Exponential functionals of Brownian motion. II: Some related diffusion processes.
Exponential martingales and CIR model
With the use of exponential martingales and the Girsanov theorem we show how to calculate bond prices in a large variety of square root processes. We clarify and correct several errors that abound in financial literature concerning these processes. The most important topics are linear risk premia, the Longstaff double square model, and calculations concerning correlated CIR processes.
Extinction for two parabolic stochastic PDE's on the lattice
Fast deterministic pricing of options on Lévy driven assets
Arbitrage-free prices of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the -scheme in time and a wavelet Galerkin method with degrees of freedom in log-price space. The dense matrix for can be replaced by a sparse matrix in the wavelet basis, and the linear...
Fast deterministic pricing of options on Lévy driven assets
Arbitrage-free prices u of European contracts on risky assets whose log-returns are modelled by Lévy processes satisfy a parabolic partial integro-differential equation (PIDE) . This PIDE is localized to bounded domains and the error due to this localization is estimated. The localized PIDE is discretized by the θ-scheme in time and a wavelet Galerkin method with N degrees of freedom in log-price space. The dense matrix for can be replaced by a sparse matrix in the wavelet basis, and the...
Fermeture de et de
Forward-backward stochastic differential equations and PDE with gradient dependent second order coefficients
We consider a system of fully coupled forward-backward stochastic differential equations. First we generalize the results of Pardoux-Tang [7] concerning the regularity of the solutions with respect to initial conditions. Then, we prove that in some particular cases this system leads to a probabilistic representation of solutions of a second-order PDE whose second order coefficients depend on the gradient of the solution. We then give some examples in dimension 1 and dimension 2 for which the assumptions...
Functional integro-differential stochastic evolution equations in Hilbert space.
Fundamental solutions to Kolmogorov equations via reduction to canonical form.
Galerkin approximation and the strong solution of the Navier-Stokes equation.
Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach
In this paper, a singular semi-linear parabolic PDE with locally periodic coefficients is homogenized. We substantially weaken previous assumptions on the coefficients. In particular, we prove new ergodic theorems. We show that in such a weak setting on the coefficients, the proper statement of the homogenization property concerns viscosity solutions, though we need a bounded Lipschitz terminal condition.
Homogenization of random walk in asymmetric random environment.
Homogenization of semilinear PDEs with discontinuous averaged coefficients.
Incompleteness of the bond market with Lévy noise under the physical measure
The problem of completeness of the forward rate based bond market model driven by a Lévy process under the physical measure is examined. The incompleteness of market in the case when the Lévy measure has a density function is shown. The required elements of the theory of stochastic integration over the compensated jump measure under a martingale measure are presented and the corresponding integral representation of local martingales is proven.
Intrinsic location parameter of a diffusion process.
Itô-Henstock integral and Itô's formula for the operator-valued stochastic process
In this paper, we introduce the Itô-Henstock integral of an operator-valued stochastic process and formulate a version of Itô's formula.
Large population limit and time behaviour of a stochastic particle model describing an age-structured population
We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound...
Local volatility in the Heston model: a Malliavin calculus approach.
Localization of spectrum bottom for the Stokes operator in a random porous medium.
Log-optimal investment in the long run with proportional transaction costs when using shadow prices
We consider a non-consuming agent interested in the maximization of the long-run growth rate of a wealth process investing either in a money market and in one risky asset following a geometric Brownian motion or in futures following an arithmetic Brownian motion. The agent faces proportional transaction costs, and similarly as in [17] where the case of stock trading is considered, we show how the log-optimal optimal policies in the long run can be derived when using the technical tool of shadow...