the matrix exponential there are many different methods to calculate the exponential of a matrix series methods differential equations methods polynomial methods matrix decomposition methods and splitting methods none of which is entirely satisfactory from either a theoretical or a computational point of view how then should the matrix exponential be introduced in an elementary differential equations course for engineering students for example with a minimum of mathematical prerequisites in this note a method is given that uses the students knowledge of homogeneous linear differential equations with constant coefficients and the cayley hamilton theorem the method is not new what is new is the approach

max norm optimization for robust matrix recovery this paper studies the matrix completion problem under arbitrary sampling schemes we propose a new estimator incorporating both max norm and nuclear norm regularization based on which we can conduct efficient low rank matrix recovery using a random subset of entries observed with additive noise under general non uniform and unknown sampling distributions this method significantly relaxes the uniform sampling assumption imposed for the widely used nuclear norm penalized approach and makes low rank matrix recovery feasible in more practical settings theoretically we prove that the proposed estimator achieves fast rates of convergence under different settings computationally we propose an alternating direction method of multipliers algorithm to efficiently compute the estimator which bridges a gap between theory and practice of machine learning methods with max norm regularization further we provide thorough numerical studies to evaluate the proposed method using both simulated and real datasets

performance analysis for a changepoint problem nonlinear stochastic differential equations are used to model a version of the changepoint problem state estimates of the minimum mean square error type are used in likelihood ratio tests to detect the time of change system performance is evaluated by deriving theoretical expressions for the false alarm probability and a special case of the miss error probability the approach uses a fokker planck equation

optimal selection of reduced rank estimators of high dimensional matrices we introduce a new criterion the rank selection criterion rsc for selecting the optimal reduced rank estimator of the coefficient matrix in multivariate response regression models the corresponding rsc estimator minimizes the frobenius norm of the fit plus a regularization term proportional to the number of parameters in the reduced rank model the rank of the rsc estimator provides a consistent estimator of the rank of the coefficient matrix in general the rank of our estimator is a consistent estimate of the effective rank which we define to be the number of singular values of the target matrix that are appropriately large the consistency results are valid not only in the classic asymptotic regime when n the number of responses and p the number of predictors stay bounded and m the number of observations grows but also when either or both n and p grow possibly much faster than m we establish minimax optimal bounds on the mean squared errors of our estimators our finite sample performance bounds for the rsc estimator show that it achieves the optimal balance between the approximation error and the penalty term furthermore our procedure has very low computational complexity linear in the number of candidate models making it particularly appealing for large scale problems we contrast our estimator with the nuclear norm penalized least squares nnp estimator which has an inherently higher computational complexity than rsc for multivariate regression models we show that nnp has estimation properties similar to those of rsc albeit under stronger conditions however it is not as parsimonious as rsc we offer a simple correction of the nnp estimator which leads to consistent rank estimation we verify and illustrate our theoretical findings via an extensive simulation study

a characterization of the spatial poisson process and changing time watanabe proved that if x t is a point process such that x t t is a martingale then x t is a poisson process and this result was generalized by bremaud for doubly stochastic poisson processes here we define two parameter point processes and extend this property without needing the strong martingale condition using this characterization we study the problem of transforming a two parameter point process into a two parameter poisson process by means of a family of stopping lines as a time change nualart and sanz gave conditions in order to transform a square integrable strong martingale into a wiener process here we do the same for the poisson process by a similar method but under more general conditions

a multiplicative seasonal component in commodity derivative pricing abstract in this paper we focus on a seasonal jump diffusion model to price commodity derivatives we propose a novel approach to estimate the functions of the risk neutral processes directly from data in the market even when a closed form solution for the model is not known then this new approach is applied to price some natural gas derivative contracts traded at new york mercantile exchange nymex moreover we use nonparametric estimation techniques in order to avoid arbitrary restrictions on the model after applying this approach we find that a jump diffusion model allowing for seasonality outperforms a standard jump diffusion model to price natural gas futures furthermore we also show that there are considerable differences in the option prices and the risk premium when we consider seasonality or not these results have important implications for practitioners in the market

sampling from large matrices an approach through geometric functional analysis we study random submatrices of a large matrix a we show how to approximately compute a from its random submatrix of the smallest possible size o rlog r with a small error in the spectral norm where r e vav2f vav22 is the numerical rank of a the numerical rank is always bounded by and is a stable relaxation of the rank of a this yields an asymptotically optimal guarantee in an algorithm for computing low rank approximations of a we also prove asymptotically optimal estimates on the spectral norm and the cut norm of random submatrices of a the result for the cut norm yields a slight improvement on the best known sample complexity for an approximation algorithm for max 2csp problems we use methods of probability in banach spaces in particular the law of large numbers for operator valued random variables

efron stein inequalities for random matrices this paper establishes new concentration inequalities for random matrices constructed from independent random variables these results are analogous with the generalized efron stein inequalities developed by boucheron et al the proofs rely on the method of exchangeable pairs

