For a large class of operator inclusions, including those generated by maps of pseudomonotone type, we obtain a general theorem on existence of solutions. We apply this result to some particular examples. This theorem is proved using the method of difference approximations.

We investigate the conditions on a hedger, who overestimates the (time- and level-dependent) volatility, to superreplicate a convex claim on several underlying assets. It is shown that the classic Black-Scholes model is the only model, within a large class, for which overestimation of the volatility yields the desired superreplication property. This is in contrast to the one-dimensional case, in which it is known that overestimation of the volatility with any time- and level-dependent model guarantees superreplication of convex claims.

In this article, we analyse the error induced by the Euler scheme combined with a symmetry procedure near the boundary for the simulation of diffusion processes with an oblique reflection on a smooth boundary. This procedure is easy to implement and, in addition, accurate: indeed, we prove that it yields a weak rate of convergence of order 1 with respect to the time-discretization step.

Using a representation in terms of a two-type branching particle system, we prove that positive solutions of the system remain bounded for suitable bounded initial conditions, provided A and B generate processes with independent increments and one of the processes is transient with a uniform power decay of its semigroup. For the case of symmetric stable processes on R 1,this answers a question raised in [4].

In Section 1 of this note we will construct an example of a subset of R × Rn such that the parabolic capacity with respect to the heat equation is zero although its orthogonal projection onto {0} × Rn is the whole space. Such examples were already given by R. Kaufman and J.-M. Wu in [5] and [6]. However, our probabilistic approach seems to be more transparent since it does not depend on explicit formulas for Green functions.

In this paper, we consider an extended model of a coupled nonlinear reaction-diffusion equations with Neumann-Neumann boundary conditions. We obtain upper linear growth bound for one of the components. We also find the corresponding bound for the case of Dirichlet-Dirichlet boundary conditions.

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].

The Stefan problem is a particular free boundary problem for the heat equation which arises in the investigation of the melting of solids. In the case of one space dimension there are numerous results available concerning the existence, uniqueness, and stability of the solution [c.f. 6]. However the case of several space variables is considerably more difficult. This is due in large part to the fact that the geometry of the problem can become quite complicated, and smooth initial and boundary data do not necessarily lead to smooth solutions. In particular, under heating, a connected solid can melt into two (or more) disconnected solids, thus leading to a problem in which the free boundary varies in a discontinuous manner. These difficulties have motivated several researchers to look for “weak” solutions to the Stefan problem [c.f. 1, 3, 5]. Although this approach is quite general and leads to numerical schemes for solving the problem under consideration, there are several drawbacks to this method, among them being the fact that no information is obtained concerning the structure of the interphase boundary, nor is there much information on the regularity of these weak solutions.