Analyzes non-Markovian environments in stochastic approximation.
problem Understanding learning mechanisms in non-ergodic, non-Markovian settings.
method Analytic framework for transformer learning and continual learning.
result Proposes a new approach to transformer and continual learning.
Transformers struggle to learn Markovian dynamics, showing NP-hard optimization challenges.
problem Understanding transformers' limitations in learning Markovian dynamical functions.
method Investigated through a structured ICL setup, analyzing loss landscapes and parameter optimization.
result Recovering optimal transformer parameters for Markovian functions is NP-hard.
State spaces of multifactor approximations of nonnegative Volterra processes are linear transformations of the nonnegative orthant.
problem Characterizing state spaces of multifactor approximations of nonnegative Volterra processes.
method Explicit linear transformation of the nonnegative orthant.
result State spaces of multifactor approximations of nonnegative Volterra processes are given by explicit linear transformation of the nonnegative orthant.
Study develops numerical schemes for non-Markovian volatility models with memory.
problem Existence and uniqueness of strong solutions for non-Markovian SDEs.
method Functional quantization scheme based on Lamperti transformation.
result Theoretical foundation for numerical schemes applied to specific models.
Paper improves neural ODEs for forecasting non-Markovian processes.
problem Forecasting irregularly observed time series with incomplete data.
method Path-dependent Neural Jump ODEs with signature transform.
result Path-dependent NJ-ODE outperforms original framework in non-Markovian data.
The paper develops a deep signature approach for option pricing under non-Markovian stochastic volatility models.
problem Pricing options under non-Markovian stochastic volatility models is challenging due to the dependence on historical paths.
method Reformulate the asset dynamics as a rough stochastic differential equation and represent rough paths via signatures. Apply standard analytical tools to solve the transformed equation.
result The deep signature approach provides a theoretically grounded and computationally efficient framework for option pricing.
New method transforms complex stochastic equations into simpler ones for efficient simulation.
problem Efficient simulation of complex path-dependent stochastic processes.
method Transforms Volterra-type SDEs into standard diffusion processes using convolution kernels.
result Proposes a numerical simulation scheme with a strong convergence rate of 1/2.
Transformers can solve complex filtering problems for non-Gaussian signals.
problem Non-linear and non-Markovian filtering problems for conditionally Gaussian signals.
method Continuous-time transformer models called filterformers.
result Filterformers can approximate the conditional law of non-Markovian and conditionally Gaussian signal processes.
Study small-time CLTs for stochastic Volterra equations with various kernels.
problem Understanding the behavior of stochastic Volterra equations with different kernels.
method Proved convergence of finite-dimensional distributions, functional CLT, and limit theorems for smooth transformations.
result Derived asymptotic pricing formulae for digital calls in rough volatility models.
Develops quasi-likelihood analysis for marked point processes and applies it to Hawkes processes.
problem Analyzing multivariate marked point processes and their applications.
method Quasi-likelihood analysis for a general class of multivariate marked point processes, with focus on marked Hawkes processes.
result The quasi-likelihood analysis for marked Hawkes processes provides explicit conditions for ergodicity and Markovian transformation.
The risk premium is one of main concepts in mathematical finance. It is a measure of the trade-offs investors make between return and risk and is defined by the excess return relative to the risk-free interest rate that is earned from an asset per one unit of risk. The purpose of this article is to determine upper and …
Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.
problem Understanding the scaling behavior of non-Markovian point processes.
method Analyzed a confined fractional Brownian motion-driven point process and compared it to a nonlinear Markovian process.
result A nonlinear Markovian process can reproduce the power-law scaling behavior of a non-Markovian point process.
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
This paper extends Markovian projections to semimartingales with jumps.
problem Extending Markovian projections to semimartingales with jumps.
method Using Markovian projections to match marginal laws of Itô semimartingales with jumps.
result Existence of Markovian projections for Itô semimartingales with jumps.
This paper solves the inversion problem for jump processes using Markovian projections.
problem Calibrating jump-diffusion models with both local and stochastic features.
method Inverting Markovian projections for pure jump processes.
result Constructs calibrated local stochastic intensity (LSI) models for credit risk applications.
We provide a new proof for regularity of affine processes on general state spaces by methods from the theory of Markovian semimartingales. On the way to this result we also show that the definition of an affine process, namely as stochastically continuous time-homogeneous Markov process with exponential affine Fourier-…
Paper introduces PRMs to learn non-Markovian stochastic rewards for reinforcement learning.
problem Lack of structured representation for non-Markovian stochastic rewards in reinforcement learning.
method Introduces probabilistic reward machines (PRMs) and presents an algorithm to learn them from decision processes.
result Algorithm proves correct and convergent for learning PRMs from decision processes.
