unequal error protection using partially regular ldpc codes in this paper we propose a scheme to construct low density parity check ldpc codes that are suitable for unequal error protection uep we derive density evolution de formulas for the proposed unequal error protecting ldpc ensembles over the binary erasure channel bec using the de formulas we optimize the codes for the finite length cases we compare our codes with some other ldpc codes the time sharing method and a previous work on uep using ldpc codes simulation results indicate the superiority of the proposed design methodology for uep

syndrome decoding of reed solomon codes beyond half the minimum distance based on shift register synthesis in this paper a new approach for decoding low rate reed solomon codes beyond half the minimum distance is considered and analyzed the maximum error correcting radius coincides with the error correcting radius of the sudan algorithm published in 1997 however unlike the sudan algorithm the approach described here is not a list decoding algorithm and is not based on polynomial interpolation the algorithm in this paper is rather syndrome based like classical algebraic decoding algorithms the computational complexity of the new algorithm is of the same order as the complexity of the well known berlekamp massey algorithm to decode errors beyond half the minimum distance the new decoder is allowed to fail for some high weight error patterns with a very small probability

on a class of optimal nonbinary linear unequal error protection codes for two sets of messages several authors have addressed the problem of designing good linear unequal error protection luep codes however very little is known about good nonbinary luep codes the authors present a class of optimal nonbinary luep codes for two different sets of messages by combining t error correcting reed solomon rs codes and shortened nonbinary hamming codes they obtain nonbinary luep codes that protect one set of messages against any t or fewer symbol errors and the remaining set of messages against any single symbol error for t spl ges 2 they show that these codes are optimal in the sense of achieving the hamming lower bound on the number of redundant symbols of a nonbinary luep code with the same parameters

stochastic optimal control of ultradiffusion processes with application to dynamic portfolio management we consider theoretical and approximation aspects of the stochastic optimal control of ultradiffusion processes in the context of a prototype model for the selling price of a european call option within a continuous time framework the dynamic management of a portfolio of assets is effected through continuous or point control activation costs and phase delay the performance index is derived from the unique weak variational solution to the ultraparabolic hamilton jacobi equation the value function is the optimal realization of the performance index relative to all feasible portfolios an approximation procedure based upon a temporal box scheme finite element method is analyzed numerical examples are presented in order to demonstrate the viability of the approach

improved upper bounds for codes with unequal error protection asymptotic nonexistence bounds for unequal error protecting codes with two protection levels are considered we show that the improved estimates on the possible distance distributions for codes may sometimes yield sharper upper bounds than the previously known ones on the higher significance protection level of both nonlinear and linear codes having two protection levels

extension of the blahut arimoto algorithm for maximizing directed information in this paper we extend the blahut arimoto algorithm for maximizing massey s directed information the algorithm can be used for estimating the capacity of channels with delayed feedback where the feedback is a deterministic function of the output in order to maximize the directed information we apply the ideas from the regular blahut arimoto algorithm i e the alternating maximization procedure to our new problem we provide both upper and lower bound sequences that converge to the optimum global value our main insight in this paper is that in order to find the maximum of the directed information over a causal conditioning probability mass function one can use a backward index time maximization combined with the alternating maximization procedure we give a detailed description of the algorithm showing its complexity and the memory needed and present several numerical examples

a simple converse of burnashev s reliability function in a remarkable paper published in 1976 burnashev determined the reliability function of variable length block codes over discrete memoryless channels dmcs with feedback subsequently an alternative achievability proof was obtained by yamamoto and itoh via a particularly simple and instructive scheme their idea is to alternate between a communication and a confirmation phase until the receiver detects the codeword used by the sender to acknowledge that the message is correct we provide a converse that parallels the yamamoto itoh achievability construction besides being simpler than the original the proposed converse suggests that a communication and a confirmation phase are implicit in any scheme for which the probability of error decreases with the largest possible exponent the proposed converse also makes it intuitively clear why the terms that appear in burnashev s exponent are necessary

new list decoding algorithms for reed solomon and bch codes in this paper we devise a rational curve fitting algorithm and apply it to the list decoding of reed solomon and bose chaudhuri hocquenghen bch codes the resulting list decoding algorithms exhibit the following significant properties

implicit explicit runge kutta methods for financial derivatives pricing models abstract implicit explicit runge kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach the methods are showed to be an alternative to other existing procedures for the numerical valuation of american type contracts we follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements finite differences and finite volume methods some numerical experiments are presented

