In this paper we consider the problem of averaging for a class of piecewise deterministic Markov processes (PDMPs) whose dynamic is constrained by the presence of a boundary. On reaching the boundary, the process is forced to jump away from it. We assume that this boundary is attractive for the process in question in the sense that its averaged flow is not tangent to it. Our averaging result relies strongly on the existence of densities for the process, allowing us to study the average number of crossings of a smooth hypersurface by an unconstrained PDMP and to deduce from this study averaging results for constrained PDMPs.

We prove existence and uniqueness for the solution of a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized Itô rule valid for functions with an absolutely continuous derivative and applicable to solutions of mixed fractional stochastic differential equations with Lipschitz coefficients, which plays a key role in our proof of existence and uniqueness. The proof of such a formula is new and relies on showing the existence of a density of the law under mild assumptions on the diffusion coefficient.

Motivated by applications to wireless networks, cloud computing, data centers, etc., stochastic processing networks have been studied in the literature under various asymptotic regimes. In the heavy traffic regime, the steady-state mean queue length is proved to be $\Theta({1}/{\epsilon})$ , where $\epsilon$ is the heavy traffic parameter (which goes to zero in the limit). The focus of this paper is on obtaining queue length bounds on pre-limit systems, thus establishing the rate of convergence to heavy traffic. For the generalized switch, operating under the MaxWeight algorithm, we show that the mean queue length is within $\textrm{O}({\log}({1}/{\epsilon}))$ of its heavy traffic limit. This result holds regardless of the complete resource pooling (CRP) condition being satisfied. Furthermore, when the CRP condition is satisfied, we show that the mean queue length under the MaxWeight algorithm is within $\textrm{O}({\log}({1}/{\epsilon}))$ of the optimal scheduling policy. Finally, we obtain similar results for the rate of convergence to heavy traffic of the total queue length in load balancing systems operating under the ‘join the shortest queue’ routeing algorithm.

We present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.

Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

Nearly fifty years after the introduction of skew Brownian motion by Itô and McKean (1963), the first passage time distribution remains unknown. In this paper we first generalize results of Pitman and Yor (2011) and Csáki and Hu (2004) to derive formulae for the distribution of ranked excursion heights of skew Brownian motion, and then use these results to derive the first passage time distribution.

The limit behaviour in probability of realised quadratic variation is discussed under a relatively simple ambit process setting. The relation of this to the underlying volatility/intermittency field is in focus, especially as concerns the question of no volatility/intermittency memory.

We study the last passage time and its asymptotic distribution for minimum contrast estimators defined through the minimization of a convex criterion function based on U-functionals. This includes cases of non-smooth estimators for vector valued parameters. We also derive a Bahadur-type representation and the law of iterated logarithms for such estimators.

We obtain characterizations of densities on the real line and provide solutions to stochastic equations using the Gibbs sampler. Particular stochastic equations considered are of the type X =dB(X+C) and X =dBX+C.

A Cox risk process with a piecewise constant intensity is considered where the sequence (Li) of successive levels of the intensity forms a Markov chain. The duration σi of the level Li is assumed to be only dependent via Li. In the small-claim case a Lundberg inequality is obtained via a martingale approach. It is shown furthermore by a Lundberg bound from below that the resulting adjustment coefficient gives the best possible exponential bound for the ruin probability. In the case where the stationary distribution of Li contains a discrete component, a Cramér–Lundberg approximation can be obtained. By way of example we consider the independent jump intensity model (Björk and Grandell 1988) and the risk model in a Markovian environment (Asmussen 1989).

It is shown that the critical two-level (2, d, 1, 1)-superprocess is persistent in dimensions d greater than 4. This complements the extinction result of Wu (1994) and implies that the critical dimension is 4.

We consider two independent Dawson-Watanabe super-Brownian motions, Y1 and Y2. These processes are diffusions taking values in the space of finite measures on ℝd. We show that if d ≤ 5 then with positive probability there exist times t such that the closed supports of intersect; whereas if d > 5 then no such intersections occur. For the case d ≤ 5, we construct a continuous, non-decreasing measure–valued process L(Y1, Y2), the collision local time, such that the measure defined by , is concentrated on the set of times and places at which intersections occur. We give a Tanaka-like semimartingale decomposition of L(Y1, Y2). We also extend these results to a certain class of coupled measurevalued processes. This extension will be important in a forthcoming paper where we use the tools developed here to construct coupled pairs of measure-valued diffusions with point interactions. In the course of our proofs we obtain smoothness results for the random measures that are uniform in t. These theorems use a nonstandard description of Yi and are of independent interest.