reputation in continuous time games we study reputation dynamics in continuous time games in which a large player e g government faces a population of small players e g households and the large player s actions are imperfectly observable the major part of our analysis examines the case in which public signals about the large player s actions are distorted by a brownian motion and the large player is either a normal type who plays strategically or a behavioral type who is committed to playing a stationary strategy we obtain a clean characterization of sequential equilibria using ordinary differential equations and identify general conditions for the sequential equilibrium to be unique and markovian in the small players posterior belief we find that a rich equilibrium dynamics arises when the small players assign positive prior probability to the behavioral type by contrast when it is common knowledge that the large player is the normal type every public equilibrium of the continuous time game is payoff equivalent to one in which a static nash equilibrium is played after every history finally we examine variations of the model with poisson signals and multiple behavioral types

decoding movement trajectories through a t maze using point process filters applied to place field data from rat hippocampal region ca1 firing activity from neural ensembles in rat hippocampus has been previously used to determine an animal s position in an open environment and separately to predict future behavioral decisions however a unified statistical procedure to combine information about position and behavior in environments with complex topological features from ensemble hippocampal activity has yet to be described here we present a two stage computational framework that uses point process filters to simultaneously estimate the animal s location and predict future behavior from ensemble neural spiking activity first in the encoding stage we linearized a two dimensional t maze and used spline based generalized linear models to characterize the place field structure of different neurons all of these neurons displayed highly specific position dependent firing which frequently had several peaks at multiple locations along the maze when the rat was at the stem of the t maze the firing activity of several of these neurons also varied significantly as a function of the direction it would turn at the decision point as detected by anova second in the decoding stage we developed a state space model for the animal s movement along a t maze and used point process filters to accurately reconstruct both the location of the animal and the probability of the next decision the filter yielded exact full posterior densities that were highly nongaussian and often multimodal our computational framework provides a reliable approach for characterizing and extracting information from ensembles of neurons with spatially specific context or task dependent firing activity

a simpler approach to matrix completion this paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix these results improve on prior work by candes and recht 2009 candes and tao 2009 and keshavan et al 2009 the reconstruction is accomplished by minimizing the nuclear norm or sum of the singular values of the hidden matrix subject to agreement with the provided entries if the underlying matrix satisfies a certain incoherence condition then the number of entries required is equal to a quadratic logarithmic factor times the number of parameters in the singular value decomposition the proof of this assertion is short self contained and uses very elementary analysis the novel techniques herein are based on recent work in quantum information theory

smoothing point processes as a means to increase throughput unlike the leaky bucket scheme which regulates the input rate the filters we study in this paper reduce the variability of interarrival times subject to a maximum delay constraint for each customer these filters are called smoothing filters for point processes by considering the output processes of various queueing systems we show that an infinite server queue acts as a smoothing filter for a doubly stochastic poisson process and a single server queue with deterministic service times acts as a smoothing filter for a stationary and ergodic point process based on the second point we provide a smoothing algorithm that satisfies a maximum delay constraint the algorithm is shown to be robust and consistent we then consider two examples where these filters can be applied to increase throughput one is the single server loss system with exponential service times the other is the leaky bucket scheme with token buffer size 1 numerical examples and simulations are also given for comparing the tradeoff between delay and throughput our approach is based on the variability ordering and a cut criterion for the majorization ordering the criterion appears to be new and of independent interest

numerical solution of stochastic quantum master equations using stochastic interacting wave functions abstract we develop a new approach for solving stochastic quantum master equations with mixed initial states first we obtain that the solution of the jump diffusion stochastic master equation is represented by a mixture of pure states satisfying a system of stochastic differential equations of schrodinger type then we design three exponential schemes for these coupled stochastic schrodinger equations which are driven by brownian motions and jump processes hence we have constructed efficient numerical methods for the stochastic master equations based on quantum trajectories the good performance of the new numerical integrators is illustrated by simulations of two quantum measurement processes

unique representation of an inverse kinetic theory for incompressible newtonian fluids abstract fundamental aspects of inverse kinetic theories for the incompressible navier stokes equations m tessarotto m ellero rgd24 italy july 10 16 2004 aip conf proc 762 2005 108 m ellero m tessarotto physica a 355 2005 233 include the possibility of defining uniquely the kinetic equation underlying such models and furthermore the construction of a kinetic theory implying also the energy equation the latter condition is consistent with the requirement that fluid fields result classical solutions of the fluid equations these issues appear of potential relevance both from the mathematical viewpoint and for the physical interpretation of the theory purpose of this work is to prove that under suitable prescriptions the inverse kinetic theory can be determined to satisfy such requirements

