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...

The existence of a weak solution and the uniqueness in law are assumed for the equation, the coefficients $b$ and $\sigma$ being generally $C\left({ℝ}^{+}\right)$-progressive processes. Any weak solution $X$ is called a $(b,\sigma )$-stock price and Girsanov Theorem jointly with the DDS Theorem on time changed martingales are applied to establish the probability distribution ${\mu }_{\sigma }$ of $X$ in $C\left({ℝ}^{+}\right)$ in the special case of a diffusion volatility $\sigma (X,t)=\tilde{\sigma }\left(X\left(t\right)\right).$ A martingale option pricing method is presented.

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium...

One shows that the linearized Navier-Stokes equation in $𝒪\subset R^d,d\ge 2$, around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller ${\displaystyle V(t,\xi )=\sum _{i=1}^N{V}_i\left(t\right){\psi }_i\left(\xi \right){\dot{\beta }}_i\left(t\right)}$, $\xi \in 𝒪$, where ${\left\{{\beta }_i\right\}}_{i=1}^N$ are independent Brownian motions in a probability space and ${\left\{{\psi }_i\right\}}_{i=1}^N$ is a system of functions on $𝒪$ with support in an arbitrary open subset ${𝒪}_0\subset 𝒪$. The stochastic control input ${\left\{V_i\right\}}_{i=1}^N$ is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability the equilibrium solution. ...

The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Itô integral, and a simpler and more direct proof of the Itô Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive...

The Cauchy problem for a stochastic partial differential equation with a spatial correlated Gaussian noise is considered. The "drift" is continuous, one-sided linearily bounded and of at most polynomial growth while the "diffusion" is globally Lipschitz continuous. In the paper statements on existence and uniqueness of solutions, their pathwise spatial growth and on their ultimate boundedness as well as on asymptotical exponential stability in mean square in a certain Hilbert space of weighted functions...

We apply an approximation by means of the method of lines for hyperbolic stochastic functional partial differential equations driven by one-dimensional Brownian motion. We study the stability with respect to small $L^2$-perturbations.