Markov processes conditioned on their location at large exponential times

For $i = 1, \cdots ,d$, let $B_{s_i }^i $ be a one-dimensional Brownian motion on the interval $[0,a_i ]$ with absorption at the endpoints. At each instant in time, one must decide to run some subset of these d Brownian motions while holding the others fixed at their current state. The resulting process evolves in the rectangle $D = [0,a_1 ] \times \cdots \times [0,a_d ]$. If, at some instant, one decides to freeze all of the Brownian motions, then a reward is received in accordance with this final position. Two types of reward functions are considered. First, it is assumed that the reward is zero everywhere in D, except along the d edges that correspond to the coordinate axes. Along these edges, it is given by $C^3 $ strictly concave functions $\gamma _i (x_i )$, which are zero at the endpoints 0 and $a_i $ of their domains. The optimal control for this problem has a simple description. Let \[\Gamma _i \left( {x_i } \right) = - \int_0^{x_i } {u\gamma ''_i } (u)du\] and put \[M_i = \left\{ {x \in D:\Gamma _i \left( {x_i } \right) = \mathop {\max }\limits_j \Gamma _j \left( {x_i } \right)} \right\}.\] It is proved that the optimal control is: On $M_i $ run any Brownian motion except the ith and stop the first time an edge is reached. The second class of reward functions are assumed to be zero everywhere except on the facets of D that meet at the origin. On the ith such facet (i.e., where $x_i = 0$), the reward function is the product of $\gamma _j (x_j )$ for $j \ne i$. Put \[N_i = \left\{ {x \in D:\Gamma _i \left( {x_i } \right) = \mathop {\min }\limits_j \Gamma _j \left( {x_j } \right)} \right\}.\] The optimal control is: On $N_i $ run the ith Brownian motion and stop when a facet of D is reached.

Suppose that S is a subordinator with a nonzero drift and W is an independent 1-dimensional Brownian motion. We study the subordinate Brownian motion X defined by X t = W(S t ). We give sharp bounds for the Green function of the process X killed upon exiting a bounded open interval and prove a boundary Harnack principle. In the case when S is a stable subordinator with a positive drift, we prove sharp bounds for the Green function of X in (0, ∞ ), and sharp bounds for the Poisson kernel of X in a bounded open interval.

This paper examines a retrial inventory system with an orbit of infinite size. An arriving customer, finding the stock out in the system, proceeds to the orbit with probability a and is lost forever with probability (1 – a). The customers in the orbit are assumed not only to retry for their demand, may renege from the orbit at random time. We model this situation by assuming Poisson arrivals for customers, exponential lead time for orders placed under (s, S) ordering policy, exponential time between successive attempts of retrial of a customer and exponential times for reneging by customers at the orbit. The condition for ergodicity of the system is obtained. The joint probability distribution of the number of customers in the system and the inventory level is obtained in steady state case. The measures of system performance in the steady state are derived and the long run total expected cost rate is derived. The results are illustrated with numerical examples.

Pitman transforms and Brownian motion in the interval viewed as an affine alcove

On the Fourier structure of the zero set of fractional Brownian motion

Upper estimates for capacities are studied for symmetric Dirichlet forms which do not necessarily admit intrinsic metric. Firstly, a general method of obtaining a sharp estimate from upper estimates for cut-off functions are provided in the local and non-local cases. Secondly, a capacitary upper estimate is established for the skew product of two symmetric Dirichlet forms for which suitable capacitary estimates are given. Several examples of capacitary upper inequalities in the local and non-local cases are also given. Especially, an upper estimate for the capacity is proved for the symmetric Dirichlet form associated to the Markov process subordinated (in the Bochner sense) to the skew product of two one-dimensional Brownian motions with respect to the local time of the first Brownian motion at the origin. This estimate is used in the recent work by the present author to establish the sharp criteria for recurrence and transience of the above-mentioned process.

We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability distribution that the sum of two one-dimensional Brownian motions will exit from the unit interval before time t, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.

We give best tests and corresponding power functions for testing the hypothesis of no drift versus the alternative of positive linear drift in a one-dimensional Brownian motion process. The tests are based on a simple random sample from a first passage time distribution associated with the Brownian motion process. In reliability applications the first passage times would be the lifetimes of objects which fail when some noisy physical parameter reaches a critical value. In this case the life distribution is determined by the assumption that the physical parameter behaves like a Brownian motion and then the problem is equivalent to testing the hypothesis of infinite mean life versus the alternative of finite mean life. These tests are also applicable (and remain best) in the case of a single realization of Brownian motion for which only certain crossing time data is available.

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.

Let B be a one-dimensional Brownian motion and \(f: {\mathbb{R}}{\longrightarrow}{\mathbb{R}}\) be a Borel function that is locally integrable on \(\mathbb{R}\backslash \{0\}\). We present necessary and sufficient conditions (in terms of the function f) for the existence of the limit $$\lim_{\varepsilon\downarrow 0}\int_0^t f(B_s)I(\vert B_s\vert > \varepsilon)ds$$ in probability and almost surely. This limit (if it exists) can be called the principal value of the integral \(\int_0^tf(B_s)ds\).

With the help of several different methods, a closed formula is obtained for the laws of the exponential functionals of Brownian motion with drift, taken at certain random times, particularly exponential times, which are assumed to be independent of the Brownian motion.

We obtain closed-form expressions for the value of the joint Laplace transform of the running maximum and minimum of a diffusion-type process stopped at the first time at which the associated drawdown or drawup process hits a constant level before an independent exponential random time. It is assumed that the coefficients of the diffusion-type process are regular functions of the current values of its running maximum and minimum. The proof is based on the solution to the equivalent inhomogeneous ordinary differential boundary-value problem and the application of the normal-reflection conditions for the value function at the edges of the state space of the resulting three-dimensional Markov process. The result is related to the computation of probability characteristics of the take-profit and stop-loss values of a market trader during a given time period.

One obtains the analogues of the “Ray's description” of the local time for one-dimensional Brownian motion, valid for arbitrary homogeneous Markov chains with discrete time and countable state space. Contrary to the case of the Brownian motion, one establishes the absence of the Markov property for the process of the occupation time in the case of the simplest one-dimensional symmetric random walk.

Stochastic processes with random reinforced relocations have been introduced in a series of papers by Boyer and co-authors (Boyer and Solis Salas 2014, Boyer and Pineda 2016, Boyer, Evans and Majumdar 2017) to model animal foraging behaviour. Such a process evolves as a Markov process, except at random relocation times, when it chooses a time at random in its whole past according to some ‘memory kernel’, and jumps to its value at that random time. We prove a quenched large deviation principle for the value of the process at large times. The difficulty in proving this result comes from the fact that the process is not Markovian due to relocations. Furthermore, the random inter-relocation times act as a random environment.