exponential inequalities for martingales with application to maximum likelihood estimation for counting processes we obtain an exponential probability inequality for martingales and a uniform probability inequality for the process gdn where n is a counting process and where g varies within a class of predictable functions g for the latter we use techniques from empirical process theory the uniform inequality is shown to hold under certain entropy conditions on g as an application we consider rates of convergence for nonparametric maximum likelihood estimators for counting processes a similar result for discrete time observations is also presented

on the matrix powers and exponential by the r generalized fibonacci sequences methods the companion matrix case properties of some r generalized fibonacci sequences in the algebra of square matrices gl r c allow us to derive some explicit formulas for the matrix powers an n 0 and exponential eta for every a gl r c connection with the verde star method is displayed for a companion matrix a we present some explicit combinatorial formulas for an n 0 and eta furthermore the chen louck s theorem is derived

concentration inequalities for matrix martingales in continuous time this paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time these results extend previously established freedman and bernstein inequalities for series of random matrices to the class of continuous time processes our analysis relies on a new supermartingale property of the trace exponential proved within the framework of stochastic calculus we provide also several examples that illustrate the fact that our results allow us to recover easily several formerly obtained sharp bounds for discrete time matrix martingales

a martingale approach to point processes in the plane on donne une definition rigoureuse des processus ponctuels a deux parametres comme une distribution d une quantite denombrable de points aleatoires du plan on obtient une caracterisation avec des droites d arret et une relation avec la predictibilite

signal processing with levy information levy processes which have stationary independent increments are ideal for modelling the various types of noise that can arise in communication channels if a levy process admits exponential moments then there exists a parametric family of measure changes called esscher transformations if the parameter is replaced with an independent random variable the true value of which represents a message then under the transformed measure the original levy process takes on the character of an information process in this paper we develop a theory of such levy information processes the underlying levy process which we call the fiducial process represents the noise type each such noise type is capable of carrying a message of a certain specification a number of examples are worked out in detail including information processes of the brownian poisson gamma variance gamma negative binomial inverse gaussian and normal inverse gaussian type although in general there is no additive decomposition of information into signal and noise one is led nevertheless for each noise type to a well defined scheme for signal detection and enhancement relevant to a variety of practical situations

exact simulation of point processes with stochastic intensities point processes with stochastic arrival intensities are ubiquitous in many areas including finance insurance reliability health care and queuing they can be simulated from a poisson process by time scaling with the cumulative intensity the paths of the cumulative intensity are often generated with a discretization method however discretization introduces bias into the simulation results the magnitude of the bias is difficult to quantify this paper develops a sampling method that eliminates the need to discretize the cumulative intensity the method is based on a projection argument and leads to unbiased simulation estimators it is exemplified for a point process whose intensity is a function of a jump diffusion process and the point process itself in this setting the method facilitates the exact sampling of both the point process and the driving jump diffusion process numerical experiments demonstrate the effectiveness of the method

mutual information relative entropy and estimation in the poisson channel let be a nonnegative random variable and let the conditional distribution of a random variable given be poisson for a parameter we identify a natural loss function such that 1 the derivative of the mutual information between and with respect to is equal to the minimum mean loss in estimating based on regardless of the distribution of 2 when is estimated based on by a mismatched estimator that would have minimized the expected loss had the integral over all values of of the excess mean loss is equal to the relative entropy between and for a continuous time setting where is a nonnegative stochastic process and the conditional law of given is that of a non homogeneous poisson process with intensity function under the same loss function 1 the minimum mean loss in causal filtering when is equal to the expected value of the minimum mean loss in noncausal filtering smoothing achieved with a channel whose parameter is uniformly distributed between 0 and bridging the two quantities is the mutual information between and 2 this relationship between the mean losses in causal and noncausal filtering holds also in the case where the filters employed are mismatched i e optimized assuming a law on which is not the true one bridging the two quantities in this case is the sum of the mutual information and the relative entropy between the true and the mismatched distribution of thus relative entropy quantifies the excess estimation loss due to mismatch in this setting these results are parallel to those recently found for the gaussian channel the i mmse relationship of guo the relative entropy and mismatched estimation relationship of verdu and the relationship between causal and noncasual mismatched estimation of weissman

on the drawdowns and drawups in diffusion type models with running maxima and minima we obtain closed form expressions for the values of joint laplace transforms of the running maximum and minimum of a diffusion type process stopped at the first time at which the associated drawdown and drawup processes hit constant levels it is assumed that the coefficients of the diffusion type process are regular functions of the running values of the process itself its maximum and minimum as well as its maximum drawdown and maximum drawup processes the proof is based on the solution to the equivalent boundary value problems and application of the normal reflection conditions for the value functions at the edges of the state space of the resulting five dimensional markov process we show that the joint laplace transforms represent linear combinations of solutions to the systems of first order partial differential equations arising from the application of the normal reflection conditions

