A stochastic differential equation of Wiener-Poisson type is considered in a $d$-dimensional bounded region. By using a penalization argument on the domain, we are able to prove the existence and uniqueness of solutions in the strong sense. The main assumptions are Lipschitzian coefficients, either convex or smooth domains and a regular outward reflecting direction. As a direct consequence, it is verified that the reflected diffusion process with jumps depends on the initial date in a Lipschitz fashion.

Let $x(t)$ be a Feller process on a complete separable metric space $A$ and consider the occupation measure $L_t(\omega, \cdot) = \int^t_0 \chi_{(\cdot)}(x(s)) ds$. The $I$-function is defined for $\mu \in \mathscr{P}(A)$, the set of probability measures on $A$, by $I(\mu) = -\inf_{u\in\mathscr{D}^+} \int_A (Lu/u)d\mu$ where $(L, \mathscr{D})$ is the generator of the process and $\mathscr{D}^+ \subset \mathscr{D}$ consists of the strictly positive functions in $\mathscr{D}$. The $I$-function determines the asymptotic rate of decay of $P((1/t)L_t(\omega, \cdot) \in G)$ for $G \subset \mathscr{P}(A)$. The first difficulty encountered in evaluating $I(\mu)$ is that the domain $\mathscr{D}$ is generally not known explicitly. In this paper, we prove a theorem which allows us to restrict the calculation of the infimum to a nice subdomain. We then apply this general result to diffusion processes with boundaries.

Let $X(t)$ be a diffusion process on $R^d$ with generator $L = (1/2)\nabla \cdot a\nabla + b\nabla$ and let $\{P_x\}, x \in R^d$, be the corresponding measures on paths. Pick $0 < t < T < \infty$ and consider the process on the time interval $\lbrack 0, t\rbrack$ conditioned to remain in a certain open, connected bounded region $G$ up to time $T$. We obtain a new process $Y^T(s), 0 \leqq s \leqq t$. Let $\tau_G = \inf\{s: X(s) \not\in G\}$. With certain hypotheses on $P_x(\tau_G > s)$ (which are always satisfied if $a^{-1}b$ is a gradient function), we show that $Y^T(s)$ is an inhomogeneous diffusion process and that as $T \rightarrow \infty, Y^T(s), 0 \leq s \leq t$ converges to a limiting homogeneous positive recurrent diffusion $Y(s), 0 \leq s \leq t$, with state space $G$. Since $t$ is arbitrary, we actually obtain a limiting process $Y(s), 0 \leq s < \infty$. The generator of the limiting process may be written in the form $L_G = (1/2)\nabla \cdot a\nabla + b\nabla + a(\nabla g_0/g_0)\nabla - a\nabla h_{g_0} \nabla$ where $g_0$ is the square root of the density of a measure $\mu_0$ which minimizes the $I$-function for the process, over all $\gamma \in \mathscr{P}(\bar{G})$, the set of probability measures on $\bar{G}$. The function $h_{g_0}$ appears in the explicit calculation of $I(\mu_0)$ and solves a certain variational equation. The invariant measure for the process is $\mu_0$.

In this paper it is shown that contrary to "linear" homogeneous nondegenerate diffusion processes, "nonlinear" diffusion processes can have a time-periodic law. We study four qualitatively different simple examples: a law-dependent Ornstein-Uhlenbeck process, a deterministic example, a one-dimensional example with a "semilinear" drift and a two-dimensional example with a linear drift.

It is proved that if $X_t$ is a diffusion generated by the operator $L = 1/2 \sum a_{ij}(x)\partial^2/\partial x_i\partial x_j + \sum u_0b_i(x)\partial/\partial x_i$ having periodic coefficients, then $\lambda^{-1/2}(X_{\lambda t} - \lambda u_0 \bar{b}t), t \geq 0$, converges in distribution to a Brownian motion as $\lambda \rightarrow \infty$. Here $\bar{b}$ is the mean of $b(x) = (b_1(x), \cdots, b_k(x))$ with respect to the invariant distribution for the diffusion induced on the torus $T^k = \lbrack 0, 1)^k$. The dispersion matrix of the limiting Brownian motion is also computed. In case $\bar{b} = 0$ this result was obtained by Bensoussan, Lions and Papanicolaou (1978). (See Theorem 4.3, page 401, as well as the author's remarks on page 529.) The case $\bar{b} \neq 0$ is of interest in understanding how solute dispersion in a porous medium behaves as the liquid velocity increases in magnitude.

We consider a storage process $X(t)$ having a compound Poisson process as input and general release rules, and a nonnegative additive functional $Z(t) = \int^t_0 f(X(s)) ds$. Under the situation that the input rate is equal to the maximal output rate, it is shown for a suitable class of functions of $f$ that an appropriate normalization of the process $Z(t)$ converges weakly to a process which is represented as a constant times the local time of a Bessel process at zero.

We study three rules for the development of a sequence of finite subtrees $\{t_n\}$ of an infinite $m$-ary tree $t$. Independent realizations $\{\omega(n)\}$ of a stationary ergodic process $\{\omega\}$ on $m$ letters are used to trace out paths in $t$. In the first rule, $t_n$ is formed by adding a node to $t_{n - 1}$ at the first location where the path defined by $\omega (n)$ leaves $t_{n - 1}$. The second and third rules are similar, but more complicated. For each rule, the height $L_n$ of the added node is shown to grow, in probability, as $\ln n$ divided by $h$ the entropy per symbol of the generic process. A typical retrieval time has the same behavior. On the other hand, $\lim \inf_nL_n/\ln n = \sigma_1, \lim \sup_n L_n/\ln n = \sigma_2$ a.s., where the constants $\sigma_1, \sigma_2$, are, in general, different, depend on the rule in use, and $\sigma_1 < 1/h < \sigma_2$. It is proven along the way that the height of $t_n$ grows as $\sigma_2\ln n$ with probability one.

We study the weak convergence in $D\lbrack 0, 1\rbrack$ of the quadratic form $\sum^{\lbrack Nt\rbrack}_{j = 1} \sum^{\lbrack Nt\rbrack}_{k = 1} a_{j - k} H_m (X_j)H_m(X_k)$, adequately normalized. Here $a_s, -\infty < s < \infty$ is a symmetric sequence satisfying $\sum |a_s| < \infty, H_m$ is the $m$th Hermite polynomial and $\{X_j\}, j \geq 1$, is a normalized Gaussian sequence with covariances $r_k \sim k^{-D} L(k)$ as $k \rightarrow \infty$, where $0 < D < 1$ and $L$ is slowly varying. We prove that, for all $m \geq 1$, the limit is Brownian motion when $1/2 < D < 1$ and it is the non-Gaussian Rosenblatt process when $0 < D < 1/2$.

Let $(\mathscr{K} (\mathbb{B}), \delta)$ denote the nonempty compact subsets of a separable Banach space $\mathbb{B}$ topologized by the Hausdorff metric. Let $K, K_1, K_2$ be i.i.d. random compact convex sets in $\mathbb{B}. K$ is called $p$-stable if for each $\alpha, \beta \geq 0$ there exist compact convex sets $C$ and $D$ such that $\mathscr{L}(\alpha K_1 + \beta K_2 + C) = \mathscr{L}((\alpha^p + \beta^p)^{1/p}K + D)$ where + refers to Minkowski sum. A characterization of the support function for a compact convex set is provided and then utilized to determine all $p$-stable random compact convex sets. If $1 \leq p \leq 2$, they are trivial, merely translates of a fixed compact convex set by a $p$-stable $\mathbb{B}$-valued random variable. For $0 < p < 1$, they are translates of stochastic integrals with respect to nonnegative independently scattered $p$-stable measures on the unit ball of $\operatorname{co} \mathscr{K}(\mathbb{B})$. Deconvexification is also discussed. The domains of attraction of $p$-stable random compact convex sets with $0 < p < 1$ are completely characterized. The case $1 < p \leq 2$ is considered in Gine, Hahn and Zinn (1983). Precedents: Lyashenko (1983) and Vitale (1983) characterize the Gaussian random compact sets in $\mathbb{R}^d$.

Let $Z^{(n)}_m$ represent the $m$th largest order statistic in a random sample of size $n$ from a distribution $F$. If $m = m(n)$ is an intermediate sequence such that $m \rightarrow \infty$ and $m/n \rightarrow 0$ as $n \rightarrow \infty$, the intermediate order statistics of the form $Z^{(n)}_{\lbrack mt_1\rbrack}, \cdots, Z^{(n)}_{\lbrack mt_k\rbrack}$, for $0 < t_1 < \cdots < t_k$, can be used jointly for making statistical inferences about the upper tail of $F$. We find the asymptotic joint distribution of order statistics of this form, for various types of underlying distributions $F$, by determining the limit (weak convergence) of a stochastic process of the form $(Z^{(n)}_{\lbrack mt\rbrack} - \beta^{(n)}_{mt})/\alpha^{(n)}_{mt}, t > 0$, for appropriate normalizing functions $\alpha^{(n)}_{mt}, > 0, \beta^{(n)}_{mt}$.

We prove that the critical probability for site percolation on the three-dimensional cubic lattice satisfies the inequality $p^{(3)}_c < 1/2$. An application to the three-dimensional Ising model is given.

We study excursion laws of two Markov processes $X, \hat{X}$ in duality, out of closed, homogeneous optional sets $M, \hat{M}$ generated by a pair of dual terminal times $R$ and $\hat{R}$. The duality assumptions enable one to compute the laws of $(R, X_R)$ using the pair of exit systems of the two processes. With this, one is able to compute the conditional laws of the excursions, given the boundary process. It turns out that these depend only on the values of the boundary process at the beginning and end of the excursions. We obtain for all $x, y(x \neq y)$ the laws $p^{x,y}$ of the excursions conditioned to start at $x$ and end at $y$. Under these laws, the excursion process is a homogeneous Markov process. It's transition laws are computed. We use the results above to treat excursions that straddle perfect terminal times.

Existence and uniqueness of solutions is established for stochastic Volterra integral equations driven by right continuous semimartingales. This resolves (in the affirmative) a conjecture of M. Berger and V. Mizel.

Previously we established that the martingales $M^\theta(t) = \bigg(\theta, Y(t) - Y(0) - \frac{1}{2} \int^t_0 \int_\Xi A^2(\xi) Y(s)\mu (d\xi) ds\bigg),$ with quadratic variation process $V^\theta(t) = \int^t_0 \int_\Xi (\theta, A(\xi) Y(s))^2\mu (d\xi) ds,$ characterize the limit process for a sequence of random evolutions. This paper shows the equivalence of this presentation to the questions of existence and uniqueness of the stochastic integral equation $Y(t) = Y(0) + \frac{1}{2} \int^t_0 \int_\Xi A^2(\xi) Y(s)\mu (d\xi) ds + \int^t_0 \int_\Xi A(\xi) Y(s) W(d\xi) ds).$ The paper proceeds in giving strong existence and uniqueness theorems for this integral equation.

This paper provides an explanation of Siegmund's duality for absorbing and reflecting Markov processes by means of a graphical representation of the type used in the analysis of infinite particle systems. It is shown that coupled realisations of a Markov process conditioned to start at each of the points of the state space can be generated on the same probability space in such a way that their ordering is preserved. Using the same probability space a specific construction is then given for the dual process.

The prophet inequality for a sequence of independent nonnegative random variables shows that the ratio of the mean of the maximum of the sequence to the optimal expected return using stopping times is always bounded by 2; i.e., on average, the proportional advantage of a prophet with complete foresight over a gambler using nonanticipating stopping rules is at most 2. Here, an inequality linking the mean of the sum of the $k$ largest order statistics of the sequence and the optimal expected return is derived. This implies that if the $k$ largest order statistics are close to the maximum in mean then the proportional advantage of the prophet is at most of order $(k + 1)/k$. An extension of the additive prophet inequality for uniformly bounded independent random variables is also given.

A multivariate generalization of Shaked's bivariate families which are dependent by total positivity (DTP) is introduced. Some interrelationships and inequalities are generalized. Monotonicity properties of conditional hazard rate and mean residual life functions of some multivariate DTP families are investigated. Relationships with other positive dependence concepts are given.

A family of concepts of dependence by reverse regular (DRR) rule is introduced. Interrelationships among DRR random variables are investigated. Relationships with other concepts of negative dependence are discussed.

The classes of the $s$-selfdecomposable and decomposable probability measures are related to the limit distributions of sequences of random variables deformed by some nonlinear or linear transformations respectively. Both are characterized in many different ways, among others as distributions of some random integrals. In particular we get that each selfdecomposable probability measure is $s$-selfdecomposable. This and other relations between these two classes seem to be rather unexpected.

Let $F(c_0w)$ be a functional of the Wiener process with variance parameter $c^2_0$ and let $F(cw)$ be an extension of $F(c_0w)$ to $F(cw), c \in (0, c_0)$. Relations are derived between the Malliavin derivatives, between the derivatives with respect to the scale parameter $(\partial F(\rho cw)/\partial\rho)_{p = 1}$ and `noncoherent derivatives' such as $(dE(F(cw + \sqrt\varepsilon c\tilde{w}) \mid w)/d\varepsilon)_{\varepsilon = 0}$ where $\tilde{w}$ is another Wiener process independent of $w$ and between the generator of the nontime-homogeneous Ornstein-Uhlenbeck process.

Suppose $X_t$ is a diffusion, reflecting at 0, with speed measure $m(dx)$. We show, under a mild regularity condition on $m$, that $\lim\sup_{t\rightarrow 0} X_t/h^{-1}(t) = c$, a.s., where $c$ is a nonzero constant and $h(t) = tm\lbrack 0, t\rbrack/\log|\log t|$. The analogue to Chung's law of the iterated logarithm is also obtained.

A local central limit theorem of the type due to L. A. Shepp is proved for certain stationary sequences of associated random variables.

In connection with a random vector $(X, Y)$ in the unit square $Q$ and a couple $(m, n)$ of positive integers, we consider all discretizations of the continuous probability distribution of $(X, Y)$ that are obtained by an $m \times n$ cartesian decomposition of $Q$. We prove that $Y$ is a (continuous and invertible) function of $X$ if and only if for each $m, n$ the maximum entropy of the finite distributions equals $\log(m + n - 1)$