Related papers: Conic Martingales from Stochastic Integrals

We propose a new weak convergence theorem for martingales, under gentler conditions than the usual convergence in probability of the sequence of associated quadratic variations. Its proof requires the combined use of Skorohod's…

This paper contributes to the study of class $(\Sigma^{r})$ as well as the c\`adl\`ag semi-martingales of class $(\Sigma)$, whose finite variational part is c\`adl\`ag instead of continuous. The two above-mentioned classes of stochastic…

Let $\mathfrak{z}$ be a stochastic exponential, i.e., $\mathfrak{z}_t=1+\int_0^t\mathfrak{z}_{s-}dM_s$, of a local martingale $M$ with jumps $\triangle M_t>-1$. Then $\mathfrak{z}$ is a nonnegative local martingale with $\E\mathfrak{z}_t\le…

We study a notion of local time for a continuous path, defined as a limit of suitable discrete quantities along a general sequence of partitions of the time interval. Our approach subsumes other existing definitions and agrees with the…

Recursive stochastic algorithms have gained significant attention in the recent past due to data driven applications. Examples include stochastic gradient descent for solving large-scale optimization problems and empirical dynamic…

The concept of stochastic Lagrangian and its use in statistical dynamics is illustrated theoretically, and with some examples. Dynamical variables undergoing stochastic differential equations are stochastic processes themselves, and their…

We study the persistence in a class of continuous stochastic processes that are stationary only under integer shifts of time. We show that under certain conditions, the persistence of such a continuous process reduces to the persistence of…

This papers addresses the stock option pricing problem in a continuous time market model where there are two stochastic tradable assets, and one of them is selected as a num\'eraire. It is shown that the presence of arbitrarily small…

Discrete multiplicative turbulent cascades are described using a formalism involving infinitely divisible random measures. This permits to consider the continuous limit of a cascade developed on a continuum of scales, and to provide the…

This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due to external influence. We focus on the case…

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends…

We study the martingale property and moment explosions of a signature volatility model, where the volatility process of the log-price is given by a linear form of the signature of a time-extended Brownian motion. Excluding trivial cases, we…

The stationary state of a stochastic process on a ring can be expressed using traces of monomials of an associative algebra defined by quadratic relations. If one considers only exclusion processes one can restrict the type of algebras and…

This work is about a new class of martingales: the vertical martingales. We construct the vertical martingale for smooth submersions and we develop a stochastic calculus for one. Furthermore, we gives a stochastic characterization for…

Over the last 50 years a steady stream of accounts have been written on the separation principle of stochastic control. Even in the context of the linear-quadratic regulator in continuous time with Gaussian white noise, subtle difficulties…

This paper explores a mathematical technique for deriving dynamical invariants (i.e. constants of motion) in time-dependent gravitational potentials. The method relies on the construction of a canonical transformation that removes the…

Suppose that $d\geq1$ and $\alpha\in (1, 2)$. Let $Y$ be a rotationally symmetric $\alpha$-stable process on $\R^d$ and $b$ a $\R^d$-valued measurable function on $\R^d$ belonging to a certain Kato class of $Y$. We show that $\rd X^b_t=\rd…

Self-normalized processes arise naturally in statistical applications. Being unit free, they are not affected by scale changes. Moreover, self-normalization often eliminates or weakens moment assumptions. In this paper we present several…

The main goal of this paper is to understand the formation of hexagonal patterns from the dynamical transition theory point of view. We consider the transitions from a steady state of an abstract nonlinear dissipative system. To shed light…

In the present paper, we study the chaotic representation property for certain families of square integrable martingales. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic…