Showing papers in "Theory of Probability and Its Applications in 2004"
TL;DR: In this article, the Hartman-Watson approach is used for valuing Asian options in terms of Bessel processes, and the Laplace transform results are shown to be equivalent to the results in this paper.
Abstract: This paper is motivated by questions about averages of stochastic processeswhich originate in mathematical finance, originally in connection with valuing the so-called Asian options. Starting with [M. Yor, {\em Adv. Appl. Probab.}, 24 (1992), pp. 509--531], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the Hartman--Watson theory of [M. Yor, {\em Z. Wahrsch. Verw.\ Gebiete}, 53 (1980), pp. 71--95]. Consequences of this approach for valuing Asian options proper have been spelled out in [H. Geman and M. Yor, {\em Math. Finance}, 3 (1993), pp. 349--375] whose Laplace transform results were in fact regarded as a significant advance. Unfortunately, a number of difficulties with the key results of this last paper have surfaced which are now addressed in this paper. One of them in particular is of a principal nature and originates with the Hartman--Watson approach itself: this approach is in general applicable without modifications only if it...
77 citations
TL;DR: This paper fills a gap in previous works by proving the conjectured formula for the classical entanglement-assisted capacity of a quantum channel with additive constraint (such as the Bosonic Gaussian channel) and gives sufficient conditions under which suprema in the capacity formulas are achieved.
Abstract: In this paper we fill a gap in previous works by proving the conjectured formula for the classical entanglement-assisted capacity of a quantum channel with additive constraint (such as the Bosonic Gaussian channel). Our main tools are the coding theorem for classical-quantum constrained channels and a finite-dimensional approximation of the input density operators for entanglement-assisted capacity. We also give sufficient conditions under which suprema in the capacity formulas are achieved.
64 citations
TL;DR: In this paper, the authors consider the problem of solving linear operator equations from noisy data under the assumptions that the singular values of the operator decrease exponentially fast and that the underlying solution is also exponentially smooth in the Fourier domain.
Abstract: We consider the problem of solving linear operator equations from noisy data under the assumptions that the singular values of the operator decrease exponentially fast and that the underlying solution is also exponentially smooth in the Fourier domain. We suggest an estimator of the solution based on a running version of block thresholding in the space of Fourier coefficients. This estimator is shown to be sharp adaptive to the unknown smoothness of the solution.
49 citations
TL;DR: In this paper, the relation between the relative entropy minimization problem and statistical physics, information theory, and financial mathematics is discussed, and the explicit form of the minimal entropy martingale measure in the general discrete-time asset price model is provided.
Abstract: This paper deals with some problems related to the relative entropy minimization under linear constraints. We discuss the relation between this problem and statistical physics, information theory, and financial mathematics. Furthermore, in financial mathematics we provide the explicit form of the minimal entropy martingale measure in the general discrete-time asset price model. We also give the explicit solution of the problem of the exponential utility maximization in the general discrete-time asset price model.
47 citations
TL;DR: In this paper, the authors introduced the concept of localization in the general theory of stochastic processes and derived a canonical representation of local martingales as for local Martingales.
Abstract: This paper introduces the concept of {\em $\sigma$-localization}, which is a generalization of localization in the general theory of stochastic processes. The $\sigma$-localized class derived from the set of martingales is the class of {\em $\sigma$-martingales}, which plays an important role in mathematical finance. These processes and the corresponding {\em $\sigma$-martingale measures} are considered in detail. By extending the stochastic integral with respect to compensated random measures, a canonical representation of $\sigma$-martingales as for local martingales is derived.
37 citations
TL;DR: In this article, a closed form of the ruin probability for Levy processes with arbitrary, positive, and mixed exponentially negative jumps is given, with arbitrary positive and mixed negative jumps being possible at a constant rate.
Abstract: We give a closed form of the ruin probability for Levy processes, possible killed at a constant rate, with arbitrary, positive, and mixed exponentially negative jumps.
35 citations
TL;DR: In this paper, it was shown that if the number of individuals at time n in a branching process in a random environment generated by independent identically distributed random probability generating functions is critical, then there exists a limit in distribution.
