VTeX: Solutions for Science Publishing

A single jump filtration ${({\mathcal{F}_{t}})_{t\in {\mathbb{R}_{+}}}}$ generated by a random variable γ with values in ${\overline{\mathbb{R}}_{+}}$ on a probability space $(\Omega ,\mathcal{F},\mathsf{P})$ is defined as follows: a set $A\in \mathcal{F}$ belongs to ${\mathcal{F}_{t}}$ if $A\cap \{\gamma >t\}$ is either ∅ or $\{\gamma >t\}$. A process M is proved to be a local martingale with respect to this filtration if and only if it has a representation ${M_{t}}=F(t){\mathbb{1}_{\{t<\gamma \}}}+L{\mathbb{1}_{\{t\geqslant \gamma \}}}$, where F is a deterministic function and L is a random variable such that $\mathsf{E}|{M_{t}}|<\infty $ and $\mathsf{E}({M_{t}})=\mathsf{E}({M_{0}})$ for every $t\in \{t\in {\mathbb{R}_{+}}:\mathsf{P}(\gamma \geqslant t)>0\}$. This result seems to be new even in a special case that has been studied in the literature, namely, where $\mathcal{F}$ is the smallest σ-field with respect to which γ is measurable (and then the filtration is the smallest one with respect to which γ is a stopping time). As a consequence, a full description of all local martingales is given and they are classified according to their global behaviour.

A general class of probability density functions is considered, containing as particular case the Barenblatt solutions arising, for instance, in the study of nonlinear heat equations. Alternative probabilistic representations of the Barenblatt-type solutions $u(x,t)$ are proposed. In the one-dimensional case, by means of this approach, $u(x,t)$ can be connected with the wave propagation.

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.

The problem of (pathwise) large deviations for conditionally continuous Gaussian processes is investigated. The theory of large deviations for Gaussian processes is extended to the wider class of random processes – the conditionally Gaussian processes. The estimates of level crossing probability for such processes are given as an application.

This paper is devoted to investigation of supremum of averaged deviations $|X(t)-f(t)-\int _{\mathbb{T}}(X(u)-f(u))\hspace{0.1667em}\mathrm{d}\mu (u)/\mu (\mathbb{T})|$ of a stochastic process from Orlicz space of random variables using the method of majorizing measures. An estimate of distribution of supremum of deviations $|X(t)-f(t)|$ is derived. A special case of the $L_{q}$ space is considered. As an example, the obtained results are applied to stochastic processes from the $L_{2}$ space with known covariance functions.

This paper is devoted to investigation of supremum of averaged deviations $|X(t)-f(t)-\int _{\mathbb{T}}(X(u)-f(u))\hspace{0.1667em}\mathrm{d}\mu (u)/\mu (\mathbb{T})|$ of a stochastic process from Orlicz space of random variables using the method of majorizing measures. An estimate of distribution of supremum of deviations $|X(t)-f(t)|$ is derived. A special case of the $L_{q}$ space is considered. As an example, the obtained results are applied to stochastic processes from the $L_{2}$ space with known covariance functions.

In this paper, we consider the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a “truncated” CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that this “truncated” process does not hit zero under the same condition considered for the original nontruncated process.

In this paper, we consider the Cox–Ingersoll–Ross (CIR) process in the regime where the process does not hit zero. We construct additive and multiplicative discrete approximation schemes for the price of asset that is modeled by the CIR process and geometric CIR process. In order to construct these schemes, we take the Euler approximations of the CIR process itself but replace the increments of the Wiener process with iid bounded vanishing symmetric random variables. We introduce a “truncated” CIR process and apply it to prove the weak convergence of asset prices. We establish the fact that this “truncated” process does not hit zero under the same condition considered for the original nontruncated process.