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.

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...

Let $u(x,y)$ be a harmonic function in the half-plane ${\mathbf{R}}_{+}^{n+1}$, $n\ge 2$. We define a family of functionals $D(u;r),-\infty \>r\>\infty$, that are analogs of the family of local times associated to the process $u(x_t,y_t)$ where $(x_t,y_t)$ is Brownian motion in ${\mathbf{R}}_{+}^{n+1}$. We show that $D\left(u\right)={sup}_{r}D(u;r)$ is bounded in $L^p$ if and only if $u(x,y)$ belongs to $H^p$, an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.

We consider a discrete-time version of the parabolic Anderson model. This may be described as a model for a directed $(1+d)$-dimensional polymer interacting with a random potential, which is constant in the deterministic direction and i.i.d. in the $d$ orthogonal directions. The potential at each site is a positive random variable with a polynomial tail at infinity. We show that, as the size of the system diverges, the polymer extremity is localized almost surely at one single point which grows ballistically....

In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter $\theta \>0$) by replacing the entries equal to one by more general non-vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very explicitly computed by using the cycle structure of the permutations....