subset source coding this paper studies the fundamental limits of storage for structured data where statistics and structure are both critical to the application accordingly a framework is proposed for optimal lossless and lossy compression of subsets of the possible realizations of a discrete memoryless source dms for the lossless subset compression problem it turns out that the optimal source code may not index the conventional source typical sequences but rather index certain subset typical sequences consistent with the subset of interest building upon an achievability and a strong converse an analytic expression is given based on the shannon entropy relative entropy and subset entropy which identifies such subset typical sequences for a broad class of subsets of a dms intuitively subset typical sequences belong to those typical sets which highly intersect the subset of interest but are still closest to the source distribution in the sense of relative entropy for the lossy subset compression problem an upper bound is derived on the subset rate distortion function in terms of the subset mutual information optimized over the set of conditional distributions that satisfy the expected distortion constraint with respect to the subset typical distribution and over a set of certain auxiliary subsets by proving a strong converse result this upper bound is shown to be tight for a class of symmetric subsets as shown in our numerical examples more often than not one achieves a gain in the fundamental limits in that the optimal compression rate for the subset in both the lossless and lossy settings can be strictly smaller than the source entropy and the source rate distortion function respectively although exceptions are also possible

numerical analysis of american option pricing in a jump diffusion model we discuss pricing formulae for american options in merton s jump diffusion model with the help of variational inequalities we derive some regularity properties of price functions using the finite difference method a discretization scheme is presented and a convergence theorem for the first order derivatives is proved numerical methods and results are also discussed

on the use of boundary conditions for variational formulations arising in financial mathematics the general intractability of derivative security valuation models to present techniques both analytic and numerical arguably remains one of the preeminant problem of mathematical finance it is the focus of this paper to examine the applicability of a promising recent development namely radial basis functions rbf to the problem of option valuation a black scholes framework is considered for american and european options written on a one and two risky assets the performance of rbf and finite differencing algorithms are examined with respect to artificial boundary conditions computational domain domain decomposition and mesh scaling

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

a modified blahut algorithm for decoding reed solomon codes beyond half the minimum distance a modification of the blahut algorithm is proposed for decoding of reed solomon codes beyond half the minimum distance an effective method is offered for the searching of unknown discrepancies needed for analytical continuation of the berlekamp massey algorithm through two additional iterations this reduces the search time by 2 q 1 n n t 1 n t times compared with the blahut algorithm an architecture of a searcher for unknown discrepancies is given the coding gain of the proposed algorithm is shown for some practical codes

adaptive finite element methods for parabolic problems i a linear model problem this paper is the first part in a series of papers on adaptive finite element methods for parabolic problems in this paper an adaptive algorithm is presented and analyzed for choosing the space a

constructing linear unequal error protection codes from algebraic curves we show that the concept of generalized algebraic geometry codes which was introduced by xing niederreiter and lam see ibid vol 45 p 2498 2501 nov 1999 gives a natural framework for constructing linear unequal error protection codes

bit wise unequal error protection for variable length block codes with feedback the bit wise unequal error protection problem for the case when the number of groups of bits l is fixed is considered for variable length block codes with feedback an encoding scheme based on fixed length block codes with erasures is used to establish inner bounds to the achievable performance for finite expected decoding time a new technique for bounding the performance of variable length block codes is used to establish outer bounds to the performance for a given expected decoding time the inner and the outer bounds match one another asymptotically and characterize the achievable region of rate exponent vectors completely the single message message wise unequal error protection problem for variable length block codes with feedback is also solved as a necessary step on the way

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

random access and source channel coding error exponents for multiple access channels a new universal coding decoding scheme for random access with collision detection is given in case of two senders the result is used to give achievable joint source channel coding error exponents for multiple access channel and independent sources in a modified model admitting zero rate communication between the senders an improved exponent is derived of form similar to csiszar s joint source channel exponent for the one sender case

unequal message protection asymptotic and non asymptotic tradeoffs we study a form of unequal error protection that we term unequal message protection ump the message set of a ump code is a union of m disjoint message classes each class has its own error protection requirement with some classes needing better error protection than others we analyze the tradeoff between rates of message classes and the levels of error protection our analysis reveals new tradeoffs which were not captured by prior works on ump codes to obtain our results we generalize finite block length achievability and converse bounds due to polyanskiy poor verdu we evaluate our bounds for the binary symmetric and binary erasure channels and analyze the asymptotic characteristic of the bounds in the fixed error and moderate deviations regimes in addition we consider two questions related to the practical construction of ump codes first we study a header construction that prefixes the message class into a header followed by data protection using a standard homogeneous classical code we show that in general this construction is not optimal at finite block lengths we further demonstrate that our main ump achievability bound can be obtained using coset codes which suggests a path to implementation of tractable ump codes

