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.
We study (backward) stochastic differential equations with noise coming from a finite state Markov chain. We show that, for the solutions of these equations to be `Markovian', in the sense that they are deterministic functions of the state of the underlying chain, the integrand must be of a specific form. This allows u…
New model for insurance states using Markov jump processes with non-countable state space.
problem Modeling insurance states with non-countable state spaces.
method Developed a new Thiele's differential equation for continuous time rehabilitation rates.
result Allows for consistent calculation of reserves in disability insurance.
Investor selects portfolios based on news attention in a hidden Markov model.
problem Mean-variance portfolio selection in a dynamic attention context.
method Closed-loop equilibrium strategies via extended HJB equation and Markov chain approximation.
result Equilibrium strategies found through iterative algorithm and numerical examples.
New method infers hidden states in continuous-time phenomena better than traditional models.
problem Traditional HSMM's are limited to discrete time grids and cannot handle irregularly spaced data.
method Formulated integro-differential forward and backward equations for CTSMC's, introduced scalable Viterbi-type algorithm.
result Efficiently solved equations for posterior marginals and path estimates.
Solves optimal stopping for Gauss-Markov bridges using time-space transformation.
problem Optimal stopping problem of a Gauss-Markov bridge.
method Time-space transformation approach, Picard iteration algorithm.
result Lipschitz continuity of the optimal stopping boundary and its characterization.
Unified proof of Aigner's conjectures using geodesics.
problem Proving conjectures related to Markov numbers.
method Using geodesics on the punctured torus.
result Unified proof of Aigner's conjectures.
New algebraic numbers defined by a specific equation.
problem No direct problem stated; focuses on new algebraic numbers.
method Definition of superalgebraic Markov numbers via a Grassmann integer equation.
result Introduced new algebraic numbers with applications in Teichmüller spaces.
We review some developments concerning Markov and Feller processes with jumps in geometric settings. These include stochastic differential equations in Markus canonical form, the Courrège theorem on Lie groups, and invariant Markov processes on manifolds under both transitive and more general Lie group actions.
We develop algorithms for the numerical computation of the quadratic hedging strategy in incomplete markets modeled by pure jump Markov process. Using the Hamilton-Jacobi-Bellman approach, the value function of the quadratic hedging problem can be related to a triangular system of parabolic partial integro-differential…
We show by explicit closed form calculations that a Hurst exponent H that is not 1/2 does not necessarily imply long time correlations like those found in fractional Brownian motion. We construct a large set of scaling solutions of Fokker-Planck partial differential equations where H is not 1/2. Thus Markov processes, …
New approach reveals causal and probabilistic relationships from equations.
problem Understanding causal and probabilistic relationships from sets of equations.
method Simon's causal ordering algorithm and Markov ordering graph construction.
result Implied conditional independences and causal relations without solving equations.
Recent studies have suggested that the cognitive process of the human brain is realized as probabilistic inference and can be further modeled by probabilistic graphical models like Markov random fields. Nevertheless, it remains unclear how probabilistic inference can be implemented by a network of spiking neurons in th…
Model stock price dynamics using semi-Markov processes.
problem Model stock price dynamics through a semi-Markov process.
method Use semi-Markov process with Poisson random measure, establish existence and uniqueness of solution, derive HJB equation.
result Obtain expressions for optimal controls and value function using HJB equation.
In this paper we propose a semi-Markov modulated model of interest rates. We assume that the switching process is a semi-Markov process with finite state space E and the modulated process is a diffusive process. We derive recursive equations for the higher order moments of the discount factor and we describe a Monte Ca…
We analyze a new Markov chain model for better sampling and optimization.
problem Developing a new Markov chain model for improved sampling and optimization.
method We introduce a new class of Ito chains with arbitrary noise and inexact drift/diffusion coefficients, proving a bound in W2-distance. result Our analysis provides improved or first results for various applications like SGLD, sampling, and boosting.
