The aim of this paper is to take an in-depth look at the long time behaviour of some continuous time markovian dynamical systems and at its numerical analysis. We first propose a short overview of the main ergodicity properties of time continuous homogeneous Markov processes (stability, positive recurrence). The basic tool is a Lyapunov function. Then, we investigate if these properties still hold for the time discretization of these processes, either with constant or decreasing step (ODE method...

The aim of this paper is to take an in-depth look at the long time behaviour of some continuous time Markovian dynamical systems and at its numerical analysis. We first propose a short overview of the main ergodicity properties of time continuous homogeneous Markov processes (stability, positive recurrence). The basic tool is a Lyapunov function. Then, we investigate if these properties still hold for the time discretization of these processes, either with constant or decreasing step (ODE...

This paper continues the research started in [J. Štěpán and P. Dostál: The $\mathrm{d}X\left(t\right)=Xb\left(X\right)\mathrm{d}t+X\sigma \left(X\right)\mathrm{d}W$ equation and financial mathematics I. Kybernetika 39 (2003)]. Considering a stock price $X\left(t\right)$ born by the above semilinear SDE with $\sigma (x,t)=\tilde{\sigma }\left(x\left(t\right)\right),$ we suggest two methods how to compute the price of a general option $g\left(X\right(T\left)\right)$. The first, a more universal one, is based on a Monte Carlo procedure while the second one provides explicit formulas. We in this case need an information on the two dimensional distributions of $ℒ\left(Y\right(s),\tau (s\left)\right)$ for $s\ge 0,$ where $Y$ is the exponential...

We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.

We derive and analyze adaptive solvers for boundary value problems in which the differential operator depends affinely on a sequence of parameters. These methods converge uniformly in the parameters and provide an upper bound for the maximal error. Numerical computations indicate that they are more efficient than similar methods that control the error in a mean square sense.