arXiv research

The study extends stochastic completeness to landmark spaces with any number of landmarks.

Using a deep criteria due to Pigola, Rigoli and Setti, we prove that a geodesically complete, properly immersed submanifold M of a stochastically complete Riemannian manifold N is stochastically complete. This implies that the weak Omori-Yau maximum principle holds on M. As geometric application, we prove sectional cur…

The paper establishes maximum principles and stochastic completeness for pseudo-Hermitian manifolds.

Based on ideas of Pigolla and Setti \cite{PS} we prove that immersed submanifolds with bounded mean curvature of Cartan-Hadamard manifolds are Feller. We also consider Riemannian submersions $π\colon M \to N$ with compact minimal fibers, and based on various criteria for parabolicity and stochastic completeness, see \c…

Study on Brownian motion on discrete curve spaces, proving stochastic completeness.

The paper characterizes stochastic completeness on Riemannian manifolds using nonlocal conditions.

Paper shows $L^\infty$-positivity and stochastic completeness are equivalent.

We present a criterion for the stochastic completeness of a submanifold in terms of its distance to a hypersurface in the ambient space. This relies in a suitable version of the Hessian comparison theorem. In the sequel we apply a comparison principle with geometric barriers for establishing mean curvature estimates fo…

Characterizes geodesic completeness for landmark spaces.

It has been suggested in 1999 that a certain volume growth condition for geodesically complete Riemannian manifolds might imply that the manifold is stochastically complete. This is motivated by a large class of examples and by a known analogous criterion for recurrence of Brownian motion. We show that the suggested im…

In this thesis, we analyze the stochastic completeness of a heat kernel on graphs which is a function of three variables: a pair of vertices and a continuous time, for infinite, locally finite, connected graphs. For general graphs, a sufficient condition for stochastic completeness is given in terms of the maximum vale…

The paper explores the $L^1$-Liouville property on graphs and its connections to stochastic completeness.

Stochastic regularization of neural networks (e.g. dropout) is a wide-spread technique in deep learning that allows for better generalization. Despite its success, continuous-time models, such as neural ordinary differential equation (ODE), usually rely on a completely deterministic feed-forward operation. This work pr…

Study shows submanifolds can't be immersed in certain spaces.

The paper optimizes financial derivatives for market completion in SV models.

Based on a criterion of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions …

Develops multifactor approximations for SVEs with completely monotone kernels.

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb{R}^+$ or $\mathbb{R}$ 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 rigidity results for specific hypersurfaces in spacetimes.

Signature volatility models are analyzed for existence, arbitrage, completeness, and hedging-error decomposition.

Study of stochastic differential equations on non-compact manifolds, solving open problem on strong completeness.

Let $S^F$ be a $\mathbb{P}$-martingale representing the price of a primitive asset in an incomplete market framework. We present easily verifiable conditions on model coefficients which guarantee the completeness of the market in which in addition to the primitive asset one may also trade a derivative contract $S^B$. B…

We show that asymptotically, completely asynchronous stochastic gradient procedures achieve optimal (even to constant factors) convergence rates for the solution of convex optimization problems under nearly the same conditions required for asymptotic optimality of standard stochastic gradient procedures. Roughly, the n…

A method to complete incomplete correlation matrices using maximum entropy.

This work studies nonnegativity-preserving kernels for stochastic equations and their applications.

Stochastic Schwarz lemma on Kähler manifolds via couplings.

In this article, we prove gradient estimates under Bakry-Emery curvature bounds for unbounded graph Laplacians which satisfy an ellipticity assumption. As applications, we study completeness and finiteness of stochastically complete graphs under Bakry-Emery curvature bounds.

Study on radial solutions of Lane-Emden system on Cartan-Hadamard manifolds.

A classical result by Alexander Grigor'yan states that on a stochastically complete manifold the non-negative superharmonic $L^1$-functions are necessarily constant. In this paper we address the question of whether and to what extent the reverse implication holds.

Mathematical models for financial asset prices which include, for example, stochastic volatility or jumps are incomplete in that derivative securities are generally not replicable by trading in the underlying. In earlier work (2004) the first author provided a geometric condition under which trading in the underlying a…

Deep neural networks solve stochastic control problems with delay.

The paper proves new comparison theorems for sub-Laplacian in foliations with minimal leaves.

The study proves theorems about minimal and H-surfaces in 3D space.

Study proves no minimal surfaces can be contained in certain half-spaces or cones.

Many recent Markov chain Monte Carlo (MCMC) samplers leverage continuous dynamics to define a transition kernel that efficiently explores a target distribution. In tandem, a focus has been on devising scalable variants that subsample the data and use stochastic gradients in place of full-data gradients in the dynamic s…

Matrix SMD converges to unique solution minimizing Bregman divergence.

The asymptotic behavior of the heat kernel of a Riemannian manifold gives rise to the classical concepts of parabolicity, stochastic completeness (or conservative property) and Feller property (or $C^{0}$-diffusion property). Both parabolicity and stochastic completeness have been the subject of a systematic study whic…

Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.

Method learns SDEs from one trajectory using GP priors and randomized cross-validation.

New findings show pure strategy equilibria are more robust in a war of attrition game.

An algorithm is proposed for solving stochastic and finite sum minimization problems. Based on a trust region methodology, the algorithm employs normalized steps, at least as long as the norms of the stochastic gradient estimates are within a specified interval. The complete algorithm---which dynamically chooses whethe…

In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and $L^p$-Liouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scal…

New algorithm reduces policy regret in tallying bandits.

We study the exponential Ornstein-Uhlenbeck stochastic volatility model and observe that the model shows a multiscale behavior in the volatility autocorrelation. It also exhibits a leverage correlation and a probability profile for the stationary volatility which are consistent with market observations. All these featu…

The paper analyzes insurance risks using stochastic models.

In this paper, we focus on approaches to parallelizing stochastic gradient descent (SGD) wherein data is farmed out to a set of workers, the results of which, after a number of updates, are then combined at a central master node. Although such synchronized SGD approaches parallelize well in idealized computing environm…

We apply stochastic Perron's method to a singular control problem where an individual targets at a given consumption rate, invests in a risky financial market in which trading is subject to proportional transaction costs, and seeks to minimize her probability of lifetime ruin. Without relying on the dynamic programming…

I discuss some theoretical results with a view to motivate some practical choices in portfolio optimization. Even though the setting is not completely general (for example, the covariance matrix is assumed to be non-singular), I attempt to highlight the features that have practical relevance. The mathematical setting i…