moment inequalities for sums of random matrices and their applications in optimization in this paper we consider various moment inequalities for sums of random matrices which are well studied in the functional analysis and probability theory literature and demonstrate how they can be used to obtain the best known performance guarantees for several problems in optimization first we show that the validity of a recent conjecture of nemirovski is actually a direct consequence of the so called non commutative khintchine s inequality in functional analysis using this result we show that an sdp based algorithm of nemirovski which is developed for solving a class of quadratic optimization problems with orthogonality constraints has a logarithmic approximation guarantee this improves upon the polynomial approximation guarantee established earlier by nemirovski furthermore we obtain improved safe tractable approximations of a certain class of chance constrained linear matrix inequalities secondly we consider a recent result of delage and ye on the so called data driven distributionally robust stochastic programming problem one of the assumptions in the delage ye result is that the underlying probability distribution has bounded support however using a suitable moment inequality we show that the result in fact holds for a much larger class of probability distributions given the close connection between the behavior of sums of random matrices and the theoretical properties of various optimization problems we expect that the moment inequalities discussed in this paper will find further applications in optimization

methods for the analysis of sampled cohort data in the cox proportional hazards model methods are provided for regression parameter and cumulative baseline hazard estimation in the cox proportional hazards model when the cohort is sampled according to a predictable sampling probability law the key to the methodology is to define counting processes which count joint failure and sampled risk sets occurrences and to derive the appropriate intensities for these counting processes leading to estimation methods parallel to those for full cohort data these techniques are illustrated for a number of sampling designs including three novel designs counter matching with additional randomly sampled controls quota sampling of controls and nested case control sampling with number of controls dependent on the failure s exposure status general asymptotic theory is developed for the maximum partial likelihood estimator and cumulative baseline hazard estimator and is used to derive the asymptotic distributions for estimators from a large class of designs

filtering a markov modulated random measure we develop a new exact filter when a hidden markov chain influences both the sizes and times of a marked point process an example would be an insurance claims process where we assume that both the stochastic intensity of the claim arrivals and the distribution of the claim sizes depend on the states of an economy we also develop the robust filter based and smoother based em algorithms for the on line recursive estimates of the unknown parameters in the markov modulated random measure our development is in the framework of modern theory of stochastic processes

closed form likelihood approximation and estimation of jump diffusions with an application to the realignment risk of the chinese yuan this paper provides closed form likelihood approximations for multivariate jump diffusion processes widely used in finance for a fixed order of approximation the maximum likelihood estimator mle computed from this approximate likelihood achieves the asymptotic efficiency of the true yet uncomputable mle as the sampling interval shrinks this method is used to uncover the realignment probability of the chinese yuan since february 2002 the market implied realignment intensity has increased fivefold the term structure of the forward realignment rate which completely characterizes future realignment probabilities is hump shaped and peaks at mid 2004 the realignment probability responds quickly to economic news releases and government interventions r 2007 elsevier b v all rights reserved

on galerkin approximations for the zakai equation with diffusive and point process observations this paper studies galerkin approximations applied to the zakai equation of stochastic filtering the basic idea of this approach is to project the infinite dimensional zakai equation onto some finite dimensional subspace generated by smooth basis functions this leads to a finite dimensional system of stochastic differential equations that can be solved numerically the contribution of the paper is twofold on the theoretical side existing convergence results are extended to filtering models with observations of point process or mixed type on the applied side various issues related to the numerical implementation of the method are considered in particular we propose working with a subspace that is constructed from a basis of hermite polynomials the paper closes with a numerical case study

stein s method for concentration inequalities we introduce a version of stein s method for proving concentration and moment inequalities in problems with dependence simple illustrative examples from combinatorics physics and mathematical statistics are provided

restricted strong convexity and weighted matrix completion optimal bounds with noise we consider the matrix completion problem under a form of row column weighted entrywise sampling including the case of uniformentrywise sampling as a special case we analyze the associated random observation operator and prove that with high probability it satisfies a form of restricted strong convexity with respect to weighted frobenius norm using this property we obtain as corollaries a number of error bounds on matrix completion in the weighted frobenius norm under noisy sampling and for both exact and near low rank matrices our results are based on measures of the spikiness and low rankness of matrices that are less restrictive than the incoherence conditions imposed in previous work our technique involves an m estimator that includes controls on both the rank and spikiness of the solution and we establish non asymptotic error bounds in weighted frobenius norm for recovering matrices lying with lq balls of bounded spikiness using information theoretic methods we show that no algorithm can achieve better estimates up to a logarithmic factor over these same sets showing that our conditions on matrices and associated rates are essentially optimal

operator khintchine inequality in non commutative probability abstract for the operator t sum n k 1 a k otimes r k where a k belongs to the schatten class s 2n and where r k are non commutative random variables with mixed moments satisfying a specific condition we prove the following khintchine inequality vert t vert 2n leq d 2n left left vert left sum n k001 a ka k right frac 1 2 right vert s 2n quad quad left vert left sum n k 1 a k a k right frac 1 2 right vert s 2n right we find the optimal constants d 2n in the case when r k are the q gaussian and circular random variables moreover we show that the moments of any probability symmetric measure appear as the optimal constants for some random variables

