This paper develops and analyzes a time-dependent optimal stopping problem and its application to the decision making process concerning organ transplants. Offers (organs for transplant) appear at jump times of a Poisson process. The values of the offers are i.i.d. random variables with a known distribution function. These values express the degree of histocompatibility between the donor and the recipient. The sequence of offers is independent of the jump times of the Poisson process. The decision...

Suppose that the process $X=\{X_n,n\in \mathbb{N}\}$ is observed sequentially. There are two random moments of time ${\theta }_1$ and ${\theta }_2$, independent of X, and X is a Markov process given ${\theta }_1$ and ${\theta }_2$. The transition probabilities of X change for the first time at time ${\theta }_1$ and for the second time at time ${\theta }_2$. Our objective is to find a strategy which immediately detects the distribution changes with maximal probability based on observation of X. The corresponding problem of double optimal stopping is constructed. The optimal strategy is found...

Niemiro and Zieliński (2007) have recently obtained uniform asymptotic normality for the Bernoulli scheme. This paper concerns a similar problem. We show the uniform central limit theorem for a sequence of stationary random variables.

Given a two-dimensional fractional multiplicative process ${\left(F_t\right)}_{t\in [0,1]}$ determined by two Hurst exponents $H_1$ and $H_2$, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0,1]$ by $F$ if and only if $H_1=H_2$.

There are two kinds of universal schemes for estimating residual waiting times, those where the error tends to zero almost surely and those where the error tends to zero in some integral norm. Usually these schemes are different because different methods are used to prove their consistency. In this note we will give a single scheme where the average error is eventually small for all time instants, while the error itself tends to zero along a sequence of stopping times of density one.

Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate $\mathbb{P}\left(su{p}_{t\ge 0}\right|Y_t\left|\ge 1\right)\le 3.375...\parallel X\parallel ₁$. As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.

Let α ∈ [0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate $\parallel Y{\parallel }_W\le \left(2(\alpha +1)²\right)/(2\alpha +1)\parallel X{\parallel }_{L^{\infty }}$. Here W is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Let $\left(X_j\right)$ be a sequence of independent identically distributed random operators on a Banach space. We obtain necessary and sufficient conditions for the Abel means of $X_n...X_2{X}_1$ to belong to Hardy and Lipschitz spaces a.s. We also obtain necessary and sufficient conditions on the Fourier coefficients of random Taylor series with bounded martingale coefficients to belong to Lipschitz and Bergman spaces.

We offer the quantitative estimation of stability of risk-sensitive cost optimization in the problem of optimal stopping of Markov chain on a Borel space $X$. It is supposed that the transition probability $p(·|x)$, $x\in X$ is approximated by the transition probability $\tilde{p}(·|x)$, $x\in X$, and that the stopping rule ${\tilde{f}}_{*}$ , which is optimal for the process with the transition probability $\tilde{p}$ is applied to the process with the transition probability $p$. We give an upper bound (expressed in term of the total variation distance: ${sup}_{x\in X}\parallel p(·|x)-\tilde{p}(·|x)\parallel )$ for...

We give a relation between the sign of the mean of an integer-valued, left bounded, random variable $X$ and the number of zeros of $1-\Phi \left(z\right)$ inside the unit disk, where $\Phi$ is the generating function of $X$, under some mild conditions

In this paper we solve the basic fractional analogue of the classical linear-quadratic gaussian regulator problem in continuous time. For a completely observable controlled linear system driven by a fractional brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion.

In this paper we solve the basic fractional analogue of the classical linear-quadratic Gaussian regulator problem in continuous time. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes a quadratic performance criterion.

We study actuarial methods of option pricing in a fractional Black-Scholes model with time-dependent volatility. We interpret the option as a potential loss and we show that the fair premium needed to insure this loss coincides with the expectation of the discounted claim payoff under the average risk neutral measure.