Abstract: Let $Z_n$~be the number of individuals at time n in a branching process in a random environment generated by independent identically distributed random probability generating functions~$f_0(s), f_1(s),\ldots, f_n(s),\ldots\;$. Let \begin{eqnarray*} X_i&=&\log f_{i-1}^{\prime}(1),\qquad i=0,1,2,\ldots\,; \\ S_0 &=&0,\quad S_n=X_1+\cdots+X_n,\qquad n\ge 1. \end{eqnarray*} It is shown that if $Z_n$ is, in a sense, ``critical," then there exists a limit in distribution $$ \lim_{n\to\infty}\exp\Big\{-\min_{0\le j\le n}S_j\Big\}\, {\bf P}\{Z_n > 0\,\big|\, f_0,\ldots,f_{n-1}\}=\zeta, $$ where $\zeta$ is a proper random variable positive with probability~1. In addition, it is shown that for a ``typical" realization of the environment the number of individuals~$Z_n$ given $\{Z_n > 0\}$ grows as $\exp\{S_n-\min_{0\le j\le n}S_j\}$ (up to a positive finite random multiplier).
33 citations
TL;DR: In this article, a distribution of the first passage time over a level for the Ornstein-Uhlenbeck process with jumps is shown to be exponentially bounded for the case of absence of positive jumps.
Abstract: Using martingale technique, we show that a distribution of the first-passage time over a level for the Ornstein--Uhlenbeck process with jumps is exponentially bounded. In the case of absence of positive jumps, the Laplace transform for this passage time is found. Further, the maximal inequalities are also given when the marginal distribution is stable.
26 citations
TL;DR: The asymptotic behavior of the survival probability of an intermediate subcritical branching process in a random environment was studied in this article, where a transformation of the reproduction law of the offspring number is attracted to a stable law.
Abstract: The asymptotic behavior of the survival probability of an intermediate subcritical branching process $Z_n$ in a random environment is found when a transformation of the reproduction law of the offspring number is attracted to a stable law $\alpha\in (1,2]$. It is shown that the distribution of the random variable $\{Z_n\}$ given $Z_n>0$ converges to a nondegenerate distribution as $n\to\infty$.
22 citations
TL;DR: In this paper, the authors propose a method for finding stochastic integral representations based on the It\^o formula for the Stochastic integral associated with the Brownian motion.
Abstract: For functionals $S=S(\omega)$ of the Brownian motion~B, we propose a method for finding stochastic integral representations based on the It\^o formula for the stochastic integral associated with~B. As an illustration of the method, we consider functionals of the ``maximal" type: $S_T$, $S_{T_{-a}}$, $S_{g_{T}}$, and $S_{\theta_T}$, where $S_T=\max_{t\le T}B_t$ , $S_{T_{-a}}=\max_{t\le T_{-a}}B_t$ with $T_{-a}=\inf\{{t>0:}\allowbreak B_t=-a\}$, $a>0$, and $S_{g_{T}}=\max_{t\le g_{T}} B_t$, $S_{\theta_T}=\max_{t\le \theta_T}B_t$, $g_{ T}$ and $\theta_T$ are {\em non}-Markov times: $g_{T}$~is the time of the last zero of Brownian motion on $[0, T]$ and $\theta_T$~is a time when the Brownian motion achieves its maximal value on $[0,T]$.
17 citations
TL;DR: In this article, an upper bound for the total variation distance between the generalized polynomial distribution and a finite signed measure, which is the convolution of two finite signed measures, one of which is of Kornya-Presman type, was presented.
Abstract: We present an upper bound for the total variation distance between the generalized polynomial distribution and a finite signed measure, which is the convolution of two finite signed measures, one of which is of Kornya--Presman type. In the one-dimensional Poisson case, such a finite signed measure was first considered by K. Borovkov and D. Pfeifer [{\em J. Appl.\ Probab.}, 33 (1996), pp. 146--155].We give asymptotic relations in the one-dimensional case, and, as an example, the independent identically distributed record model is investigated.It turns out that here the approximation is of order $O(n^{-s}(\ln n)^{-{(s+1)/2}})$ for s being a fixed positive integer, whereas in the approximation with simple Kornya--Presman signed measures, we only have the rate $O((\ln n)^{-(s+1)/2})$.