adaptive multilevel methods for obstacle problems the authors consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies the linear problems are solved iteratively by preconditioned conjugate gradient iterations the proposed preconditioners are treated theoretically as abstract additive schwarz methods and are implemented as truncated hierarchical basis preconditioners to allow for local mesh refinement semilocal and local a posteriors error estimates are derived providing lower and upper estimates for the discretization error the theoretical results are illustrated by numerical computations

pricing options in jump diffusion models an extrapolation approach we propose a new computational method for the valuation of options in jump diffusion models the option value function for european and barrier options satisfies a partial integrodifferential equation pide this pide is commonly integrated in time by implicit explicit imex time discretization schemes where the differential diffusion term is treated implicitly while the integral jump term is treated explicitly in particular the popular imex euler scheme is first order accurate in time second order accuracy in time can be achieved by using the imex midpoint scheme in contrast to the above approaches we propose a new high order time discretization scheme for the pide based on the extrapolation approach to the solution of odes that also treats the diffusion term implicitly and the jump term explicitly the scheme is simple to implement can be added to any pide solver based on the imex euler scheme and is remarkably fast and accurate we demonstrate our approach on the examples of merton s and kou s jump diffusion models the diffusion extended variance gamma model as well as the two dimensional duffie pan singleton model with correlated and contemporaneous jumps in the stock price and its volatility by way of example pricing a one year double barrier option in kou s jump diffusion model our scheme attains accuracy of 10 5 in 72 time steps in 0 05 seconds in contrast it takes the first order imex euler scheme more than 1 3 million time steps in 873 seconds and the second order imex midpoint scheme 768 time steps in 0 49 seconds to attain the same accuracy our scheme is also well suited for bermudan options combining simplicity of implementation and remarkable gains in computational efficiency we expect this method to be very attractive to financial engineering modelers

identification via channels the authors main finding is that any object among doubly exponentially many objects can be identified in blocklength n with arbitrarily small error probability via a discrete memoryless channel dmc if randomization can be used for the encoding procedure a novel doubly exponential coding theorem is presented which determines the optimal r that is the identification capacity of the dmc as a function of its transmission probability matrix this identification capacity is a well known quantity namely shannon s transmission capacity for the dmc

lower bounds on parameter modulation estimation under bandwidth constraints the problem of modulating the value of a parameter onto a band limited signal to be transmitted over a continuous time additive white gaussian noise awgn channel and then estimating this parameter at the receiver is considered the performance is measured by the mean power alpha error mp alpha text e which is defined as the worst case alpha th order moment of the absolute estimation error the optimal exponential decay rate of the mp alpha text e as a function of the transmission time is investigated two upper converse bounds on the mp alpha text e exponent are derived on the basis of known bounds for the awgn channel of inputs with unlimited bandwidth the bounds are computed for typical values of the error moment and the signal to noise ratio snr and the snr asymptotics of the different bounds are analyzed the new bounds are compared with known converse and achievability bounds which were derived from channel coding considerations

on the approximation of optimal stopping problems with application to financial mathematics the determination of the value function associated with a given reward and stochastic process represents an important class of stochastic control problem in particular the expectations of such processes may be represented as solutions of variational inequalities of evolutionary type typically characterized by their high number of degrees of freedom unbounded domains and lack of natural boundary conditions in this paper we introduce two methodologies for computing the value function of optimal stopping associated with general stochastic processes our results are implemented utilizing finite elements and are validated using problems taken from financial mathematics

a discontinuoushpfinite element method for diffusion problems we present an extension of the discontinuous galerkin method which is applicable to the numerical solution of diffusion problems the method involves a weak imposition of continuity conditions on the solution values and on fluxes across interelement boundaries within each element arbitrary spectral approximations can be constructed with different orderspin each element we demonstrate that the method is elementwise conservative a property uncharacteristic of high order finite elements for clarity we focus on a model class of linear second order boundary value problems and we developpriorierror estimates convergence proofs and stability estimates the results of numerical experiments onh andp convergence rates for representative two dimensional problems suggest that the method is robust and capable of delivering exponential rates of convergence

nonuniform error correction using low density parity check codes this correspondence introduces a framework to design and analyze low density parity check ldpc codes over nonuniform channels we study ldpc codes for channels with nonuniform noise distributions rate adaptive coding and unequal error protection first we propose a technique to design ldpc codes for volume holographic memory vhm systems for which the noise distribution is nonuniform we show that the proposed coding scheme has an easy design procedure and results in efficient codes for holographic memories an important property of the proposed technique is the design of the codes that have a low error floor and low variable node degrees while maintaining performance close to the shannon limit we then show that punctured ldpc codes can be studied as a special case of our design methodology for nonuniform channels finally we propose a method to generate ldpc codes that can provide unequal error protection in addition to having a good overall performance moreover the highly protected bits can be decoded without requiring the entire word to be decoded

