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In this paper, we consider a joint drift rate control and two-sided impulse control problem in which the system manager adjusts the drift rate as well as the instantaneous relocation for a Brownian motion, with the objective of minimizing the total average state-related cost and control cost. The system state can be negative. Assuming that instantaneous upward and downward relocations take a different cost structure, which consists of both a setup cost and a variable cost, we prove that the optimal control policy takes an 
$\left\{ {\!\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right),\!\left\{ {{\mu ^{\ast}}(x)\,:\,x \in [ {{s^{\ast}},{S^{\ast}}}]} \right\}} \right\}$ form. Specifically, the optimal impulse control policy is characterized by a quadruple 
$\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right)$, under which the system state will be immediately relocated upwardly to 
${q^{\ast}}$ once it drops to 
${s^{\ast}}$ and be immediately relocated downwardly to 
${Q^{\ast}}$ once it rises to 
${S^{\ast}}$; the optimal drift rate control policy will depend solely on the current system state, which is characterized by a function 
${\mu ^{\ast}}\!\left( \cdot \right)$ for the system state staying in 
$[ {{s^{\ast}},{S^{\ast}}}]$. By analyzing an associated free boundary problem consisting of an ordinary differential equation and several free boundary conditions, we obtain these optimal policy parameters and show the optimality of the proposed policy using a lower-bound approach. Finally, we investigate the effect of the system parameters on the optimal policy parameters as well as the optimal system’s long-run average cost numerically.
We establish higher moment formulae for Siegel transforms on the space of affine unimodular lattices as well as on certain congruence quotients of 
$\mathrm {SL}_d({\mathbb {R}})$. As applications, we prove functional central limit theorems for lattice point counting for affine and congruence lattices using the method of moments.
In the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and 
$a-z$, respectively, where 
$a>0$ and 
$0< z < a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and 
$1-p$. The game continues until one of the two players is ruined. For even a and 
$0<z\leq {a}/{2}$, the family of distributions of the duration (total number of rounds) of the game indexed by 
$p \in [0,{\frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for 
${a}/{2} \leq z<a$, the family of distributions of the duration indexed by 
$p \in [{\frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for 
$z={a}/{2}$, in terms of the likelihood ratio order, the distribution of the duration is maximized over 
$p \in [0,1]$ by 
$p={\frac{1}{2}}$. The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.
The Brownian bridge or Lévy–Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. We focus on the uniform error. In particular, we show constructively that at level N, at which there are 
$d=2^N$ points evaluated on the Brownian path, the uniform error and its square, and the uniform error of geometric Brownian motion, have upper bounds of order 
$\mathcal {O}(\sqrt {\ln d/d})$, matching the known orders. We apply the results to an option pricing example.
Let 
${\mathrm{d}} X(t) = -Y(t) \, {\mathrm{d}} t$, where Y(t) is a one-dimensional diffusion process, and let 
$\tau(x,y)$ be the first time the process (X(t), Y(t)), starting from (x, y), leaves a subset of the first quadrant. The problem of computing the probability 
$p(x,y)\,:\!=\, \mathbb{P}[X(\tau(x,y))=0]$ is considered. The Laplace transform of the function p(x, y) is obtained in important particular cases, and it is shown that the transform can at least be inverted numerically. Explicit expressions for the Laplace transform of 
$\mathbb{E}[\tau(x,y)]$ and of the moment-generating function of 
$\tau(x,y)$ can also be derived.
The Chow–Robbins game is a classical, still partly unsolved, stopping problem introduced by Chow and Robbins in 1965. You repeatedly toss a fair coin. After each toss, you decide whether you take the fraction of heads up to now as a payoff, otherwise you continue. As a more general stopping problem this reads 
$V(n,x) = \sup_{\tau }\mathbb{E} \left [ \frac{x + S_\tau}{n+\tau}\right]$
, where S is a random walk. We give a tight upper bound for V when S has sub-Gaussian increments by using the analogous time-continuous problem with a standard Brownian motion as the driving process. For the Chow–Robbins game we also give a tight lower bound and use these to calculate, on the integers, the complete continuation and the stopping set of the problem for 
$n\leq 489\,241$
.
We consider the near-critical Erdős–Rényi random graph G(n, p) and provide a new probabilistic proof of the fact that, when p is of the form 
$p=p(n)=1/n+\lambda/n^{4/3}$
and A is large,
where 
\begin{equation*}\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp A^{-3/2}e^{-\frac{A^3}{8}+\frac{\lambda A^2}{2}-\frac{\lambda^2A}{2}},\end{equation*}
$\mathcal{C}_{\max}$
is the largest connected component of the graph. Our result allows A and 
$\lambda$
to depend on n. While this result is already known, our proof relies only on conceptual and adaptable tools such as ballot theorems, whereas the existing proof relies on a combinatorial formula specific to Erdős–Rényi graphs, together with analytic estimates.
