Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs

We address a general optimal switching problem over finite horizon for a stochastic system described by a differential equation driven by Brownian motion. The main novelty is the fact that we allow for infinitely many modes (or regimes, i.e. the possible values of the piecewise-constant control process). We allow all the given coefficients in the model to be path-dependent, that is, their value at any time depends on the past trajectory of the controlled system. The main aim is to introduce a suitable (scalar) backward stochastic differential equation (BSDE), with a constraint on the martingale part, that allows to give a probabilistic representation of the value function of the given problem. This is achieved by randomization of control, i.e. by introducing an auxiliary optimization problem which has the same value as the starting optimal switching problem and for which the desired BSDE representation is obtained. In comparison with the existing literature we do not rely on a system of reflected BSDE nor can we use the associated Hamilton\u2013Jacobi\u2013Bellman equation in our non-Markovian framework

Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian

We consider Hamilton Jacobi Bellman equations in an inifinite dimensional Hilbert space, with quadratic (respectively superquadratic) hamiltonian and with continuous (respectively lipschitz continuous) final conditions. This allows to study stochastic optimal control problems for suitable controlled Ornstein Uhlenbeck process with unbounded control processes

Major adverse kidney events in pediatric continuous kidney replacement therapy

IMPORTANCE: Continuous kidney replacement therapy (CKRT) is increasingly used in youths with critical illness, but little is known about longer-term outcomes, such as persistent kidney dysfunction, continued need for dialysis, or death. OBJECTIVE: To characterize the incidence and risk factors, including liberation patterns, associated with major adverse kidney events 90 days after CKRT initiation (MAKE-90) in children, adolescents, and young adults. DESIGN, SETTING, AND PARTICIPANTS: This international, multicenter cohort study was conducted among patients aged 0 to 25 years from The Worldwide Exploration of Renal Replacement Outcomes Collaborative in Kidney Disease (WE-ROCK) registry treated with CKRT for acute kidney injury or fluid overload from 2015 to 2021. Exclusion criteria were dialysis dependence, concurrent extracorporeal membrane oxygenation use, or receipt of CKRT for a different indication. Data were analyzed from May 2 to December 14, 2023. EXPOSURE: Patient clinical characteristics and CKRT parameters were assessed. CKRT liberation was classified as successful, reinstituted, or not attempted. Successful liberation was defined as the first attempt at CKRT liberation resulting in 72 hours or more without return to dialysis within 28 days of CKRT initiation. MAIN OUTCOMES AND MEASURES: MAKE-90, including death or persistent kidney dysfunction (dialysis dependence or ≥25% decline in estimated glomerular filtration rate from baseline), were assessed. RESULTS: Among 969 patients treated with CKRT (529 males [54.6%]; median [IQR] age, 8.8 [1.7-15.0] years), 630 patients (65.0%) developed MAKE-90. On multivariable analysis, cardiac comorbidity (adjusted odds ratio [aOR], 1.60; 95% CI, 1.08-2.37), longer duration of intensive care unit admission before CKRT initiation (aOR for 6 days vs 1 day, 1.07; 95% CI, 1.02-1.13), and liberation pattern were associated with MAKE-90. In this analysis, patients who successfully liberated from CKRT within 28 days had lower odds of MAKE-90 compared with patients in whom liberation was attempted and failed (aOR, 0.32; 95% CI, 0.22-0.48) and patients without a liberation attempt (aOR, 0.02; 95% CI, 0.01-0.04). CONCLUSIONS AND RELEVANCE: In this study, MAKE-90 occurred in almost two-thirds of the population and patient-level risk factors associated with MAKE-90 included cardiac comorbidity, time to CKRT initiation, and liberation patterns. These findings highlight the high incidence of adverse outcomes in this population and suggest that future prospective studies are needed to better understand liberation patterns and practices

Differentiability of backward stochastic differential equations in Hilbert spaces with monotone generators

The aim of the present paper is to study the regularity properties of the solution of a backward stochastic differential equation with a monotone generator in infinite dimension. We show some applications to the nonlinear Kolmogorov equation and to stochastic optimal control

Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in Rd\R^d

We consider a class of second order linear nonautonomous parabolic equations in R^d with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator G(t,s) be compact in C_b(R^d) for t>s, and describe the asymptotic behavior of G(t,s)f as t-s goes to infinity in terms of a family of measures mu_s, s in R, solution of the associated Fokker-Planck equation