On the exponential integrability of the derivative of intersection and self-intersection local time for Brownian motion and related processes

Renormalization and convergence in law for the derivative of intersection local time in [formula omitted

Let (Xt)t>=0 denote the measure-valued critical branching Brownian motion on Rd with initial state being Lebesgue measure. A strong ergodic theorem is proved for (Xt)t>=0 when d>=3, while a weak ergodic theorem is proved for d = 2. Also a weak local occupation time (an analogue of the local time for Brownian motion) is shown to exist in dimensions d=1,2 and 3.

Given that each term of the multiple Wiener integral expansion for the renormalized self-intersection local time of higher dimensional Brownian motion converges in law to another, independent Brownian motion we resum the leading, martingale parts of these terms in closed form and also represent this sum as a stochastic integral.

We discuss the weak compactness problem related to the selfintersection local time of Brownian motion. We also propose a regular renormalization for self-intersection local time of higher dimensional Brownian motion.

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area)(T) covered by the process in the time interval [0,T]. The Laplace transform of(T) follows as a consequence. The main proof involves a double Laplace transform of(T) and is based on excursion theory and local time for Brownian motion.

Rosen (Seminaire de Probabilites XXXVIII, 2005) proved the existence of a process known as the derivative of the intersection local time of Brownian motion in one dimension. The purpose of this paper is to use the methods developed in Nualart and Vives (Publicacions Matematiques 36(2):827–836, 1992) in order to give a simple new proof of the existence of this process. Some related theorems and conjectures are discussed.

We consider the intersection local time of Brownian motion without renormalization through Itô-Wiener expansions. In order to recognize the existence, we extend the Watanabe space. We also discuss how to substitute Wiener functionals for parameters of a generalized Wiener functional. As a consequence a relationship between the unrenormalized intersection local time and the local time is clarified.

Chapter 10 - Brownian Motion and Stationary Processes

In his 1939 paper [1], P. Lévy introduced the notion of local time for Brownian motion as the limit of the occupation time of the space interval (0, є) blown up by a factor 1/є: $${L_\varepsilon }\left( t \right) = m\left\{ {s \in \left( {0,t} \right]|0 < B\left( s \right) < \varepsilon } \right\}/\varepsilon \to L\left( t \right)for\varepsilon \to 0 + $$ (1).

Sufficient conditions are given for a family of local times |L t µ | ofd-dimensional Brownian motion to be jointly continuous as a function oft and μ. Then invariance principles are given for the weak convergence of local times of lattice valued random walks to the local times of Brownian motion, uniformly over a large family of measures. Applications included some new results for intersection local times for Brownian motions on ℝ2 and ℝ2.

Stochastic differential equation for Brox diffusion

During this article we will discuss the problem behaviour of the value of the underlying asset, highlighting a central issue, what is the impact of different models that describe the behaviour of the value of underlying assets on the value of the oil project? The literature on real options is based on an assumption that the value of the underlying variable, cost of production of hydrocarbons or other relevant variables stochastic, following a process of geometric Brownian motion, GBM (Paddock, Siegel and Smith, 1988). In fact, most empirical research: Consider the initial investment decision but neglect the flexibility of operating the project; Suppose that the price of oil is stochastic, but unaware of the uncertainty about the costs of production and reserves of hydrocarbons; Suppose that oil prices follows a process of Brownian motion. The evolution of three variables, prices, production costs and quantities of reserves of hydrocarbons are presented in this article. The value of managerial flexibility is appreciated in this article using data on planned exploration and production of hydrocarbons by a TUNISIAN oil firm.

McGill showed that the intrinsic local time process $$\tilde L$$ (t, x), t ≧ 0, x ∈ ℝ, of one-dimensional Brownian motion is, for fixedt>0, a supermartingale in the space variable, and derived an expression for its Doob-Meyer decomposition. This expression referred to the derivative of some process which was not obviously differentiable. In this paper, we provide an independent proof of the result, by analysing the local time of Brownian motion on a family of decreasing curves. The ideas involved are best understood in terms of stochastic area integrals with respect to the Brownian local time sheet, and we develop this approach in a companion paper. However, the result mentioned above admits a direct proof, which we give here; one is inevitably drawn to look at the local time process of a Dirichlet process which is not a semimartingale.

Chapter 7 - One-Dimensional Diffusions and Their Convergence in Distribution