We prove that the variance swap rate (fair strike) equals the price of a co-terminal European-style contract when the underlying is an exponential Markov process, time-changed by an arbitrary continuous stochastic clock, which has arbitrary correlation with the driving Markov process, provided that the payoff function …
Sum of Lagrange numbers equals a specific formula.
problem Proving the Markov Uniqueness Conjecture (MUC).
method Combining McShane's identity and Schmutz's work.
result MUC is equivalent to the given sum formula.
Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
Study approximates financial market with discrete-time models.
problem Approximating continuous-time financial market models with discrete-time.
method Constructs discrete-time market models with Markov switching and proves convergence.
result Discrete-time models converge to continuous-time Black-Scholes model with Markov switching.
We study the high frequency price dynamics of traded stocks by a model of returns using a semi-Markov approach. More precisely we assume that the intraday return are described by a discrete time homogeneous semi-Markov process and the overnight returns are modeled by a Markov chain. Based on this assumptions we derived…
Optimizes control of hybrid systems with multiple switching processes.
problem Optimal control of hybrid systems with multiple Markov switching processes.
method Combines two separate Markov chains into one synthetic chain, derives HJB equations, and solves the portfolio choice problem.
result Derives explicit solutions and value functions for the optimal control problem.
Uniform TD(0) bound derived for function approximation with Markov noise.
problem Uniform concentration bound for TD(0) with function approximation.
method Contractive stochastic approximation, martingale and Markov noises, Poisson equation, relaxed concentration inequalities.
result Uniform all-time concentration bound for TD(0) with linear function approximation.
Fenrir uses probabilistic numerics to simplify solving initial value problems.
problem Solving initial value problems in ordinary differential equations.
method Probabilistic numerics and Gauss--Markov regression.
result The method simplifies parameter estimation in ODEs, making it easier and more robust.
Optimizes consumption under regime-switching economic states with risk-sensitive preferences.
problem Optimizing consumption in an economy with uncertain states and random shocks.
method Risk-sensitive optimization of consumption-utility with a Markov chain model of economic states and i.i.d. random shocks.
result Existence of unique optimal policy and value function in stationary policies.
We present a stochastic analysis of a data set consisiting of 10^6 quotes of the US Doller - German Mark exchange rate. Evidence is given that the price changes x(tau) upon different delay times tau can be described as a Markov process evolving in tau. Thus, the tau-dependence of the probability density function (pdf) …
New methods reduce computational cost for Gaussian Markov Random Fields with sparse constraints.
problem Inference and simulation of GMRFs are computationally prohibitive with many constraints.
method Proposes a basis transformation into blocks of constrained and non-constrained subspaces.
result Significantly outperforms existing alternatives in computational cost.
The article presents a general discrete time dividend valuation model when the dividend growth rate is a general continuous variable. The main assumption is that the dividend growth rate follows a discrete time semi-Markov chain with measurable space. The paper furnishes sufficient conditions that assure finiteness of …
Deriving option prices from operational-time Markov lattices
problem Option pricing
method Operational-time Markov lattice
result Derives option-pricing equations from an operational-time Markov lattice
In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on R+ or R with values in a general Riemannian maifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper …
New neural method for inferring Markov jump processes.
problem Inference in Markov jump processes is challenging.
method Variational inference using neural ODEs and backpropagation.
result Trains neural representations of data to approximate process rates.
Proposes a new method combining Reservoir Computing and Normalizing Flow for predicting stochastic dynamical systems.
problem Predicting and capturing long-term behaviors of stochastic dynamical systems.
method Data-driven framework combining Reservoir Computing and Normalizing Flow, integrating error modeling and both approaches virtues.
result Successfully predicts the long-term evolution of stochastic dynamical systems and replicates dynamical behaviors.