phase space lagrangian dynamics of incompressible thermofluids phase space lagrangian dynamics in ideal fluids i e continua is usually related to the so called ideal tracer particles the latter which can in principle be permitted to have arbitrary initial velocities are understood as particles of infinitesimal size which do not produce significant perturbations of the fluid and do not interact among themselves an unsolved theoretical problem is the correct definition of their dynamics in ideal fluids the issue is relevant in order to exhibit the connection between fluid dynamics and the classical dynamical system underlying a prescribed fluid system which uniquely generates its time evolution the goal of this paper is to show that the tracer particle dynamics can be exactly established for an arbitrary incompressible fluid uniquely based on the construction of an inverse kinetic theory ikt m tessarotto m ellero bull am phys soc 45 9 2000 40 m tessarotto m ellero aip conf proc 762 2005 108 rgd24 italy july 10 16 2004 m ellero m tessarotto physica a 355 2005 233 m tessarotto m ellero physica a 373 2007 142 arxiv physics 0602140 m tessarotto m ellero in m s ivanov a k rebrov eds proc 25th rgd international symposium on rarefied gas dynamics st petersburg russia july 21 28 2006 novosibirsk publ house of the siberian branch of the russian academy of sciences 2007 p 1001 arxiv physics 0611113 m tessarotto c cremaschini strong solutions of the incompressible navier stokes equations in external domains local existence and uniqueness arxiv 0809 5164v1 math ph 2008 as an example the case of an incompressible newtonian thermofluid is considered here

distribution of singular values of random band matrices marchenko pastur law and more we consider the limiting spectral distribution of matrices of the form frac 1 2b n 1 r x r x where x is an n times n band matrix of bandwidth b n and r is a non random band matrix of bandwidth b n we show that the stieltjes transform of esd of such matrices converges to the stieltjes transform of a non random measure and the limiting stieltjes transform satisfies an integral equation for r 0 the integral equation yields the stieltjes transform of the marchenko pastur law

renewal theory properties of ion channels this paper uses a type of data to attack the problem of deciding whether or not the channels in a membrane are operating independently and if so estimating the distribution of the sojourn periods in emitting and nonomitting states

mathematical properties of the navier stokes dynamical system for incompressible newtonian fluids abstract the connection between fluid dynamics and classical statistical mechanics has motivated in the past mathematical investigations of the incompressible navier stokes ns equations inse by means of an asymptotic kinetic theory this feature has suggested the search for possible alternative exact approaches based on the construction of a suitable inverse kinetic theory ikt which can avoid the asymptotic character and the intrinsic mathematical difficulty of direct kinetic theories in this paper the fundamental mathematical properties of the ns phase space dynamical system underlying inse and determined by ikt are investigated in particular an equivalence theorem with the inse problem and a global existence theorem are proved to hold for the ns dynamical system

zero rate reliability of the exponential server timing channel we determine the reliability function of the exponential server timing channel estc in the limit as the data rate approaches zero the limit shows that at low rates the estc is strictly more reliable than the poisson channel without dark current answering a question arikan posed in these transactions the proof employs a distance metric over inputs to timing channels that parallels euclidean and hamming distance for conventional channels a consequence of the proof is that bounded distance decoding with distance measured according to this metric is exponentially optimum for the estc in the low rate regime we also prove the straight line bound for the channel and a bound on the reliability of timing channels with general service distributions in the limit as the data rate approaches zero

matrix concentration inequalities via the method of exchangeable pairs this paper derives exponential concentration inequalities and polynomial moment inequalities for the spectral norm of a random matrix the analysis requires a matrix extension of the scalar concentration theory developed by sourav chatterjee using stein s method of exchangeable pairs when applied to a sum of independent random matrices this approach yields matrix generalizations of the classical inequalities due to hoeffding bernstein khintchine and rosenthal the same technique delivers bounds for sums of dependent random matrices and more general matrix valued functions of dependent random variables

multidimensional residual analysis of point process models for earthquake occurrences residual analysis methods for examining the fit of multidimensional point process models are applied to point process models for the space time magnitude distribution of earthquake occurrences using in particular the multidimensional version of ogata s epidemic type aftershock sequence etas model and a 30 year catalog of 580 earthquakes occurring in bear valley california one method involves rescaled residuals obtained by transforming points along one coordinate to form a homogeneous poisson process inside a random irregular boundary another method involves thinning the point process according to the conditional intensity to form a homogeneous poisson process on the original untransformed space the thinned residuals suggest that the fit of the model may be significantly improved by using an anisotropic spatial distance function in the estimation of the spatially varying background rate using rescaled residuals it is shown that the temporal magnitude distribution of aftershock activity is not