TL;DR: In this article, it was shown that the optimal feedback control in the standard nonhomogeneous LQG problem with infinite horizon has the following property: there is a constant $b_*$ such that, whatever $b> b_*, the deficiency process of optimal control with respect to any possible control u, i.e., the difference $J_T(\widehat u\hspace*{0.2pt})- J_T(u)$ between the optimal cost process and the cost process corresponding to controlling u, is majorated at infinity
Abstract: We show that the optimal feedback control $\widehat u$ in the standard nonhomogeneous LQG-problem with infinite horizon has the following property. There is a constant $b_*$ such that, whatever $b> b_*$ is, the deficiency process of optimal control with respect to any possible control u, i.e., the difference $J_T(\widehat u\hspace*{0.2pt})- J_T(u)$ between the optimal cost process $J_T(\widehat u\hspace*{0.2pt})$ and the cost process corresponding to control u, is majorated at infinity by a deterministic function $b\log T$. In other words, $b\log T$ is an upper function for any deficiency process. This result, combined with an example of an LQG-regulator where, for certain $b>0$, the function $b\log T$ is not an upper function for certain deficiency processes, gives an answer to the long-standing open problem about the best possible rate function for sensitive probabilistic criteria. Our setting covers the optimal tracking problem.
TL;DR: In this article, it was shown that the norming sequence in strong limit theorems for increments of sums is determined by a behavior of the inverse function to the function of deviations.
Abstract: We investigate the almost surely asymptotic behavior of increments of sums of independent identically distributed random variables satisfying the one-sided Cram\'er condition. We establish that, irrespective of the length of the increments, the norming sequence in strong limit theorems for increments of sums is determined by a behavior of the inverse function to the function of deviations. This allows for unifying the following well-known results for increments of sums: the strong law of large numbers, the Erd\H{o}s--R\'enyi law and Mason's extension of this law, the Shepp law, the Cs\"{o}rg\H{o}--R\'{e}v\'{e}sz theorems, and the law of the iterated logarithm. In the case of large increments, we derive new results for random variables from the domain of attraction of a stable law with index $\alpha\in (1,2]$ and the parameter of symmetry $\beta=-1$.
TL;DR: In this paper, the authors show that either the above-mentioned linear averaging holds, or a nonlinear averaging of the form $f(y)=\sum\alpha_if_i(x_i) is sufficient for a stock price breakout under an uncertainty factor.
Abstract: In the presence of an uncertainty factor, that is, if some variable X assumes several values $x_1,\ldots,x_n$ rather than a single value, one usually performs an averaging over these values with some coefficients (measures) $\alpha_i$ such that $\sum_{i=1}^n\alpha_i=1$ and sets $y=\sum\alpha_ix_i$. For an equity market, there arises a nonlinear averaging for y. We consider an averaging of the form $f(y)=\sum\alpha_if_i(x_i)$. Starting from four natural axioms, we prove that either the above-mentioned linear averaging holds, or $y=\log\sum_{i=1}^ne^{x_i}$. An example of a stock price breakout under this summation is given.
TL;DR: In this paper, the lower bounds of efficiency for the moderate deviation probabilities of tests and estimators were studied for both the logarithmic and strong asymptotics of the standard normal distribution.
Abstract: We study the lower bounds of efficiency for the moderate deviation probabilities of tests and estimators. These bounds cover both the logarithmic and strong asymptotics. For the problems of hypothesis testing we propose a natural representation for the lower bounds of type I and type II error probabilities in terms of inverse function of the standard normal distribution. The lower bounds for the moderate deviation probabilities of estimators are deduced easily from the corresponding bounds in hypothesis testing.
TL;DR: In this article, the Poisson limit theorem for large excursions of the modulus of the Gaussian vector process with independent identically distributed increments was proved for a large number of excursions.
Abstract: This paper proves the Poisson limit theorem for a number of large excursions of the modulus of the Gaussian vector process with independent identically distributed increments.
