Modern Stochastics: Theory and Applications

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.

In this paper the fractional Cox–Ingersoll–Ross process on ${\mathbb{R}_{+}}$ for $H<1/2$ is defined as a square of a pointwise limit of the processes ${Y_{\varepsilon }}$, satisfying the SDE of the form $d{Y_{\varepsilon }}(t)=(\frac{k}{{Y_{\varepsilon }}(t){1_{\{{Y_{\varepsilon }}(t)>0\}}}+\varepsilon }-a{Y_{\varepsilon }}(t))dt+\sigma d{B^{H}}(t)$, as $\varepsilon \downarrow 0$. Properties of such limit process are considered. SDE for both the limit process and the fractional Cox–Ingersoll–Ross process are obtained.

In this paper we define the fractional Cox–Ingersoll–Ross process as $X_{t}:={Y_{t}^{2}}\mathbf{1}_{\{t<\inf \{s>0:Y_{s}=0\}\}}$, where the process $Y=\{Y_{t},t\ge 0\}$ satisfies the SDE of the form $dY_{t}=\frac{1}{2}(\frac{k}{Y_{t}}-aY_{t})dt+\frac{\sigma }{2}d{B_{t}^{H}}$, $\{{B_{t}^{H}},t\ge 0\}$ is a fractional Brownian motion with an arbitrary Hurst parameter $H\in (0,1)$. We prove that $X_{t}$ satisfies the stochastic differential equation of the form $dX_{t}=(k-aX_{t})dt+\sigma \sqrt{X_{t}}\circ d{B_{t}^{H}}$, where the integral with respect to fractional Brownian motion is considered as the pathwise Stratonovich integral. We also show that for $k>0$, $H>1/2$ the process is strictly positive and never hits zero, so that actually $X_{t}={Y_{t}^{2}}$. Finally, we prove that in the case of $H<1/2$ the probability of not hitting zero on any fixed finite interval by the fractional Cox–Ingersoll–Ross process tends to 1 as $k\to \infty $.

We prove the asymptotic normality of the discretized maximum likelihood estimator for the drift parameter in the homogeneous ergodic diffusion model.

In the paper we establish strong uniqueness of solution of a system of stochastic differential equations with random non-Lipschitz coefficients that involve both the square integrable continuous vector martingales and centered and non-centered Poisson measures.