Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show that this value function is the smallest nonnegative viscosity supersolution ...

and Applied Analysis 3 2. Proof of Theorems Consider the more general semilinear elliptic problem −Δu f x, u , in Ω, u > 0, in Ω, u 0, on ∂Ω, 2.1 where the function f x, s is locally Hölder continuous in Ω × 0,∞ and continuously differentiable with respect to the variable s. A function u is called to be a subsolution of problem 2.1 if u ∈ C2 Ω ∩ C Ω , and −Δu ≤ fx, u, in Ω, u > 0, in Ω, u 0, on...

We consider general two-sided matching markets, so-called matching with contracts markets as introduced by Hatfield and Milgrom (2005), and analyze (Maskin) monotonic and Nash implementable solutions. We show that for matching with contracts markets the stable correspondence is monotonic and implementable (Theorems 1 and 3). Furthermore, any solution that is Pareto efficient, individually ratio...

We study the problem of the existence and nonexistence of positive solutions to a superlinear second–order divergence type elliptic equation with measurable coefficients −∇ · a · ∇u = u (∗), p > 1, in an unbounded cone–like domain G ⊂ R (N ≥ 3). We prove that the critical exponent p∗(a,G) = inf{p > 1 : (∗) has a positive supersolution at infinity in G } for a nontrivial cone– like domain is alw...

We formulate and study a one-dimensional single-species di¬usive-delay population model. The time delay is the time taken from birth to maturity. Without di¬usion, the delay di¬erential model extends the well-known logistic di¬erential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple globa...

We study integrability conditions for existence and nonexistence of a local-in-time integral solution fractional semilinear heat equations with rather general growing nonlinearities in uniformly local $$L^p$$ spaces. Our main results about this matter consist Theorems 1.4, 1.6, 5.1 5.3. introduce supersolution an equation which can be applied to nonlocal parabolic equation. When the nonlinear t...

We investigate which elliptic PDEs have the property that every viscosity supersolution belongs to a Sobolev space $$W^{1,q}_{loc}(\Omega )$$ , $$\Omega \subseteq \mathbb {R}^n$$ . The asymptotic cone of operator’s sublevel set seems be essential. It turns out much can said if we know how is related dominative p-Laplacian, with $$p = \frac{n-1}{n}q + 1$$ show that, in certain sense, this minima...