noncommutative burkholder rosenthal inequalities we investigate martingale inequalities in noncommutative lp spaces associated with a von neumann algebra equipped with a faithful normal state we prove the noncommutative analogue of the classical burkholder inequality on the conditioned or little square function and extend the noncommutative burkholder gundy inequalities from comm math phys 189 1997 667 698 to this nontracial setting we include several related results

trees under attack a ray knight representation of feller s branching diffusion with logistic growth we obtain a representation of feller s branching diffusion with logistic growth in terms of the local times of a reflected brownian motion h with a drift that is affine linear in the local time accumulated by h at its current level as in the classical ray knight representation the excursions of h are the exploration paths of the trees of descendants of the ancestors at time t 0 and the local time of h at height t measures the population size at time t we cope with the dependence in the reproduction by introducing a pecking order of individuals an individual explored at time s and living at time t h s is prone to be killed by any of its contemporaneans that have been explored so far the proof of our main result relies on approximating h with a sequence of harris paths h n which figure in a ray knight representation of the total mass of a branching particle system we obtain a suitable joint convergence of h n together with its local times and with the girsanov densities that introduce the dependence in the reproduction

pricing vulnerable power exchange options in an intensity based framework abstract in this article we study the valuation of a european vulnerable power exchange option in an intensity based framework we assume that the default by the counter party is the time of the first jump of a doubly stochastic poisson process whose intensity is modeled by a jump diffusion process the dynamics of the two assets are assumed to be driven by correlated jump diffusion processes all the three processes are assumed to be correlated in continuous part as well as in the jump part in the proposed framework employing the measure change technique we obtain the explicit formula for the price of the power exchange option with counter party risk furthermore sensitivity analysis is given to illustrate the effects of counterparty risk on the price of the option and effect of various parameters on the option prices

asymptotic control for a class of piecewise deterministic markov processes associated to temperate viruses we aim at characterizing the asymptotic behavior of value functions in the control of piecewise deterministic markov processes pdmp of switch type under nonexpansive assumptions for a particular class of processes inspired by temperate viruses we show that uniform limits of discounted problems as the discount decreases to zero and time averaged problems as the time horizon increases to infinity exist and coincide the arguments allow the limit value to depend on initial configuration of the system and do not require dissipative properties on the dynamics the approach strongly relies on viscosity techniques linear programming arguments and coupling via random measures associated to pdmp as an intermediate step in our approach we present the approximation of discounted value functions when using piecewise constant in time open loop policies

strong converse for identification via quantum channels we present a simple proof of the strong converse for identification via discrete memoryless quantum channels based on a novel covering lemma the new method is a generalization to quantum communication channels of ahlswede s 1979 1992 approach to classical channels it involves a development of explicit large deviation estimates to the case of random variables taking values in self adjoint operators on a hilbert space this theory is presented separately in an appendix and we illustrate it by showing its application to quantum generalizations of classical hypergraph covering problems

a nonparametric dynamic additive regression model for longitudinal data in this work we study additive dynamic regression models for longitudinal data these models provide a flexible and nonparametric method for investigating the time dynamics of longitudinal data the methodology is aimed at data where measurements are recorded at random time points we model the conditional mean of responses given the full internal history and possibly time varying covariates we derive the asymptotic distribution for a new nonparametric least squares estimator of the cumulative time varying regression functions based on the asymptotic results confidence bands may be computed and inference about time varying coefficients may be drawn we propose two estimators of the cumulative regression function one estimator that involves smoothing and one that does not the latter however has twice the variance as the smoothing based estimator goodness of fit of the model is considered using martingale residuals finally we also discuss how partly conditional mean models in which the mean of the response is regressed onto selected time varying covariates may be analysed in the same framework we apply the methods to longitudinal data on height development for cystic fibrosis patients

on channel capacity per unit cost memoryless communication channels with arbitrary alphabets where each input symbol is assigned a cost are considered the maximum number of bits that can be transmitted reliably through the channel per unit cost is studied it is shown that if the input alphabet contains a zero cost symbol then the capacity per unit cost admits a simple expression as the maximum normalized divergence between two conditional output distributions the direct part of this coding theorem admits a constructive proof via stein s lemma on the asymptotic error probability of binary hypothesis tests single user multiple access and interference channels are studied

random vectors in the isotropic position abstract letybe a random vector in r n satisfying e y y id letmbe a natural number and lety1 ymbe independent copies ofy we study the question of approximation of the identity operator by finite sums of the tensorsyi yi we prove that for some absolute constantc e 1 m i 1 m y i y i id c log n m e y log m 1 log m provided that the last expression is smaller than 1 we apply this estimate to improve a result of bourgain concerning the number of random points needed to bring a convex body into a nearly isotropic position

