Related papers: It\^o Stochastic differentials
We define an abstract framework called {\it discrete finite differences embedding} which can be used to obtain discrete analogue of formal functional relations in the spirit of category theory. For ordinary differential equations we exhibit…
This article studies the solutions of time-dependent differential inclusions which is motivated by their utility in the modeling of certain physical systems. The differential inclusion is described by a time-dependent set-valued mapping…
We introduce a theory of stochastic integration with respect to a family of semimartingales depending on a continuous parameter, as a mathematical background to the theory of bond markets. We apply our results to the problem of…
We study the use of Temporal-Difference learning for estimating the structural parameters in dynamic discrete choice models. Our algorithms are based on the conditional choice probability approach but use functional approximations to…
An exact invariant is derived for $n$-degree-of-freedom Hamiltonian systems with general time-dependent potentials. The invariant is worked out in two equivalent ways. In the first approach, we define a special {\it Ansatz\/} for the…
By introducing a color filtration to the multiplicity space, we extend the quantum Ito calculus on multiple symmetric Fock space to the framework of filtered adapted biprocesses. In this new notion of adaptedness,``classical'' time…
Motivated by studies of indirect measurements in quantum mechanics, we investigate stochastic differential equations with a fixed point subject to an additional infinitesimal repulsive perturbation. We conjecture, and prove for an important…
The convergence of stochastic integrals is essential to stochastic analysis, especially in applications to mathematical finance, where they model the gains associated with a self-financing strategy. However, Fatou convergence of…
In this paper, we establish an analytic framework for studying set-valued backward stochastic differential equations (set-valued BSDE), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or…
Adding the time as a component of a stochastic process before computing its signature terminal value ensures injectivity and supports universal approximation results, but it induces linear dependence among the components of the signature…
We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this derivative is invariant under changes of chart and is thus well-defined for functions defined on a…
Stochastic volatility models describe asset prices $S_t$ as driven by an unobserved process capturing the random dynamics of volatility $\sigma_t$. Here, we quantify how much information about $\sigma_t$ can be inferred from asset prices…
This work develops new results for stochastic approximation algorithms. The emphases are on treating algorithms and limits with discontinuities. The main ingredients include the use of differential inclusions, set-valued analysis, and…
We present a Bayesian non-parametric way of inferring stochastic differential equations for both regression tasks and continuous-time dynamical modelling. The work has high emphasis on the stochastic part of the differential equation, also…
In this note we prove that the local martingale part of a convex function f of a d-dimensional semimartingale X = M + A can be written in terms of an It^o stochastic integral \int H(X)dM, where H(x) is some particular measurable choice of…
The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure.…
We develop a stochastic calculus that makes it easy to capture a variety of predictable transformations of semimartingales such as changes of variables, stochastic integrals, and their compositions. The framework offers a unified treatment…
Based on an extension of the martingale comparison method some comparison results for path-dependent functions of semimartingales are established. The proof makes essential use of the functional It\^o calculus. A main tool is an extension…
Stochastic thermodynamics is formulated for variables that are odd under time reversal. The invariance under spatial rotation of the collision rates due to the isotropy of the heat bath is shown to be a crucial ingredient. An alternative…
The main objective consists in generalizing a well-known It{\^o} formula of J. Jacod and A. Shiryaev: given a c{\`a}dl{\`a}g process S, there is an equivalence between the fact that S is a semimartingale with given characteristics (B^k , C,…