New SDEs from affine and polynomial perspectives for path-dependent processes.
problem Characterizing path-dependent stochastic processes.
method Affine and polynomial processes, signature SDEs, Fourier-Laplace transform, Riccati and linear ODEs.
result Explicit formulas for the Fourier-Laplace transform and expected values of entire functions of signature processes.
Combines Gaussian processes and polynomial chaos for stochastic control.
problem Uncertainties in dynamic models lead to performance issues in predictive control.
method Combines Gaussian processes with polynomial chaos expansions to estimate probability distributions of nonlinear functions.
result Demonstrates accurate approximation and closed-loop performance in stochastic nonlinear model predictive control.
Approximates discounted moments for financial products using polynomial expansions.
problem Approximating discounted moments of stochastic processes for financial applications.
method High-order power series expansion of the infinitesimal generator.
result Error decreases to around 10 to 100 times machine precision for higher orders.
Quantization algorithms have been successfully adopted to option pricing in finance thanks to the high convergence rate of the numerical approximation. In particular, very recently, recursive marginal quantization has been proven to be a flexible and versatile tool when applied to stochastic volatility processes. In th…
In the setting of polynomial jump-diffusion dynamics, we provide an explicit formula for computing correlators, namely, cross-moments of the process at different time points along its path. The formula appears as a linear combination of exponentials of the generator matrix, extending the well-known moment formula for p…
We introduce polynomial processes in the sense of [8] in the context of stochastic portfolio theory to model simultaneously companies' market capitalizations and the corresponding market weights. These models substantially extend volatility stabilized market models considered by Robert Fernholz and Ioannis Karatzas in …
Paper identifies reductive MDPs, solving them in polynomial time.
problem Computational hardness of general MDPs and tractability of finite-horizon MDPs.
method Defines reductivity, a new class of SSPs, and develops a polynomial-time solution.
result Optimal policies can be found in polynomial time for reductive SSPs and MDPs.
New method samples from time-integrated stochastic bridges using neural networks.
problem Sampling from time-integrated stochastic bridges with high accuracy and speed.
method Polynomial chaos expansion and artificial neural networks.
result Robust, data-driven Monte Carlo sampling with thousands of samples in milliseconds.
The model uses signatures to accurately calibrate SPX and VIX options without jumps or rough volatility.
problem Joint calibration of SPX and VIX options without jumps or rough volatility.
method The approach uses a stochastic volatility model with signatures of polynomial diffusions to price and calibrate SPX and VIX options.
result Highly accurate calibration results for SPX and VIX options without adding jumps or rough volatility.
A new volatility model calibrates SPX & VIX smiles with 6 parameters.
problem Joint calibration of SPX and VIX smiles with a simple model.
method Quintic Ornstein-Uhlenbeck volatility model with polynomial volatility process.
result Remarkable joint fits of SPX-VIX smiles with only 6 parameters.
Neural Chaos uses neural networks instead of polynomials for stochastic modeling.
problem Challenges in constructing surrogate models with uncertainty quantification for complex or high-dimensional stochastic processes.
method Adopting spectral expansion formalism with neural network basis functions, identifying them data-drivenly without prior assumptions.
result Demonstrates effectiveness of the proposed scheme through numerical examples of varying complexity.
Paper provides a method to price electricity storage contracts using COS technique.
problem Valuation of electricity storage contracts considering physical and operational constraints.
method Uses Fourier-based COS method to price contracts based on stochastic polynomial process.
result The COS method accurately and efficiently prices electricity storage contracts.
We consider a stochastic volatility model with Lévy jumps for a log-return process Z=(Zt)t≥0 of the form Z=U+X, where U=(Ut)t≥0 is a classical stochastic volatility process and X=(Xt)t≥0 is an independent Lévy process with absolutely continuous Lévy measure ν. Small-time expansio…
Typically flat filling, linear or polynomial interpolation methods to generate missing historical data. We introduce a novel optimal method for recreating data generated by a diffusion process. The results are then applied to recreate historical data for stocks.
Extends Feller theory to non-locally compact spaces for stochastic equations.
problem Stochastic partial differential equations and fractional processes.
method Extended Feller processes and proofs of folklore results.
result No condition of generalized Feller semigroups can be dropped.
Develops polynomial diffusion models for multi-factor commodity futures dynamics.
problem Modeling futures prices using latent state variables for short and long-term stochastic factors.
method Polynomial diffusion models to incorporate non-linear effects, two filtering methods for estimation.
result Accurate estimation of futures prices despite parameter identification issues in polynomial diffusion models.