This paper studies transformer learning dynamics and initialization.
problem Understanding how transformers learn Markov chains and the role of initialization.
method First-order Markov chains and single-layer transformers, proving learning dynamics and conditions for convergence.
result Transformer parameters can converge to global or local minima based on initialization and Markovian data properties.
The fundamental task of general density estimation p(x) has been of keen interest to machine learning. In this work, we attempt to systematically characterize methods for density estimation. Broadly speaking, most of the existing methods can be categorized into either using: \textit{a}) autoregressive models to estim…
Stochastic differential equation approximation for linear TD(0) under Markovian noise
problem Temporal-difference learning with linear function approximation
method Stochastic differential equation approximation
result Explains the constant-stepsize error floor
Paper establishes convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.
problem Analyzing convergence rates and concentration bounds for stochastic approximation and reinforcement learning with Markovian noise.
method Novel discretization of the mean ODE of stochastic approximation algorithms using intervals with diminishing length.
result First almost sure convergence rate and maximal concentration bound with exponential tails for contractive stochastic approximation algorithms with Markovian noise.
Signature portfolios approximate optimal wealth in non-Markovian markets.
problem Approximating optimal wealth in non-Markovian markets.
method Linear path-functional portfolios based on signatures of market weights.
result Signature portfolios can uniformly approximate any continuous portfolio function.
Inference, prediction and control of complex dynamical systems from time series is important in many areas, including financial markets, power grid management, climate and weather modeling, or molecular dynamics. The analysis of such highly nonlinear dynamical systems is facilitated by the fact that we can often find a…
Unified analytical tool for non-Markovian jump processes.
problem Analyzing history-dependent jump processes with non-Markovian behavior.
method Developed a standard form of master equations using Laplace-space embedding and asymptotic solution.
result Unified analytical toolset for general non-Markovian processes, leading to the GLE approximation.
Study improves covariance estimation for SGD under Markovian data, matching best rates.
problem Improving covariance estimation for SGD in Markovian data settings.
method Online overlapping batch-means covariance estimator for SGD under Markovian sampling.
result Established convergence rates for covariance estimation under Markovian sampling.
We develop a Markovian approximation for SVV models to compute hedging strategies.
problem Computing optimal hedging strategies for SVV models with non-Markovian noise.
method Develop a Markovian approximation of the Volterra noise kernel to compute hedging strategies.
result Error estimates for the approximation of volatility, prices, and optimal hedge.
A new HOM model improves forecasting of Indian base metal prices.
problem Improving accuracy in predicting base metal prices in the Indian market.
method A Higher Order Markovian (HOM) model with varying order based on market delay.
result The HOM model consistently outperforms the standard Markovian model in forecasting.
Describes state variables in sequential decision problems, linking them to Markovian and non-Markovian models.
problem Sequential decision problems, especially in active learning and POMDPs, where decisions affect what is observed and learned.
method Canonical framework and novel two-agent perspective of POMDPs, defining state variables to claim Markovian or non-Markovian models.
result Properly modeled sequential decision problems are Markovian, while real decision problems are often non-Markovian.
Deep learning solves non-Markovian FBSDEs for utility maximization.
problem Solving utility maximization problems under rough volatility.
method Deep learning-based numerical methods for non-Markovian fully coupled FBSDEs.
result Error estimates and convergence provided for the deep learning approach.
Investigates optimal consumption and investment strategies in non-Markovian markets with unbounded parameters.
problem Optimal consumption and investment strategies in non-Markovian markets with unbounded parameters.
method Martingale optimal principle and quadratic BSDEs with exponential moment.
result Establishes optimal strategies for consumption and investment.
We simplify a complex volatility model to make it easier to price options.
problem The rough Bergomi model's non-Markovian nature complicates option pricing.
method We approximate the rBergomi model with a Bergomi model that is Markovian.
result The rBergomi model can be effectively approximated by a Markovian model.
New strategy identifies best Markovian arm with fixed confidence.
problem Identifying the best arm in Markovian bandit models with fixed confidence.
method Analyzed the Track-and-Stop strategy and derived a concentration inequality for Markov chains.
result The Track-and-Stop strategy is at most a factor of four apart from the lower bound for asymptotic performance.