detection of stochastic processes this paper reviews two streams of development from the 1940 s to the present in signal detection theory the structure of the likelihood ratio for detecting signals in noise and the role of dynamic optimization in detection problems involving either very large signal sets or the joint optimization of observation time and performance this treatment deals exclusively with basic results developed for the situation in which the observations are modeled as continuous time stochastic processes the mathematics and intuition behind such developments as the matched filter the rake receiver the estimator correlator maximum likelihood sequence detectors multiuser detectors sequential probability ratio tests and cumulative sum quickest detectors are described

performance bounds for mismatched decision schemes with poisson process observations abstract this paper develops a framework for analyzing the performance loss in fixed time interval decision algorithms that are based on observations of time inhomogeneous poisson processes when some parameters characterizing the observation process are not known exactly key to the development is the formulation of an analytically computable performance metric which can be used in lieu of the true but intractable error probabilities the proposed metric is obtained by identifying analytical upper bounds on the error probabilities in terms of the uncertain parameters using these tools it is shown that performance degrades gracefully as long as the true values of the parameters remain within a neighborhood of the nominal values used in decision making the results find direct application to problems of detecting illicit nuclear materials in transit

option pricing with a general marked point process this paper examines the impact of a random number of price changes on the options valuation the model introduces the structure of the general marked point processes mpp this kind of model allows us to take account of more general distributions of interarrival times than usual jump diffusion models in particular stock price variations can be correlated with the times of transactions thus the investors can decide to trade according to the history of market values and the sizes of price variations can also depend on the past times of transactions by using the special decomposition of predictable processes with respect to a marked point process the determination of all risk neutral probabilities is detailed derivative prices are calculated in this context with different basic examples

lsv based tail inequalities for sums of random matrices the techniques of random matrices have played an important role in many machine learning models in this letter we present a new method to study the tail inequalities for sums of random matrices different from other work ahlswede winter 2002 tropp 2012 hsu kakade zhang 2012 our tail results are based on the largest singular value lsv and independent of the matrix dimension since the lsv operation and the expectation are noncommutative we introduce a diagonalization method to convert the lsv operation into the trace operation of an infinitely dimensional diagonal matrix in this way we obtain another version of laplace transform bounds and then achieve the lsv based tail inequalities for sums of random matrices

optimum bounds for the distributions of martingales in banach spaces a general device is proposed which provides for extension of exponential inequalities for sums of independent real valued random variables to those for martingales in the 2 smooth banach spaces this is used to obtain optimum bounds of the rosenthal burkholder and chung types on moments of the martingales in 2 smooth banach spaces in turn it leads to best order bounds on moments of sums of independent random vectors in any separable banach spaces although the emphasis is put on infinite dimensional martingales most of the results seem to be new even for one dimensional martingales moreover the bounds on moments of the rosenthal burkholder type seem to be to a certain extent new even for sums of independent real valued random variables analogous inequalities for one dimensional supermartingales are given

matrix infinitely divisible series tail inequalities and their applications in this paper we study tail inequalities of the largest eigenvalue of a matrix infinitely divisible i d series which is a finite sum of fixed matrices weighted by i d random variables we obtain several types of tail inequalities including bennett type and bernstein type inequalities this allows us to further bound the expectation of the spectral norm of a matrix i d series moreover by developing a new lower bound function for q s s 1 log s 1 s that appears in the bennett type inequality we derive a tighter tail inequality of the largest eigenvalue of the matrix i d series than the bernstein type inequality when the matrix dimension is high the resulting lower bound function is of independent interest and can improve any bennett type concentration inequality that involves the function q s the class of i d probability distributions is large and includes gaussian and poisson distributions among many others therefore our results encompass the existing work on matrix gaussian series as a special case lastly we show that the tail inequalities of a matrix i d series have applications in several optimization problems including the chance constrained optimization problem and the quadratic optimization problem with orthogonality constraints in addition we also use the resulting tail bounds to show that random matrices constructed from i d random variables satisfy the restricted isometry property rip when it acts as a measurement matrix in compressed sensing

estimation of continuous time markov processes sampled at random time intervals we introduce a family of generalized method of moments estimators of the parameters of a continuous time markov process observed at random time intervals the results include strong consistency asymptotic normality and a characterization of standard errors sampling is at an arrival intensity that is allowed to depend on the underlying markov process and on the parameter vector to be estimated we focus on financial applications including tick based sampling allowing for jump diffusions regime switching diffusions and reflected diffusions

non commutative martingale inequalities we prove the analogue of the classical burkholder gundy inequalites for non commutative martingales as applications we give a characterization for an ito clifford integral to be an lp martingale via its integrand and then extend the ito clifford integral theory in l2 developed by barnett streater and wilde to lp for all 1 p we include an appendix on the non commutative analogue of the classical fefferman duality between h1 and bmo

