On It\^{o}'s formula for elliptic diffusion processes

Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83--109] prove an extension of It\^{o}'s formula for $F(X_t,t)$, where $F(x,t)$ has a locally square-integrable derivative in $x$ that satisfies a mild continuity condition in $t$ and $X$ is a one-dimensional diffusion process such that the law of $X_t$ has a density satisfying certain properties. This formula was expressed using quadratic covariation. Following the ideas of Eisenbaum [Potential Anal. 13 (2000) 303--328] concerning Brownian motion, we show that one can re-express this formula using integration over space and time with respect to local times in place of quadratic covariation. We also show that when the function $F$ has a locally integrable derivative in $t$, we can avoid the mild continuity condition in $t$ for the derivative of $F$ in $x$.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6049 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Stochastic delay equations with non-negativity constraints driven by fractional Brownian motion

In this note we prove an existence and uniqueness result for the solution of multidimensional stochastic delay differential equations with normal reflection. The equations are driven by a fractional Brownian motion with Hurst parameter $H>1/2$. The stochastic integral with respect to the fractional Brownian motion is a pathwise Riemann--Stieltjes integral.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ327 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Stochastic epidemic SEIRS models with a constant latency period

In this paper we consider the stability of a class of deterministic and stochastic SEIRS epidemic models with delay. Indeed, we assume that the transmission rate could be stochastic and the presence of a latency period of $r$ consecutive days, where $r$ is a fixed positive integer, in the "exposed" individuals class E. Studying the eigenvalues of the linearized system, we obtain conditions for the stability of the free disease equilibrium, in both the cases of the deterministic model with and without delay. In this latter case, we also get conditions for the stability of the coexistence equilibrium. In the stochastic case we are able to derive a concentration result for the random fluctuations and then, using the Lyapunov method, that under suitable assumptions the free disease equilibrium is still stable