a numerical analysis of variational valuation techniques for derivative securities in this paper we consider the partial differential equations approach for valuing european and american style options on multiple assets we use a method of lines finite element implementation available in the software package femlab in order to solve the variational inequality that characterizes the american style option as well as the partial differential equation that defines the european style option for two and three state variables a detailed study of the approximation error is provided including a theoretical estimate an asymptotic analysis the space time distribution and the dependence on the size of the truncation domain

streaming data transmission in the moderate deviations and central limit regimes we consider streaming data transmission over a discrete memoryless channel a new message is given to the encoder at the beginning of each block and the decoder decodes each message sequentially after a delay of t blocks in this streaming setup we study the fundamental interplay between the rate and error probability in the central limit and moderate deviations regimes and show that 1 in the moderate deviations regime the moderate deviations constant improves over the block coding or non streaming setup by a factor of t and 2 in the central limit regime the second order coding rate improves by a factor of approximately sqrt t for a wide range of channel parameters for both the regimes we propose coding techniques that incorporate a joint encoding of fresh and previous messages in particular for the central limit regime we propose a coding technique with truncated memory to ensure that a summation of constants which arises as a result of applications of the central limit theorem does not diverge in the error analysis furthermore we explore interesting variants of the basic streaming setup in the moderate deviations regime we first consider a scenario with an erasure option at the decoder i e the decoder can output an erasure symbol instead of a message estimate and show that both the exponents of the total error and the undetected error probabilities improve by factors of t next by utilizing the erasure option we show that the exponent of the total error probability can be improved to that of the undetected error probability in the order sense at the expense of a variable decoding delay

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

bounded distance 1 soft decision reed solomon decoding we present a new reed solomon decoding algorithm which embodies several refinements of an earlier algorithm some portions of this new decoding algorithm operate on symbols of length lgq bits other portions operate on somewhat longer symbols in the worst case the total number of calculations required by the new decoding algorithm is proportional to nr where n is the code s block length and r is its redundancy this worst case workload is very similar to prior algorithms but in many applications average case workload and error correcting performance are both much better the input to the new algorithm consists of n received symbols from gf q and n nonnegative real numbers each of which is the reliability of the corresponding received symbol any conceivable errata pattern has a score equal to the sum of the reliabilities of its locations with nonzero errata values a max likelihood decoder would find the minimum score over all possible errata patterns our new decoding algorithm finds the minimum score only over a subset of these possible errata patterns the errata within any candidate errata pattern may be partitioned into errors and erasures depending on whether the corresponding reliabilities are above or below an erasure threshold different candidate errata patterns may have different thresholds each chosen to minimize its corresponding errata count which is defined as 2 spl middot number of errors number of erasures the new algorithm finds an errata pattern with minimum score among all errata patterns for which errata count spl les r 1 where r is the redundancy of the rs code conventional algorithms also require that the erasure threshold be set a priori the new algorithm obtains the best answer over all possible settings of the erasure threshold conventional cyclic rs codes have length n q 1 and their locations correspond to the nonzero elements of gf q the new algorithm also applies very naturally to rs codes which have been doubly extended by the inclusion of 0 and spl infin as additional locations

two topics on linear unequal error protection codes bounds on their length and cyclic code classes it is possible for a linear block code to provide more protection for selected positions in the input message words than is guaranteed by the minimum distance of the code linear codes having this property are called linear unequal error protection luep codes bounds on the length of a luep code that ensures a given unequal error protection are derived a majority decoding method for certain classes of cyclic binary uep codes is treated a list of short i e of length less than 16 binary luep codes of optimal i e minimal length and a list of all cyclic binary uep codes of length less than 40 are included

the awgn red alert problem consider the following unequal error protection scenario one special message dubbed the red alert message is required to have an extremely small probability of missed detection the remainder of the messages must keep their average probability of error and probability of false alarm below a certain threshold the goal then is to design a codebook that maximizes the error exponent of the red alert message while ensuring that the average probability of error and probability of false alarm go to zero as the blocklength goes to infinity this red alert exponent has previously been characterized for discrete memoryless channels this paper completely characterizes the optimal red alert exponent for additive white gaussian noise channels with block power constraints

unequal error protection an information theoretic perspective an information theoretic framework for unequal error protection is developed in terms of the exponential error bounds the fundamental difference between the bit wise and message wise unequal error protection uep is demonstrated for fixed length block codes on discrete memoryless channels dmcs without feedback effect of feedback is investigated via variable length block codes it is shown that feedback results in a significant improvement in both bit wise and message wise ueps except the single message case for missed detection the distinction between false alarm and missed detection formalizations for message wise uep is also considered all results presented are at rates close to capacity