This paper first describes a class of uncertain stochastic control systems with Markovian switching, and derives an Itô-Liu formula for Markov-modulated processes. And we characterize an optimal control law, which satisfies the generalized Hamilton-Jacobi-Bellman (HJB) equation with Markovian switching. Then, by using …
Modeling high-frequency order book data with Hawkes-Markovian process.
problem Capturing the dynamics of high-frequency order book events.
method Hawkes process with Markovian baseline intensities, LASSO regularization, and Akaike Information Criteria.
result Effective modeling of order book dynamics with reduced parameter redundancy.
Improved SGD bounds for machine learning models with Markovian noise.
problem Uniform high-probability bounds for SGD under PL condition with Markovian noise.
method Combining Poisson equation for Markovian noise and probabilistic induction for almost-sure bounds.
result Matching 1/k decay rate for expected suboptimality. This paper addresses parameter estimation for wave equations with Markovian switching.
problem Parameter estimation for wave equations with abrupt changes.
method Bayesian statistical framework using discrete sparse Bayesian learning.
result Strong performance in parameter estimation for variable coefficient PDEs.
Paper derives convergence rates and confidence intervals for LSA with Markovian noise.
problem Analyzing convergence rates and constructing confidence intervals for LSA with Markovian noise.
method Derives non-asymptotic Berry-Esseen bounds and multiplier block bootstrap procedure.
result Provides O(n−1/4) convergence rates and guarantees consistent inference. Kernel analog forecasting studied for multiscale systems.
problem Interpreting data-driven predictions in multiscale dynamical systems.
method Kernel analog forecasting methods applied to multiscale systems with varying Markovian closures.
result Guidance provided for interpreting data-driven predictions in practice.
Paper tackles robust offline RL for non-Markovian processes, improving efficiency and applicability.
problem Learning robust policies for non-Markovian decision processes with limited offline data.
method Proposes a novel algorithm with dataset distillation and LCB design for robust values, derived new dual forms, and introduces concentrability coefficients.
result Proves polynomial sample efficiency for finding ε-optimal robust policies.
Hawkes processes are a class of simple point processes that are self-exciting and have clustering effect, with wide applications in finance, social networks and many other fields. This paper considers a self-exciting Hawkes process where the baseline intensity is time-dependent, the exciting function is a general funct…
Paper provides exponential convergence guarantees for Iterative Markovian Fitting.
problem Addressing the Schrödinger Bridge problem in computational optimal transport and generative modeling.
method Develops non-asymptotic exponential convergence guarantees for Iterative Markovian Fitting.
result First non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions.
Paper examines constant stepsize in LSA for Markovian data inference.
problem Improving statistical inference with constant stepsize in LSA for Markovian data.
method Established CLT, used averaged LSA iterates, applied Richardson-Romberg extrapolation.
result Constant stepsize leads to better CI coverage, especially with limited data.
Unified approach for first-order methods with Markovian noise in stochastic optimization and variational inequalities.
problem Stochastic optimization problems with Markovian noise.
method Unified theoretical analysis of first-order gradient methods using randomized batching and multilevel Monte Carlo.
result Optimal (linear) dependence on the mixing time of the noise sequence, eliminating previous limiting assumptions.
Developed scalable Monte Carlo method for VIX option pricing.
problem VIX option pricing in stochastic Volterra rough volatility models with non-Markovian vol-of-vol.
method Infinite dimensional Markovian representation to devise scalable least squares Monte Carlo.
result Efficient VIX option pricing method for generalized models.
We show that when the price process S represents a fully incomplete market, the optimal super-replication of any Markovian claim g(ST) with g(⋅) being nonnegative and lower semicontinuous is of buy-and-hold type. Since both (unbounded) stochastic volatility models and rough volatility models are examples of …
Study finds rough volatility models underperform in SPX option pricing.
problem Inconsistency of rough volatility models with SPX option prices.
method Empirical study using SPX options data, comparing rough and Markovian models.
result Rough volatility models with H∈(0,1/2) are inconsistent with SPX smiles, especially at short maturities. We analyze SA with Markovian data and nonlinear updates, overcoming prior limitations.
problem Analyzing stochastic approximation with Markovian data and nonlinear updates.
method Fine-grained analysis of SA iterates and Markovian data, leveraging smoothness and recurrence properties.
result Established weak convergence and precise asymptotic bias of SA iterates.
The purpose of this article is to describe all possible beliefs of market participants on objective measures under Markovian environments when a risk-neutral measure is given. To achieve this, we employ the Martin integral representation of Markovian pricing kernels. Then, we offer economic and financial implications o…