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On pathwise uniqueness for stochastic differential equations driven by stable Lévy processes

Nicolas Fournier (2013)

Annales de l'I.H.P. Probabilités et statistiques

We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order α with drift and diffusion coefficients b , σ . When α ∈ ( 1 , 2 ) , we investigate pathwise uniqueness for this equation. When α ∈ ( 0 , 1 ) , we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether α ∈ ( 0 , 1 ) or α ∈ ( 1 , 2 ) and on whether the driving stable process is symmetric or not. Our assumptions...

On Poisson-Dirichlet problems with polynomial data

Henryk Gzyl (2002)

Publicacions Matemàtiques

In this note we provide a probabilistic proof that Poisson and/or Dirichlet problems in an ellipsoid in Rd, that have polynomial data, also have polynomial solutions. Our proofs use basic stochastic calculus. The existing proofs are based on famous lemma by E. Fisher which we do not use, and present a simple martingale proof of it as well.

On the regularity of stochastic currents, fractional brownian motion and applications to a turbulence model

Franco Flandoli, Massimiliano Gubinelli, Francesco Russo (2009)

Annales de l'I.H.P. Probabilités et statistiques

We study the pathwise regularity of the map φ↦I(φ)=∫0T〈φ(Xt), dXt〉, where φ is a vector function on ℝd belonging to some Banach space V, X is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A continuous version of this map, seen as a random element of the topological dual of V will be called stochastic current. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture...

Optimal control of a stochastic heat equation with boundary-noise and boundary-control

Arnaud Debussche, Marco Fuhrman, Gianmario Tessitore (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C1 regularity...

Option pricing in a CEV model with liquidity costs

Krzysztof Turek (2016)

Applicationes Mathematicae

The goal of this paper is to make an attempt to generalise the model of pricing European options with an illiquid underlying asset considered by Rogers and Singh (2010). We assume that an investor's decisions have only a temporary effect on the price, which is proportional to the square of the change of the number of asset units in the investor's portfolio. We also assume that the underlying asset price follows a CEV model. To prove existence and uniqueness of the solution, we use techniques similar...

Pricing forward-start options in the HJM framework; evidence from the Polish market

P. Sztuba, A. Weron (2001)

Applicationes Mathematicae

We show how to use the Gaussian HJM model to price modified forward-start options. Using data from the Polish market we calibrate the model and price this exotic option on the term structure. The specific problems of Central Eastern European emerging markets do not permit the use of the popular lognormal models of forward LIBOR or swap rates. We show how to overcome this difficulty.

Probabilistic models of vortex filaments

Franco Flandoli, Ida Minelli (2001)

Czechoslovak Mathematical Journal

A model of vortex filaments based on stochastic processes is presented. In contrast to previous models based on semimartingales, here processes with fractal properties between 1 / 2 and 1 are used, which include fractional Brownian motion and similar non-Gaussian examples. Stochastic integration for these processes is employed to give a meaning to the kinetic energy.

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