relative efficiency of the log rank test within a multiplicative intensity model summary for large sample clinical trials with independent individuals randomly allocated to two treatment groups in which survival times follow a log linear multiplicative intensity model with treatment group as one covariate this paper calculates the asymptotic relative efficiency of the log rank test for treatment effect as compared with the optimal score test the method is to exhibit the failure hazard intensity not of proportional hazards form obtained by ignoring all covariates other than treatment group the efficiency formulae are illustrated in two examples and estimation from data of the loss of efficiency is illustrated for two clinical trial datasets

classroom note a note on the matrix exponential this note is devoted to simplifying one method to calculate the exponential of a matrix presented by i e leonard in siam rev 38 1996 pp 507 512

jump diffusion processes in random environments abstract in this paper we investigate jump diffusion processes in random environments which are given as the weak solutions of sdes we formulate conditions ensuring existence and uniqueness in law of solutions we investigate the markov property to prove uniqueness we solve a general martingale problem for cadlag processes this result is of independent interest application of our results to generalized exponential levy model are present in the last section

matrix poincare inequalities and concentration abstract we show that any probability measure satisfying a matrix poincare inequality with respect to some reversible markov generator satisfies an exponential matrix concentration inequality depending on the associated matrix carre du champ operator this extends to the matrix setting a classical phenomenon in the scalar case moreover the proof gives rise to new matrix trace inequalities which could be of independent interest we then apply this general fact by establishing matrix poincare inequalities to derive matrix concentration inequalities for gaussian measures product measures and for strong rayleigh measures the latter represents the first instance of matrix concentration for general matrix functions of negatively dependent random variables

specification diagnostics for duration models a martingale approach abstract this paper develops specification diagnostics for discrete time duration models which are based on the martingale theory of stochastic processes the methods can be applied to parametric grouped data duration models with time varying covariates right censoring and which account for unobserved heterogeneity a residual process is defined such that under the correct model specification the process is approximately a martingale difference sequence chi square tests based on these residuals are derived results from monte carlo simulations are presented

point processes and queues martingale dynamics i martingales 1 histories and stopping times 2 martingales 3 predictability 4 square integrable martingales references solutions to exercises chapter i ii point processes queues and intensities 1 counting processes and queues 2 watanabe s characterization 3 stochastic intensity general case 4 predictable intensities 5 representation of queues 6 random changes of time 7 cryptographic point processes references solutions to exercises chapter ii iii integral representation of point process martingales 1 the structure of internal histories 2 regenerative form of the intensity 3 the representation theorem 4 hilbert space theory of poissonian martingales 5 useful extensions references solutions to exercises chapter iii iv filtering 1 the theory of innovations 2 state estimates for queues and markov chains 3 continuous states and nontrivial prehistory references solutions to exercises chapter iv v flows in markovian networks of queues 1 single station the historical results and the filtering method 2 jackson s networks 3 burke s output theorem for networks 4 cascades and loops in jackson s networks 5 independence and poissonian flows in markov chains references solutions to exercises chapter v vi likelihood ratios 1 radon nikodym derivatives and tests of hypotheses 2 changes of intensities a la girsanov 3 filtering by the method of the probability of reference 4 applications 5 the capacity of a point process channel 6 detection formula187 references189 solutions to exercises chapter vi vii optimal control 1 modeling intensity controls 2 dynamic programming for intensity controls complete observation case 3 input regulation a case study in impulsive control 4 attraction controls 5 existence via likelihood ratio references solutions to exercises chapter vii viii marked point processes 1 counting measure and intensity kernels 2 martingale representation and filtering 3 radon nikodym derivatives 4 towards a general theory of intensity references solutions to exercises chapter viii a1 background in probability and stochastic processes 1 introduction 2 monotone class theorem 3 random variables 4 expectations 5 conditioning and independence 6 convergence 7 stochastic processes 8 markov processes references a2 stopping times and point process histories 1 stopping times 2 changes of time and meyer dellacherie s integration formula 3 point process histories references a3 wiener driven dynamical systems 1 ito s stochastic integral 2 square integrable brownian martingales 3 girsanov s theorem references a4 stieltjes lebesgue calculus 1 the stieltjes lebesgue integral 2 the product and exponential formulas references general bibliography

an inverse kinetic theory for the incompressible navier stokes equations abstract an inverse kinetic theory applying specifically to incompressible newtonian fluids which permits us to avoid the n 2 algorithmic complexity of the poisson equation for the fluid pressure is presented the theory is based on the construction of a suitable kinetic equation in phase space which permits us to determine exactly the fluid equations by means of the velocity moments of the kinetic distribution function it is found that the fluid pressure can also be determined as a moment of the distribution function without solving the poisson equation as is usually required in direct solution methods for the incompressible fluid equations finally the dynamical system underlying the incompressible navier stokes equations and advancing in time the fluid fields has been also identified and proven to produce an unique set of fluid equations