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We consider the problem of an optimal consumption strategy on the infinite time horizon based on the hyperbolic absolute risk aversion utility when the interest rate is an Ornstein-Uhlenbeck process. Using the method of subsolution and supersolution we obtain the existence of solutions of the dynamic programming equation. We illustrate the paper with a numerical example of the optimal consumpti...

We consider a sequence of repeated prediction games and formally pass to the limit. The supersolutions of the resulting non-linear parabolic partial differential equation are closely related to the potential functions in the sense of N.Cesa-Bianci, G. Lugosi (2003). Any such supersolution gives an upper bound for forecaster’s regret and suggests a potentialbased prediction strategy, satisfying ...

We consider a Dirichlet problem for system of equations involving Kirchhoff type $ p_i $-Laplacian differential operators and exhibiting gradient dependence in the reaction term (convection). Using subsolution-supersolution method, we establish existence weak solutions into suitable ordered intervals. Following unified approach, also provide comparison argument to obtain positive certain models.

We provide sharp bounds for the exponential moments and $p$-moments, $1\leqslant p \leqslant 2$, of terminate distribution a martingale whose square function is uniformly bounded by one. introduce Bellman corresponding extremal problem reduce it to already known on $\mathrm{BMO}([0,1])$. In case tail estimates, similar reduction does not work exactly, so we come up with fine supersolution that ...

We provide a simple method for obtaining new Liouville-type theorems positive supersolutions of the elliptic problem -Δ p u+b(x)|∇u| pq q+1 =c(x)u q in Ω, where Ω is an exterior domain ℝ N with N≥p>1 and q≥p-1. In case q≠p-1, we mainly deal potentials type b(x)=|x| , c(x)=λ|x| σ λ>0 a,σ∈ℝ. show that do not exist some ranges parameters p,q,a,σ, which turn out to be optimal. When q=p-1, consider ...

We apply the stochastic Perron method of Bayraktar and Sîrbu to a general infinite horizon optimal control problem, where the state X is a controlled diffusion process, and the state constraint is described by a closed set. We prove that the value function v is bounded from below (resp., from above) by a viscosity supersolution (resp., subsolution) of the related state constrained problem for t...

We introduce an iterative method for computing the first eigenpair (λp, ep) for the pLaplacian operator with homogeneous Dirichlet data as the limit of (μq,uq) as q → p −, where uq is the positive solution of the sublinear Lane-Emden equation −∆puq = μqu q−1 q with same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained th...