Deep nets solve MDPs without high dimensions.
problem Solving Bellman equations for MDPs in high dimensions.
method Deep neural networks with ReLU activation approximating payoff and transition functions.
result Deep nets can approximate Q-functions in polynomially bounded parameters. We investigate probabilistic graphical models that allow for both cycles and latent variables. For this we introduce directed graphs with hyperedges (HEDGes), generalizing and combining both marginalized directed acyclic graphs (mDAGs) that can model latent (dependent) variables, and directed mixed graphs (DMGs) that c…
The paper studies risk-sensitive MDPs with recursive risk measures.
problem Risk-sensitive decision-making in MDPs with unbounded costs.
method Recursive application of static risk measures, Bellman equation derivation, existence of optimal policies.
result Existence of Markovian optimal policies for infinite planning horizons, contractive model for stationary optimal policy.
This project attempts to address the problem of asset pricing in a financial market, where the interest rates and volatilities exhibit regime switching. This is an extension of the Black-Scholes model. Studies of Markov-modulated regime switching models have been well-documented. This project extends that notion to a c…
Deep learning estimates time-varying Markov model parameters.
problem Estimating time-dependent parameters in Markov models.
method Reframes parameter estimation as an optimization problem using maximum likelihood.
result Real solution close to SDE with neural network-derived parameters under specific conditions.
This research formalizes uncertainty quantification for Universal Differential Equations models.
problem Quantifying uncertainties in Universal Differential Equations models.
method Formalized uncertainty quantification methods for UDEs, including frequentist and Bayesian approaches.
result Evaluation of ensemble, variational inference, and MCMC sampling methods for UDEs.
The study extends GBM to include stable nonzero prices and finds a pronounced potential well.
problem The standard GBM model cannot describe stable nonzero prices in financial dynamics.
method Generalized GBM with polynomial drift of order q, model selection, and Markov chain Monte Carlo ensembles of potential functions.
result The optimal model for financial data is q=2, indicating the existence of a stable price.
DCDC calculates convergence rates for Markov chains using neural networks.
problem Computing precise convergence rates for Markov chains is hard.
method Developed a neural network-based algorithm (DCDC) to bound convergence rates in Wasserstein distance.
result Demonstrated effective convergence bounds for real-world Markov chains.
We establish causal semantics for SDEs and develop methods to reason about them.
problem Understanding causal relationships in systems modeled by stochastic differential equations.
method We introduce a causal graph framework, Markov properties, and do-calculus for SDEs.
result We prove the σ-separation Markov property and do-calculus for causal SDEs. We consider an investor faced with the utility maximization problem in which the risky asset price process has pure-jump dynamics affected by an unobservable continuous-time finite-state Markov chain, the intensity of which can also be controlled by actions of the investor. Using the classical filtering theory, we redu…
Develops an actor-critic algorithm for risk-sensitive Markov decision processes.
problem Risk-sensitive cost criterion in Markov decision processes.
method Actor-critic algorithm with function approximation.
result Asymptotic convergence of the actor-critic algorithm.
The paper extends game theory using Hodge theory on graphs.
problem Generalizing Shapley's value allocation formula for cooperative games on graphs.
method Connecting stochastic path integrals to Hodge-theoretic Poisson's equations on graphs.
result The value allocation operator is the solution to Poisson's equation in combinatorial Hodge theory.
The generic identification problem is to decide whether a stochastic process (Xt) is a hidden Markov process and if yes to infer its parameters for all but a subset of parametrizations that form a lower-dimensional subvariety in parameter space. Partial answers so far available depend on extra assumptions on the pro…
We study the high frequency price dynamics of traded stocks by a model of returns using a semi-Markov approach. More precisely we assume that the intraday returns are described by a discrete time homogeneous semi-Markov which depends also on a memory index. The index is introduced to take into account periods of high a…
Revisits life insurance surplus models with new technical bases.
problem Classifying and extending life insurance surplus models.
method Using Markov models and classifying technical bases in Thiele's equation.
result Introduces a `canonical' model with three technical bases.
Paper calculates KL divergence for isotropic Gaussian-Markov fields.
problem Measuring divergence between isotropic Gaussian-Markov fields.
method Derives closed-form KL divergence expressions.
result Develops new similarity measures in image processing.