On the forward algorithm for stopping problems on continuous-time Markov chains

We revisit the forward algorithm, developed by Irle, to characterize both the value function and the stopping set for a large class of optimal stopping problems on continuous-time Markov chains. Our objective is to renew interest in this constructive method by showing its usefulness in solving some constrained optimal stopping problems that have emerged recently.

Overcoming Data Availability Attacks in Blockchain Systems: LDPC Code Design for Coded Merkle Tree

<div>In blockchain systems, full nodes store the entire blockchain ledger and validate all transactions in the system by operating on the entire ledger. However, for better scalability and decentralization of the system, blockchains also run light nodes that only store a small portion of the ledger. In blockchain systems having a majority of malicious full nodes, light nodes are vulnerable to a data availability (DA) attack. In this attack, a malicious node makes the light nodes accept an invalid block by hiding the invalid portion of the block from the nodes in the system. Recently, a technique based on LDPC codes called Coded Merkle Tree (CMT) was proposed by Yu et al. that enables light nodes to detect a DA attack by randomly requesting/sampling portions of the block from the malicious node. However, light nodes fail to detect a DA attack with high probability if a malicious node hides a small stopping set of the LDPC code. To mitigate this problem, Yu et al. used well-studied techniques to design random LDPC codes with high minimum stopping set size. Although effective, these codes are not necessarily optimal for this application. In this paper, we demonstrate that a suitable co-design of specialized LDPC codes and the light node sampling strategy can improve the probability of detection of DA attacks. We consider different adversary models based on their computational capabilities of finding stopping sets in LDPC codes. For a weak adversary model, we devise a new LDPC code construction termed as the entropy-constrained PEG (EC-PEG) algorithm which concentrates stopping sets to a small group of variable nodes. We demonstrate that the EC-PEG algorithm coupled with a greedy sampling strategy improves the probability of detection of DA attacks. For stronger adversary models, we provide a co-design of a sampling strategy called linear-programming-sampling (LP-sampling) and an LDPC code construction called linear-programming-constrained PEG (LC-PEG) algorithm. The new co-design demonstrates a higher probability of detection of DA attacks compared to approaches proposed in earlier literature.</div>

Overcoming Data Availability Attacks in Blockchain Systems: LDPC Code Design for Coded Merkle Tree

<div>In blockchain systems, full nodes store the entire blockchain ledger and validate all transactions in the system by operating on the entire ledger. However, for better scalability and decentralization of the system, blockchains also run light nodes that only store a small portion of the ledger. In blockchain systems having a majority of malicious full nodes, light nodes are vulnerable to a data availability (DA) attack. In this attack, a malicious node makes the light nodes accept an invalid block by hiding the invalid portion of the block from the nodes in the system. Recently, a technique based on LDPC codes called Coded Merkle Tree (CMT) was proposed by Yu et al. that enables light nodes to detect a DA attack by randomly requesting/sampling portions of the block from the malicious node. However, light nodes fail to detect a DA attack with high probability if a malicious node hides a small stopping set of the LDPC code. To mitigate this problem, Yu et al. used well-studied techniques to design random LDPC codes with high minimum stopping set size. Although effective, these codes are not necessarily optimal for this application. In this paper, we demonstrate that a suitable co-design of specialized LDPC codes and the light node sampling strategy can improve the probability of detection of DA attacks. We consider different adversary models based on their computational capabilities of finding stopping sets in LDPC codes. For a weak adversary model, we devise a new LDPC code construction termed as the entropy-constrained PEG (EC-PEG) algorithm which concentrates stopping sets to a small group of variable nodes. We demonstrate that the EC-PEG algorithm coupled with a greedy sampling strategy improves the probability of detection of DA attacks. For stronger adversary models, we provide a co-design of a sampling strategy called linear-programming-sampling (LP-sampling) and an LDPC code construction called linear-programming-constrained PEG (LC-PEG) algorithm. The new co-design demonstrates a higher probability of detection of DA attacks compared to approaches proposed in earlier literature.</div>

Missing at random: a stochastic process perspective

We offer a natural and extensible measure-theoretic treatment of missingness at random. Within the standard missing data framework, we give a novel characterization of the observed data as a stopping-set sigma algebra. We demonstrate that the usual missingness at random conditions are equivalent to requiring particular stochastic processes to be adapted to a set-indexed filtration. These measurability conditions ensure the usual factorization of likelihood ratios. We illustrate how the theory extends easily to incorporate explanatory variables, to describe longitudinal data in continuous time, and to admit more general coarsening of observations.

Cardinality estimation for random stopping sets based on Poisson point processes

We construct unbiased estimators for the distribution of the number of vertices inside random stopping sets based on a Poisson point process. Our approach is based on moment identities for stopping sets, showing that the random count of points inside the complement of a stopping set S has a Poisson distribution conditionally to S. The proofs do not require the use of set-indexed martingales, and our estimators have a lower variance when compared to standard sampling. Numerical simulations are presented for examples such as the convex hull and the Voronoi flower of a Poisson point process, and their complements.

On the threshold strategies in optimal stopping problems for diffusion processes

We study a problem when the optimal stopping for a one-dimensional diffusion process is generated by a threshold strategy. Namely, we give necessary and sufficient conditions (on the diffusion process and the payoff function) under which a stopping set has a threshold structure.