Influence of Noise on Force Measurements

We demonstrate how the ineluctable presence of thermal noise alters the measurement of forces acting on microscopic and nanoscopic objects. We quantify this effect exemplarily for a Brownian particle near a wall subjected to gravitational and electrostatic forces. Our results demonstrate that the force measurement process is prone to artifacts if the noise is not correctly taken into account.Comment: 4 Pages, 4 Figures, Accepte

On inversions and Doob hh-transforms of linear diffusions

Let $X$ be a regular linear diffusion whose state space is an open interval $E\subseteq\mathbb{R}$. We consider a diffusion $X^*$ which probability law is obtained as a Doob $h$-transform of the law of $X$, where $h$ is a positive harmonic function for the infinitesimal generator of $X$ on $E$. This is the dual of $X$ with respect to $h(x)m(dx)$ where $m(dx)$ is the speed measure of $X$. Examples include the case where $X^*$ is $X$ conditioned to stay above some fixed level. We provide a construction of $X^*$ as a deterministic inversion of $X$, time changed with some random clock. The study involves the construction of some inversions which generalize the Euclidean inversions. Brownian motion with drift and Bessel processes are considered in details.Comment: 19 page

Stochastic Calculus for a Time-changed Semimartingale and the Associated Stochastic Differential Equations

It is shown that under a certain condition on a semimartingale and a time-change, any stochastic integral driven by the time-changed semimartingale is a time-changed stochastic integral driven by the original semimartingale. As a direct consequence, a specialized form of the Ito formula is derived. When a standard Brownian motion is the original semimartingale, classical Ito stochastic differential equations driven by the Brownian motion with drift extend to a larger class of stochastic differential equations involving a time-change with continuous paths. A form of the general solution of linear equations in this new class is established, followed by consideration of some examples analogous to the classical equations. Through these examples, each coefficient of the stochastic differential equations in the new class is given meaning. The new feature is the coexistence of a usual drift term along with a term related to the time-change.Comment: 27 pages; typos correcte

Dry Markets and Superreplication Bounds of American Derivatives

This paper studies the impact of dry markets for underlying assets on the pricing of American derivatives, using a discrete time framework. Dry markets are characterized by the possibility of non-existence of trading at certain dates. Such non-existence may be deterministic or probabilistic. Using superreplicating strategies, we derive expectation representations for the range of arbitrage-free values of the dervatives. In the probabilistic case, if we consider an enlarged filtration induced by the price process and the market existence process, ordinary stopping times are required. If not, randomized stopping times are required. Several comparisons of the ranges obtained with the two market restrictions are performed. Finally, we conclude that arbitrage arguments are not enough to define the optimal exercise policy.N/

On arbitrages arising from honest times

In the context of a general continuous financial market model, we study whether the additional information associated with an honest time gives rise to arbitrage profits. By relying on the theory of progressive enlargement of filtrations, we explicitly show that no kind of arbitrage profit can ever be realised strictly before an honest time, while classical arbitrage opportunities can be realised exactly at an honest time as well as after an honest time. Moreover, stronger arbitrages of the first kind can only be obtained by trading as soon as an honest time occurs. We carefully study the behavior of local martingale deflators and consider no-arbitrage-type conditions weaker than NFLVR.Comment: 25 pages, revised versio

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations

Quantum noise and stochastic reduction

In standard nonrelativistic quantum mechanics the expectation of the energy is a conserved quantity. It is possible to extend the dynamical law associated with the evolution of a quantum state consistently to include a nonlinear stochastic component, while respecting the conservation law. According to the dynamics thus obtained, referred to as the energy-based stochastic Schrodinger equation, an arbitrary initial state collapses spontaneously to one of the energy eigenstates, thus describing the phenomenon of quantum state reduction. In this article, two such models are investigated: one that achieves state reduction in infinite time, and the other in finite time. The properties of the associated energy expectation process and the energy variance process are worked out in detail. By use of a novel application of a nonlinear filtering method, closed-form solutions--algebraic in character and involving no integration--are obtained for both these models. In each case, the solution is expressed in terms of a random variable representing the terminal energy of the system, and an independent noise process. With these solutions at hand it is possible to simulate explicitly the dynamics of the quantum states of complicated physical systems.Comment: 50 page

Scaled penalization of Brownian motion with drift and the Brownian ascent

We study a scaled version of a two-parameter Brownian penalization model introduced by Roynette-Vallois-Yor in arXiv:math/0511102. The original model penalizes Brownian motion with drift $h\in\mathbb{R}$ by the weight process ${\big(\exp(\nu S_t):t\geq 0\big)}$ where $\nu\in\mathbb{R}$ and $\big(S_t:t\geq 0\big)$ is the running maximum of the Brownian motion. It was shown there that the resulting penalized process exhibits three distinct phases corresponding to different regions of the $(\nu,h)$-plane. In this paper, we investigate the effect of penalizing the Brownian motion concurrently with scaling and identify the limit process. This extends a result of Roynette-Yor for the ${\nu<0,~h=0}$ case to the whole parameter plane and reveals two additional "critical" phases occurring at the boundaries between the parameter regions. One of these novel phases is Brownian motion conditioned to end at its maximum, a process we call the Brownian ascent. We then relate the Brownian ascent to some well-known Brownian path fragments and to a random scaling transformation of Brownian motion recently studied by Rosenbaum-Yor.Comment: 32 pages; made additions to Section

Large Deviations Principle for a Large Class of One-Dimensional Markov Processes

We study the large deviations principle for one dimensional, continuous, homogeneous, strong Markov processes that do not necessarily behave locally as a Wiener process. Any strong Markov process $X_{t}$ in $\mathbb{R}$ that is continuous with probability one, under some minimal regularity conditions, is governed by a generalized elliptic operator $D_{v}D_{u}$, where $v$ and $u$ are two strictly increasing functions, $v$ is right continuous and $u$ is continuous. In this paper, we study large deviations principle for Markov processes whose infinitesimal generator is $\epsilon D_{v}D_{u}$ where $0<\epsilon\ll 1$. This result generalizes the classical large deviations results for a large class of one dimensional "classical" stochastic processes. Moreover, we consider reaction-diffusion equations governed by a generalized operator $D_{v}D_{u}$. We apply our results to the problem of wave front propagation for these type of reaction-diffusion equations.Comment: 23 page