arXiv research

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

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…

Neural Jump ODEs model Itô processes without adversarial training.

Simplified SGD interpretation as Ito process for broader applicability.

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…

Developed a machine-checked Itô calculus for Brownian motion.

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

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

This paper extends Markovian projections to semimartingales with jumps.

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

Develops a new calculus for stochastic processes with occupation flows.

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

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

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

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…

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

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…

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

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 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…

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

This paper solves the inversion problem for jump processes using Markovian projections.

Neural models price financial options without assuming underlying price forms.

Atlas models are systems of Ito processes with parameters that depend on rank. We show that the parameters of a simple Atlas model can be identified by measuring the variance of the top-ranked process for different sampling intervals.

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

Formula for option pricing in a stochastic volatility model with jumps.

G-framework is presented by Peng [41] for measure risk under uncertainty. In this paper, we define fractional G-Brownian motion (fGBm). Fractional G-Brownian motion is a centered G-Gaussian process with zero mean and stationary increments in the sense of sub-linearity with Hurst index $H\in (0,1)$. This process has sta…

We present an overview of the broad class of financial models in which the prices of assets are Lévy-Ito processes driven by an $n$-dimensional Brownian motion and an independent Poisson random measure. The Poisson random measure is associated with an $n$-dimensional Lévy process. Each model consists of a pricing kerne…

New method infers and samples point processes from latent diffusion.

An explicit martingale representation for random variables described as a functional of a Levy process will be given. The Clark-Ocone theorem shows that integrands appeared in a martingale representation are given by conditional expectations of Malliavin derivatives. Our goal is to extend it to random variables which a…

A geometric reformulation of the martingale problem associated with a set of diffusion processes is proposed. This formulation, based on second order geometry and Ito integration on manifolds, allows us to give a natural and effective definition of Lie symmetries for diffusion processes.

These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distri…

The paper examines the consistency of Lasso regression applied to signature analysis of time series data.

While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessaril…

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

Proposes overnight volatility model for better market dynamics.

Develops pathwise analysis for log-optimal portfolios using rough paths theory.

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

Study uses viscosity solutions to solve control problems involving measure-valued martingales.

The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, that includes the Ito and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is…

The usual derivation of the Fokker-Planck partial differential eqn. assumes the Chapman-Kolmogorov equation for a Markov process. Starting instead with an Ito stochastic differential equation we argue that finitely many states of memory are allowed in Kolmogorov's two pdes, K1 (the backward time pde) and K2 (the Fokker…

We derive asymptotic expansions for option data to detect infinite variation volatility.

This paper provides an existence-and-uniqueness theorem characterizing the stochastic integral with respect to a Wiener process. The integral is represented as a mapping from the space of measurable and adapted pathwise locally integrable processes to the space of continuous adapted processes. It is characterized in te…

In this paper we present a slight modification of the Fourier estimation method of the spot volatility (matrix) process of a continuous Itô semimartingale where the estimators are always non-negative definite. Since the estimators are factorized, computational cost will be saved a lot.

We consider a standard optimal investment problem in a complete financial market driven by a Wiener process and derive an explicit formula for the optimal portfolio process in terms of the vertical derivative from functional It^o calculus. An advantage with this approach compared to the Malliavin calculus approach is t…

New method estimates volatility for processes with jumps of unbounded variation.

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