Study models interest rates as CTMC, pricing and replicating derivatives.
problem Modeling and pricing financial derivatives in a CTMC setting.
method Model short rate as CTMC, derive pricing and replication strategies, apply Ross Recovery Theorem.
result Derive real-world dynamics of CTMC.
LEAPS samples discrete distributions via CTMCs and locally equivariant networks.
problem Sampling from discrete distributions with known normalization.
method Continuous-time Markov chain, locally equivariant functions, attention layers, convolutional networks.
result LEAPS minimizes the variance of importance weights, improving sampling efficiency.
New methods solve complex financial equations.
problem Solving backward stochastic differential equations driven by continuous-time Markov chains.
method Multi-stage Euler-Maruyama methods and multilevel spatial discretization.
result Efficiently solved stiff Markov BSDEs.
New method links covariates to CTMCs using RKHS, improving state transitions modeling.
problem Traditional multistate models rely on linear relationships, limiting flexibility.
method Nonparametric approach using RKHS, with Frequentist and Bayesian versions.
result Effective in identifying nonlinear transition functions and predicting long-term behaviors.
Federated CTMC model estimates bridge deterioration hazards without sharing raw data.
problem Bridge inspection data privacy and cross-organizational data sharing constraints.
method Federated CTMC hazard model with local optimization and FedAvg aggregation.
result Federated model converges on global benchmark parameters without data transfer.
We consider the task of learning a parametric Continuous Time Markov Chain (CTMC) sequence model without examples of sequences, where the training data consists entirely of aggregate steady-state statistics. Making the problem harder, we assume that the states we wish to predict are unobserved in the training data. Spe…
Fluid approximations have seen great success in approximating the macro-scale behaviour of Markov systems with a large number of discrete states. However, these methods rely on the continuous-time Markov chain (CTMC) having a particular population structure which suggests a natural continuous state-space endowed with a…
This paper develops methods for pricing American Parisian options under general Markov models.
problem Pricing American Parisian options with various types and payoff functions.
method General approaches using CTMC approximation for time-inhomogeneous Markov models, including state augmentation and variational inequalities.
result Efficient algorithms for pricing American Parisian options confirmed with numerical experiments.
Improved language models using ratio-matching and KL divergence.
problem Efficiently modeling discrete data with diffusion models.
method Introduced new theorems and a novel CTMC transition-rate matrix for ratio-matching and KL divergence.
result 10-15% improvement in perplexity and faster training steps.
We develop continuous time Markov chain (CTMC) approximation of one-dimensional diffusions with a lower sticky boundary. Approximate solutions to the action of the Feynman-Kac operator associated with a sticky diffusion and first passage probabilities are obtained using matrix exponentials. We show how to compute matri…
Continuous time framework for discrete data denoising models.
problem Efficient training and sampling for discrete data denoising models.
method Formulated as Continuous Time Markov Chains (CTMCs), efficient training using continuous time ELBO, high-dimensional CTMC simulation, novel theoretical error bound.
result Continuous time treatment enables novel theoretical error bound between generated and true data distributions.
QTD integrates quantization with diffusion for efficient data generation.
problem Challenges in continuous diffusion models, especially long-range transitions and biases.
method Quantized Transition Diffusion (QTD) integrates data quantization with discrete diffusion dynamics.
result QTD achieves efficient data generation with minimal score evaluations.
Unified framework for drawdown risk computation under Markov models.
problem High computational challenges in drawdown risk metrics.
method Unified framework for computing five drawdown quantities under general Markov models, using linear systems and efficient algorithms.
result Efficient algorithms achieve same complexity as path-independent problems, validated by rigorous convergence analysis and extensive experiments.
Sampling the parameters of high-dimensional Continuous Time Markov Chains (CTMC) is a challenging problem with important applications in many fields of applied statistics. In this work a recently proposed type of non-reversible rejection-free Markov Chain Monte Carlo (MCMC) sampler, the Bouncy Particle Sampler (BPS), i…
Discrete diffusion models improve data generation for discrete data like language and graphs.
problem Adapting diffusion models to discrete state spaces for better data generation.
method Formulated as CTMCs, used uniformization of continuous Markov chains for sampling.
result Derive guarantees for sampling from any distribution on a hypercube, aligning with state-of-the-art achievements.
This paper analyzes discrete diffusion models, deriving convergence bounds for their generated samples.
problem Theoretical guarantees for discrete-state diffusion models remain under-explored.
method Continuous Time Markov Chain (CTMC) framework and discrete-time sampling algorithm.
result Convergence bounds for KL divergence and TV distance are derived, showing linear dependence on dimension.
Paper approximates rough stochastic local volatility models for efficient computation.
problem No unified method for rough stochastic local volatility models.
method Semimartingale and continuous-time Markov chain approximation.
result Fast CTMC algorithm with weak convergence proved.
New method simulates sticky boundaries in multidimensional diffusions.
problem Simulating sticky boundaries in multidimensional diffusions.
method Approximate sticky diffusion by a Markov chain, using either finite difference or matching local moments.
result Validates both construction methods for first-order simulation schemes.
Bond rating Transition Probability Matrices (TPMs) are built over a one-year time-frame and for many practical purposes, like the assessment of risk in portfolios or the computation of banking Capital Requirements (e.g. the new IFRS 9 regulation), one needs to compute the TPM and probabilities of default over a smaller…
Unified framework for pricing various debt securities.
problem Pricing of different types of debt securities under general short-rate processes.
method Unifying framework using continuous-time Markov chain approximations and bi-dimensional diffusion processes.
result Closed-form matrix expressions and efficient algorithms for pricing various debt securities.
Cai, Song and Kou (2015) [Cai, N., Y. Song, S. Kou (2015) A general framework for pricing Asian options under Markov processes. Oper. Res. 63(3): 540-554] made a breakthrough by proposing a general framework for pricing both discretely and continuously monitored Asian options under one-dimensional Markov processes. In …
Predicting stochastic cellular dynamics as emerging from the mechanistic models of molecular interactions is a long-standing challenge in systems biology: low-level chemical reaction network (CRN) models give raise to a highly-dimensional continuous-time Markov chain (CTMC) which is computationally demanding and often …
Method calculates Parisian stopping times and option prices using Markov chains.
problem Computing distribution and pricing of Parisian stopping times under Markov processes.
method Continuous-time Markov chain approximation to solve for distribution and convergence analysis.
result Sharp convergence rate and efficient method for diffusion and jump models.
New estimator for SDEs is shown to be an adjoint state method.
problem Estimating gradients for overparameterized SDEs efficiently.
method Demonstrates generator gradient estimator as an adjoint state method.
result Generator gradient estimator is an adjoint state method for SDEs.
PH-VAE models heavy-tailed data with flexible Phase-Type distributions.
problem Standard VAEs fail to capture heavy-tailed behavior in real-world data.
method PH-VAE uses Phase-Type distributions defined by continuous-time Markov chains to adaptively model tail behavior.
result PH-VAE significantly outperforms existing heavy-tail-aware VAEs in approximating diverse heavy-tailed distributions.
Unified error analysis for discrete flow models.
problem Error analysis of discrete flow models.
method Stochastic calculus theory, Girsanov theorem, generator matching, uniformization.
result First error analysis for discrete flow models.
This primer explains diffusion models in general state spaces.
problem Diffusion models in general state spaces are not well-introduced.
method Develops discrete-time and continuous-time views of diffusion models, deriving Fokker-Planck and master equations.
result Unified understanding of diffusion models across continuous and discrete domains.
GADD accelerates uniform-rate discrete diffusion models by 2 orders of magnitude.
problem Slow sampling in uniform-rate discrete diffusion models.
method Gibbs-based corrector (GADD) that constructs Gibbs posterior likelihoods directly from the concrete score function.
result Achieves an overall sampling complexity of O(polylog(ε−1)). This work improves the Euler method for masked diffusion models, providing tighter convergence guarantees.
problem Improving the convergence rates of masked diffusion models.
method Developed a direct total-variation (TV) based analysis for the Euler method, relaxing assumptions and improving parameter dependencies.
result Established convergence guarantees for the Euler sampler without requiring surrogate initialization, and provided a tight lower bound.
The paper analyzes sampling efficiency of discrete diffusion models, providing sharp and adaptive guarantees.
problem Theoretical foundations of discrete diffusion models, especially sampling efficiency.
method Continuous-time Markov chain (CTMC) formulation, τ-leaping-based samplers, effective total correlation. result The τ-leaping algorithm achieves an iteration complexity of order ildeO(d/ε) for uniform discrete diffusion, improving existing bounds by a factor of d. A discrete diffusion model learns denoising, scoring, and bridging in different coordinates.
problem Understanding what a discrete diffusion model learns in different coordinate systems.
method Rigorous derivation of continuous-time Markov chain ELBO, Oracle Distance theorem, and exact coordinates for optimizer.
result The negative ELBO is exactly equal to the data entropy plus the path KL from the oracle reverse process to the learned one.