TL;DR: In this article, lower bounds for a Cramer-type large deviation theorem for self-normalized sums are shown. But they do not imply that the bounds obtained by Jing, Shao, and Wang [{\em Ann. Probab.}, 31 (2003), pp. 2167--2215] are sharp.
Abstract: Let $X_1,X_2,\dots$ be independent random variables with zero means and finite variances. In this paper we prove lower bounds for a Cramer-type large deviation theorem for self-normalized sums which imply that the bounds obtained by Jing, Shao, and Wang [{\em Ann. Probab.}, 31 (2003), pp. 2167--2215] are sharp.
TL;DR: In this paper, the authors introduced the notion of separating time for a pair of measures and defined the criteria for the local absolute continuity, absolute continuity and singularity of the mutual arrangement of the measures.
Abstract: We introduce the notion of a {\it separating time} for a pair of measures~${\bf P}$ and~${\widetilde{\bf P}}$ on a filtered space. This notion is convenient for describing the mutual arrangement of~${\bf P}$ and~${\widetilde{\bf P}}$ from the viewpoint of the absolute continuity and singularity. Furthermore, we find the explicit form of the separating time for the case, where ${\bf P}$ and ${\widetilde{\bf P}}$ are distributions of Levy processes, solutions of stochastic differential equations, and distributions of Bessel processes. The obtained results yield, in particular, the criteria for the local absolute continuity, absolute continuity, and singularity of~${\bf P}$ and~${\widetilde{\bf P}}$.
TL;DR: In this paper, almost sure versions of limit theorems by Kruglov for the Pearson π π 2 -statistic are obtained, where π is a constant.
Abstract: Almost sure versions of limit theorems by Kruglov for the Pearson $\chi^2$-statistic are obtained.
TL;DR: In this paper, the state-dependent branching process in random environment is considered and the Lamperti-Kersting criteria for the recurrence or transience of growth models are applied to the process.
Abstract: The state-dependent branching process in random environment $Z_n$---the generalization of the Smith--Wilkinson model---is considered. The Lamperti--Kersting criteria for the recurrence or transience of growth models are applied to the process $X_n:=\log Z_n$, $Z_n\ge 3$.
TL;DR: In this paper, it was shown that such examples arise, as a rule, in the theory of non-locally compact groups, and for locally compact groups this phenomenon cannot happen.
Abstract: In the late 1940s, A. N. Kolmogorov suggested a remarkably simple example of a transitive, but not ergodic, action of the group of all permutations of positive integers. It turned out that such examples arise, as a rule, in the theory of actions of non--locally compact groups, and for locally compact groups this phenomenon cannot happen. Kolmogorov's example also helps to give a correct definition of the decomposition into ergodic components and orbit partition for actions of general groups.
TL;DR: In this paper, the authors consider independent identically distributed random variables and find conditions under which the sum of √ √ f n f n = √ n to infinity.
Abstract: Let us consider independent identically distributed random variables $X_1, X_2, \dots\,$, such that $$ U_n=\frac{S_n}{B_n} -n\,a_n \longrightarrow \xi_\alpha\qq \mbox{weakly as}\quad n\to\infty, $$ where $S_n = X_1 + \cdots + X_n$, $B_n>0$, $a_n$ are some numbers $(n\geq 1)$, and a random variable $\xi_\alpha$ has a stable distribution with characteristic exponent $\alpha\in (0, 2)$.Our basic purpose is to find conditions under which $$ \sum_n f_n {\bf P}\{|U_n|\geq\varepsilon\varphi_n\}\sim \sum_n f_n {\bf P}\{|\xi_\al|\ge\varepsilon\varphi_n\},\qquad\varepsilon\searrow 0, $$ with a positive sequence $\varphi_n$, which tends to infinity and satisfies mild additional restrictions, and with a nonnegative sequence $f_n$ such that $\sum_n f_n =\infty $.
TL;DR: In this article, the authors consider the class of all one-to-one mappings of an n-element set into itself, each of which has exactly N connected components, and prove the Poisson and local normal limit theorems for this random variable.