Stochastic gradient descent achieves polynomial convergence rates for noiseless linear models.
problem Convergence analysis of stochastic gradient descent in noiseless linear models.
method Fixed step-size stochastic gradient descent on least-square risk.
result Polynomial convergence rates depend on the regularities of the optimum and feature vectors.
We consider a general d-dimensional Levy-type process with killing. Combining the classical Dyson series approach with a novel polynomial expansion of the generator A(t) of the Levy-type process, we derive a family of asymptotic approximations for transition densities and European-style options prices. Examples of stoc…
This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the in…
A new model fits SPX and VIX volatility surfaces and term structures efficiently.
problem Calibrating SPX and VIX volatility models to market data.
method Gaussian polynomial volatility models, joint calibration, functional quantization, Neural Networks.
result A conventional one-factor Markovian model outperforms rough and non-rough models.
New algorithms reduce complexity for learning in MDPs with entropy regularization.
problem Efficient learning for MDPs with large or continuous state and action spaces.
method Multilevel Monte Carlo (MLMC) algorithms integrating fixed-point iteration and stochastic approximation of the Bellman operator.
result MLMC with unbiased approximation of the Bellman operator achieves polynomial sample complexity.
Study approximates weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
problem Approximating weak error for specific stochastic models with rough and Gaussian mean-reverting volatility.
method Used Euler type scheme with integrated kernels to study weak convergence rate.
result Obtained weak convergence rate of min(3α−1,1) for discretised rough Ornstein-Uhlenbeck process and stochastic rough volatility model. Study proposes a new model for joint survival annuity valuation.
problem Valuation of joint survival annuities and options.
method Linear-rational Wishart mortality model based on stochastic matrix affine process.
result Derives closed-form expression for joint survival annuity and option.
A new SBI framework for trawl processes efficiently estimates parameters from large datasets.
problem Challenges in estimating parameters of complex stochastic processes.
method Telescoping ratio estimation, Chebyshev polynomial approximations, amortized posterior inference.
result Accurate and efficient inference for intractable stochastic processes, even with limited data.
We study discretizations of polynomial processes using finite state Markov processes satisfying suitable moment matching conditions. The states of these Markov processes together with their transition probabilities can be interpreted as Markov cubature rules. The polynomial property allows us to study such rules using …
Two new methods solve large-scale stochastic convex problems with linear constraints.
problem Solving large-scale stochastic convex optimization problems with many linear constraints.
method Conditional gradient-based methods that process only a subset of constraints at each iteration.
result Rigorous convergence guarantees for the proposed methods.
A new neural network model uses polynomial chaos theory to improve neural signal processing.
problem Redundant neural signal representation in DANNs.
method Employing arbitrary polynomial chaos theory to construct orthonormal representations in DANNs.
result Improves neural signal processing by reducing redundancy and enhancing orthogonality.
New sampling method for Heston model reduces complexity.
problem Efficient sampling for Heston model's time integrated variance.
method Series expansion, change of measure, Chebyshev polynomial approximations.
result Strong, efficient sampling scheme established for Heston model.
Algorithm learns diffusion processes with high-dimensional state spaces.
problem Stochastic control of unbounded diffusion processes with high-dimensional state spaces.
method Adaptive partitioning and learning algorithm that refines discretization based on estimation bias and statistical confidence.
result Established regret bounds that depend on problem parameters, extending to unbounded diffusion processes.
We introduce polynomial processes taking values in an arbitrary Banach space B via their infinitesimal generator L and the associated martingale problem. We obtain two representations of the (conditional) moments in terms of solutions of a system of ODEs on the truncated tensor algebra of dual respectively bidual s…
We developed a new class of physics-informed generative adversarial networks (PI-GANs) to solve in a unified manner forward, inverse and mixed stochastic problems based on a limited number of scattered measurements. Unlike standard GANs relying only on data for training, here we encoded into the architecture of GANs th…
AL-SPCE improves reliability analysis for complex systems with active learning and SPCE.
problem Efficiently analyzing reliability of complex, computationally expensive models with intrinsic randomness.
method Active learning framework using stochastic polynomial chaos expansions (SPCE) to reduce computational burden.
result AL-SPCE maintains high accuracy in reliability estimates while significantly improving efficiency.
