arXiv subjects

Antoine Tambue

Publications and source records attributed to Antoine Tambue.

40 records · Page 3Linked to original sources

arXiv2012

A Stochastic Delay Model For Pricing Debt And Loan Guarantees: Theoretical Results

We consider that the price of a firm follows a non linear stochastic delay differential equation. We also assume that any claim value whose value depends on firm value and time follows a non linear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a Random Partial Differential Equation (RPDE) satisfied by any corporate claim whose value is a function of firm value and time. Under specific final and boundary conditions, we solve the RPDE for the debt value and loan guarantees within a single period and homogeneous class of debt.

q-fin.PR↗

arXiv2012

Efficient Simulation of Geothermal Processes in Heterogeneous Porous Media based on the Exponential Rosenbrock-Euler and Rosenbrock-type Methods

Simulation of geothermal systems is challenging due to coupled physical processes in highly heterogeneous media. Combining the exponential Rosenbrock--Euler and Rosenbrock-type methods with control-volume (two-point flux approximation) space discretizations leads to efficient numerical techniques for simulating geothermal systems. In terms of efficiency and accuracy, the exponential Rosenbrock--Euler time integrator has advantages over standard time-dicretization schemes, which suffer from time-step restrictions or excessive numerical diffusion when advection processes are dominating. Based on linearization of the equation at each time step, we make use of matrix exponentials of the Jacobian from the spatial discretization, which provide the exact solution in time for the linearized equations. This is at the expense of computing the matrix exponentials of the stiff Jacobian matrix, together with propagating a linearized system. However, using a Krylov subspace or Leja points techniques make these computations efficient. The Rosenbrock-type methods use the appropriate rational functions of the Jacobian from the spatial discretization. The parameters in these schemes are found in consistency with the required order of convergence in time. As a result, these schemes are A-stable and only a few linear systems are solved at each time step. The efficiency of the methods compared to standard time-discretization techniques are demonstrated in numerical examples.

math.NA↗

arXiv2011

Stochastic exponential integrators for finite element discretization of SPDEs for multiplicative and additive noise

We consider the numerical approximation of a general second order semi--linear parabolic stochastic partial differential equation (SPDEs) driven by space-time noise, for multiplicative and additive noise. We examine convergence of exponential integrators for multiplicative and additive noise. We consider noise that is in trace class and give a convergence proof in the mean square $L^{2}$ norm. We discretize in space with the finite element method and in our implementation we examine both the finite element and the finite volume methods. We present results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation motivated from realistic porous media flow.

math.NA↗

arXiv2010

Stochastic Exponential Integrators for a Finite Element Discretization of SPDEs

We consider the numerical approximation of general semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space-time noise. In contrast to the standard time stepping methods which uses basic increments of the noise and the approximation of the exponential function by a rational fraction, we introduce a new scheme, designed for finite elements, finite volumes or finite differences space discretization, similar to the schemes in \cite{Jentzen3,Jentzen4} for spectral methods and \cite{GTambue} for finite element methods. We use the projection operator, the smoothing effect of the positive definite self-adjoint operator and linear functionals of the noise in Fourier space to obtain higher order approximations. We consider noise that is white in time and either in $H^1$ or $H^2$ in space and give convergence proofs in the mean square $L^{2}$ norm for a diffusion reaction equation and in mean square $ H^{1}$ norm in the presence of an advection term. For the exponential integrator we rely on computing the exponential of a non-diagonal matrix. In our numerical results we use two different efficient techniques: the real fast \Leja points and Krylov subspace techniques. We present results for a linear reaction diffusion equation in two dimensions as well as a nonlinear example of two-dimensional stochastic advection diffusion reaction equation motivated from realistic porous media flow.

math.NA↗