Abstract: We consider the class of all one-to-one mappings of an n-element set into itself, each of which has exactly N connected components. Letting $n,N\to\infty$, we find that the asymptotic behavior of the mean and variance of the random variable is equal to the number of components of a given size in a mapping that is selected at random and is equiprobable among the elements of the mentioned class, and we prove the Poisson and local normal limit theorems for this random variable. Asymptotic estimates are found for the number of mappings with N components, among which there are exactly k components of a fixed size.
TL;DR: In this paper, the definition of a generalized additive homogeneous functional of a Wiener process is introduced, and it is shown that a generalized functional is uniquely specified by its characteristics.
Abstract: The definition of a generalized additive homogeneous functional of a Wiener process is introduced. It is shown that a generalized functional is uniquely specified by its characteristics. In this case the functions from the Schwartz space $S^*$ of slowly growing generalized functions play the role of generating functions.
TL;DR: In this article, the optimal stopping of sequences of random variables satisfying some asymptotic independence property is considered, assuming that the embedded planar point processes converge to a Poisson process.
Abstract: We consider optimal stopping of sequences of random variables satisfying some asymptotic independence property. Assuming that the embedded planar point processes converge to a Poisson process, we introduce some further conditions to obtain approximation of the optimal stopping problem of the discrete time sequence by the optimal stopping of the limiting Poisson process. This limiting problem can be solved in several cases. We apply this method to obtain approximations for the stopping of moving average sequences, of hidden Markov chains, and of max-autoregressive sequences. We also briefly discuss extensions to the case of Poisson cluster processes in the limit.
TL;DR: One-dimensional stochastic equations whose coefficients depend on a small parameter are considered in this article, where necessary and sufficient conditions are obtained for the weak convergence of their solutions to the solution of the unknown process.
Abstract: One-dimensional stochastic equations are considered whose coefficients depend on a small parameter. Necessary and sufficient conditions are obtained for the weak convergence of their solutions to the solution of the stochastic equation containing local time of an unknown process.
TL;DR: In this paper, a more general version of Dobrushin's result connected with an optimal coupling of two random variables is proven and an application to the problem of Poisson approximation in Abelian groups is considered.
Abstract: A more general version of Dobrushin's result connected with an optimal coupling of two random variables is proven. An application to the problem of Poisson approximation in Abelian groups is considered. In particular, an optimal coupling in Poisson approximation of empirical processes is studied.
TL;DR: It is shown that the optimal stopping time in the full-information optimal stopping problem of N independent identically distributed random variables has the asymptotic properties.
Abstract: We show that the optimal stopping time in the full-information optimal stopping problem of N independent identically distributed random variables has the asymptotic properties ${\bf E}\tau_N\approx N/3$, ${\bf D}\tau_N\approx N^2/18$.
TL;DR: In this article, a fractional distribution of an s-dimensional Gaussian random vector is considered and equilibria for the distribution deviation from the uniform distribution are proved using the Poisson summation formula and some facts from the theory of representation of integers by square forms.
Abstract: This paper considers a fractional distribution of an s-dimensional Gaussian random vector. Inequalities for the distribution deviation from the uniform distribution are proved. The proofs use the Poisson summation formula and some facts from the theory of representation of integers by square forms. The main attention of this part of the paper is devoted to the case of small values of s. The case of large values of s will be consider additionally.
TL;DR: In this paper, the authors consider the problem of estimating the first component of a random vector by observations of n independent copies of the vector given a priori known set with specified metric characteristics such as varepsilon-entropy or Kolmogorov widths.
Abstract: Let $(X,Y)$ be a random vector whose first component takes on values in a measurable space $({\mathfrak{X}},{\mathfrak{A}},\mu )$ with measure $\mu$, and let Y be a real-valued random variable. Let $$ f(x)={\bf E}\{Y\mid X=x\} $$ be the regression function of Y on X. We consider the problem of estimating $f(x)$ by observations of n independent copies of $(X,Y)$ given $f\in{\bf F}$, where ${\bf F}$ is an a priori known set with specified metric characteristics such as $\varepsilon$-entropy or Kolmogorov widths.