Polynomial processes have the property that expectations of polynomial functions (of degree n, say) of the future state of the process conditional on the current state are given by polynomials (of degree ≤n) of the current state. Here we explore the application of polynomial processes in the context of structur…
Wide networks with polynomial activations have proven asymptotic behavior.
problem Understanding the behavior of neural networks in the large width limit.
method Proving a conjecture for deep networks with polynomial activation functions.
result Tight bounds on the behavior of wide networks during stochastic gradient descent and derivation of their finite-width dynamics.
Paper proves minibatch SGD for GP inference converges and improves generalization.
problem Theoretical understanding and practical use of SGD for correlated samples in Gaussian process inference.
method Proves minibatch SGD converges to a critical point with rate O(1/K) for K iterations, under certain kernel conditions.
result Minibatch SGD for GP inference improves generalization and reduces computational burden.
A machine learning approach to compute Black-Scholes prices with uncertain volatility.
problem Approximating financial markets with continuous-time models like Black-Scholes when data is discrete.
method Generalized Polynomial Chaos (gPC) method combined with a machine learning technique called Bi-Fidelity.
result Efficient numerical method to quantify uncertainty in derivative pricing.
Efficient algorithm for matching graphs with community structure.
problem Graph matching between correlated stochastic block models with constant correlation.
method Partition trees rooted from each vertex, comparing edge statistics to different communities.
result First low-order polynomial-time algorithm achieving exact matching with high probability in dense graphs.
Develops new bounds for deterministic samplers in diffusion models.
problem Analyzing deterministic samplers in diffusion generative models.
method Operational interpretation of deterministic sampling; restoration and degradation steps.
result First polynomial convergence bounds for DDIM-type samplers.
We consider a Markov process X, which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of $Z…
Spaces of polynomials are shown to be Euclidean balls.
problem Understanding the geometry of Lorentzian and real stable polynomials.
method Refined connection between symmetric exclusion process and polynomial geometry.
result Spaces of Lorentzian and real stable polynomials are homeomorphic to closed Euclidean balls.
A new method combines SciML and UQ with physical constraints.
problem Uncertainty quantification in scientific machine learning tasks.
method Physics-constrained polynomial chaos expansion.
result Effective uncertainty quantification and SciML integration.
The paper explores asset price models using signatures of underlying processes, providing methods for calibration and pricing.
problem Developing asset price models that can approximate classical models and learn parameters from various data sources.
method Using linear functions of the signature of a primary underlying process, the paper provides conditions for absence of arbitrage and tractable option pricing formulas.
result The linearity of the model allows for fast and accurate calibrations from time-series and implied volatility data.
This work improves SGD minibatch sampling using determinantal point processes based on orthogonal polynomials.
problem Improving variance reduction in stochastic gradient descent (SGD) for large datasets.
method Orthogonal polynomial-based determinantal point processes for sampling minibatches in SGD.
result DPP minibatches lead to a smaller mean square approximation error than uniform minibatches.
Deep Gaussian Processes with polynomial kernels can collapse rapidly without proper hyperparameter tuning.
problem The collapse of Deep Gaussian Processes with polynomial kernels without careful hyperparameter tuning.
method Analysis using the Berry-Esseen Theorem and observation of prior behavior.
result The prior of a Deep Gaussian Process collapses rapidly towards zero or places negligible mass on low norm functions without proper hyperparameter tuning.
Paper characterizes equilibrium strategies for stochastic control with higher-order moments.
problem Stochastic control problems with higher-order moments.
method Novel characterization of time-consistent control problems, deriving equilibrium conditions via BSDEs.
result Derives sufficient and necessary conditions for an open-loop Nash equilibrium control (ONEC) in a novel way.
Constructs rank-based continuous semimartingales for financial markets.
problem Model financial markets using rank-based diffusions.
method Uses Dirichlet forms and Feller property to construct semimartingales.
result Establishes nonexistence of triple collisions and simplified rank process dynamics.
A dynamical system can be regarded as an information processing apparatus that encodes input streams from the external environment to its state and processes them through state transitions. The information processing capacity (IPC) is an excellent tool that comprehensively evaluates these processed inputs, providing de…
This paper develops an asymptotic expansion technique in momentum space for stochastic filtering. It is shown that Fourier transformation combined with a polynomial-function approximation of the nonlinear terms gives a closed recursive system of ordinary differential equations (ODEs) for the relevant conditional distri…