Using the dynamic programming principle in optimal stopping theory, we derive a semilinear Black and Scholes type partial differential equation set in a fixed domain for the value of an American (call/put) option. The nonlinearity in the semilinear Black and Scholes equation depends discontinuously on the American option value, so that standard theory for partial differential equation does not ...

We consider boundary value problems for semilinear evolution inclusions. We establish the existence of extremal solutions. Using that result, we show that the evolution inclusion has periodic extremal trajectories. These results are then applied to closed loop control systems. Finally, an example of a semilinear parabolic distributed parameter control system is worked out in detail.

Abstract: We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise. We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet...

In this article, well-posedness of stochastic anisotropic $p$-Laplace equation driven by L\'evy noise is shown. Such an in deterministic setting was considered Lions [7]. The results obtained article can be applied to solve a large class semilinear and quasilinear partial differential equations.

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2. In Part 1, we prove...

In 1978 De Giorgi formulated the following conjecture. Let u be a solution of u = u 3 ? u in all of R n such that juj 1 and @ n u > 0 in R n. Is it true that all level sets fu = g of u are hyperplanes, at least if n 8 ? Equivalently, does u depend only on one variable? When n = 2, this conjecture was proved in 1997 by N. Ghoussoub and C. Gui. In the present paper we prove it for n = 3. The ques...

We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the u5-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.