arXiv research

We analyze a new Markov chain model for better sampling and optimization.

Counterexample shows Ito integrand needn't be locally square integrable.

Stein's method for measuring convergence to a continuous target distribution relies on an operator characterizing the target and Stein factor bounds on the solutions of an associated differential equation. While such operators and bounds are readily available for a diversity of univariate targets, few multivariate targ…

Projects Markovian processes from Itô semimartingales with jumps.

Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes whe…

Itô maps provide a method for any-step SDE integration.

Developed a machine-checked Itô calculus for Brownian motion.

This paper extends Markovian projections to semimartingales with jumps.

A machine-checked Itô calculus for Brownian motion on $[0,T]$

A new approach to continuous-time universal portfolios using pathwise Itô calculus.

Extends Itô's formula for path-dependent functions in finance.

Neural Jump ODEs model Itô processes without adversarial training.

The paper introduces a new volatility model for state heterogeneous financial markets using high-frequency data.

In a 2006 article (\cite{A1}), Allouba gave his quadratic covariation differentiation theory for Itô's integral calculus. He defined the derivative of a semimartingale with respect to a Brownian motion as the time derivative of their quadratic covariation and a generalization thereof. He then obtained a systematic diff…

Study on stochastic mean curvature flow on networks using Ito calculus.

Study $N$-player and mean-field games in Itô-diffusion markets with competitive or homophilous interactions.

The paper develops a method for stochastic differential equations on manifolds using Schwartz morphisms and diffusion generators.

We define two new notions of projection of a stochastic differential equation (SDE) onto a submanifold: the Ito-vector and Ito-jet projections. This allows one to systematically develop low dimensional approximations to high dimensional SDEs using differential geometric techniques. The approach generalizes the notion o…

Framework learns surrogates for molecular dynamics across multiple time-scales.

Study proves existence and uniqueness for differential equations with non-Lipschitz coefficients.

Derives functional Itô formula for non-anticipative maps of rough paths.

Itô processes are the most common form of continuous semimartingales, and include diffusion processes. This paper is concerned with the nonparametric regression relationship between two such Itô processes. We are interested in the quadratic variation (integrated volatility) of the residual in this regression, over a un…

This paper gives several simple constructions of the pathwise Ito integral $\int_0^tφdω$ for an integrand $φ$ and a price path $ω$ as integrator, with $φ$ and $ω$ satisfying various topological and analytical conditions. The definitions are purely pathwise in that neither $φ$ nor $ω$ are assumed to be paths of stochast…

Options financial instruments designed to protect investors from the stock market randomness. In 1973, Fisher Black, Myron Scholes and Robert Merton proposed a very popular option pricing method using stochastic differential equations within the Ito interpretation. Herein, we derive the Black-Scholes equation for the o…

We use pathwise Itô calculus to prove two strictly pathwise versions of the master formula in Fernholz' stochastic portfolio theory. Our first version is set within the framework of Föllmer's pathwise Itô calculus and works for portfolios generated from functions that may depend on the current states of the market port…

We study a portfolio selection problem in a continuous-time Itô-Markov additive market with prices of financial assets described by Markov additive processes which combine Lévy processes and regime switching models. Thus the model takes into account two sources of risk: the jump diffusion risk and the regime switching …

This paper analyzes sampling from heavy-tailed distributions using discretized Itô diffusions.

Dupire's functional Itô calculus provides an alternative approach to the classical Malliavin calculus for the computation of sensitivities, also called Greeks, of path-dependent derivatives prices. In this paper, we introduce a measure of path-dependence of functionals within the functional Itô calculus framework. Name…

We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuatio…

Low-frequency historical data, high-frequency historical data and option data are three major sources, which can be used to forecast the underlying security's volatility. In this paper, we propose two econometric models, which integrate three information sources. In GARCH-Itô-OI model, we assume that the option-implied…

Develops a new calculus for stochastic processes with occupation flows.

Derives FPDE for equity-linked insurance pricing.

Derives formula for skew stickiness ratio in asset price and volatility dynamics.

Ito-Takimura recently defined a splice-unknotting number $u^-(D)$ for knot diagrams. They proved that this number provides an upper bound for the crosscap number of any prime knot, asking whether equality holds in the alternating case. We answer their question in the affirmative. (Ito has independently proven the same …

The paper analyzes performance criteria for competing fund managers in Ito-diffusion markets.

We explain how Itô Stochastic Differential Equations (SDEs) on manifolds may be defined using 2-jets of smooth functions. We show how this relationship can be interpreted in terms of a convergent numerical scheme. We show how jets can be used to derive graphical representations of Itô SDEs. We show how jets can be used…

Neural networks can approximate complex stochastic equations well.

NANSDE-Net models time series with memory using neural ARMA-type noise.

Study local expansions of continuous-time processes using Ito signature properties.

Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that th…

Paper introduces a new outer measure for continuous price paths with instant enforcement.

The article constructs stochastic integration in Riemannian manifolds.

We consider idealized financial markets in which price paths of the traded securities are cadlag functions, imposing mild restrictions on the allowed size of jumps. We prove the existence of quadratic variation for typical price paths, where the qualification "typical" means that there is a trading strategy that risks …

Proposes overnight volatility model for better market dynamics.

The paper clarifies the approximation of SGD with Ito SDEs for finite learning rates.

The paper proves signatures of non-geometric rough paths can approximate functionals uniformly.

In deep latent Gaussian models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. This work considers the diffusion limit of suc…

Novel approach to financial derivatives pricing using rough path theory.