS-BIUSJ-Physique recherche (751052113) / SudocSudocFranceF