Modern Stochastics: Theory and Applications

A new multi-factor short rate model is presented which is bounded from below by a real-valued function of time. The mean-reverting short rate process is modeled by a sum of pure-jump Ornstein–Uhlenbeck processes such that the related bond prices possess affine representations. Also the dynamics of the associated instantaneous forward rate is provided and a condition is derived under which the model can be market-consistently calibrated. The analytical tractability of this model is illustrated by the derivation of an explicit plain vanilla option price formula. With view on practical applications, suitable probability distributions are proposed for the driving jump processes. The paper is concluded by presenting a post-crisis extension of the proposed short and forward rate model.

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, a solution is given to reflected backward doubly stochastic differential equations when the barrier is not necessarily right-continuous, and the noise is driven by two independent Brownian motions and an independent Poisson random measure. The existence and uniqueness of the solution is shown, firstly when the coefficients are stochastic Lipschitz, and secondly by weakening the conditions on the stochastic growth coefficient.

We find the best approximation of the fractional Brownian motion with the Hurst index $H\in (0,1/2)$ by Gaussian martingales of the form ${\textstyle\int _{0}^{t}}{s^{\gamma }}d{W_{s}}$, where W is a Wiener process, $\gamma >0$.

A multivariate errors-in-variables (EIV) model with an intercept term, and a polynomial EIV model are considered. Focus is made on a structural homoskedastic case, where vectors of covariates are i.i.d. and measurement errors are i.i.d. as well. The covariates contaminated with errors are normally distributed and the corresponding classical errors are also assumed normal. In both models, it is shown that (inconsistent) ordinary least squares estimators of regression parameters yield an a.s. approximation to the best prediction of response given the values of observable covariates. Thus, not only in the linear EIV, but in the polynomial EIV models as well, consistent estimators of regression parameters are useless in the prediction problem, provided the size and covariance structure of observation errors for the predicted subject do not differ from those in the data used for the model fitting.