Approximate solutions of hybrid stochastic pantograph equations with Levy jumps

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in 퐿 2 sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milošević and Jovanović (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory

Approximate solutions of hybrid stochastic pantograph equations with Levy jumps

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in 퐿 2 sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milošević and Jovanović (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory

Approximate solutions of hybrid stochastic pantograph equations with Levy jumps

We investigate a class of stochastic pantograph differential equations with Markovian switching and Levy jumps. We prove that the approximate solutions converge to the true solutions in 퐿 2 sense as well as in probability under local Lipschitz condition and generalize the results obtained by Fan et al. (2007), Milošević and Jovanović (2011), and Marion et al. (2002) to cover a class of more general stochastic pantograph differential equations with jumps. Finally, an illustrative example is given to demonstrate our established theory

Mean Square Polynomial Stability of Numerical Solutions to a Class of Stochastic Differential Equations

The exponential stability of numerical methods to stochastic differential equations (SDEs) has been widely studied. In contrast, there are relatively few works on polynomial stability of numerical methods. In this letter, we address the question of reproducing the polynomial decay of a class of SDEs using the Euler--Maruyama method and the backward Euler--Maruyama method. The key technical contribution is based on various estimates involving the gamma function

Models and information-theoretic bounds for nanopore sequencing

Nanopore sequencing is an emerging new technology for sequencing DNA, which can read long fragments of DNA (~50,000 bases) in contrast to most current short-read sequencing technologies which can only read hundreds of bases. While nanopore sequencers can acquire long reads, the high error rates (20%-30%) pose a technical challenge. In a nanopore sequencer, a DNA is migrated through a nanopore and current variations are measured. The DNA sequence is inferred from this observed current pattern using an algorithm called a base-caller. In this paper, we propose a mathematical model for the "channel" from the input DNA sequence to the observed current, and calculate bounds on the information extraction capacity of the nanopore sequencer. This model incorporates impairments like (non-linear) inter-symbol interference, deletions, as well as random response. These information bounds have two-fold application: (1) The decoding rate with a uniform input distribution can be used to calculate the average size of the plausible list of DNA sequences given an observed current trace. This bound can be used to benchmark existing base-calling algorithms, as well as serving a performance objective to design better nanopores. (2) When the nanopore sequencer is used as a reader in a DNA storage system, the storage capacity is quantified by our bounds