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## Section: Research Program

### Numerical Probability

Our project team is very much involved in numerical probability, aiming at pushing numerical methods towards the effective implementation. This numerical orientation is supported by a mathematical expertise which permits a rigorous analysis of the algorithms and provides theoretical support for the study of rates of convergence and the introduction of new tools for the improvement of numerical methods. This activity in the MathRisk team is strongly related to the development of the Premia software.

#### Simulation of stochastic differential equations

##### - Weak convergence of the Euler scheme in optimal transport distances.

With A. Kohatsu-Higa, A. Alfonsi and B. Jourdain [4] have proved using optimal transport tools that the Wasserstein distance between the time marginals of an elliptic SDE and its Euler discretization with $N$ steps is not larger than $C\sqrt{log\left(N\right)}/N$. The logarithmic factor may is removed when the uniform time-grid is replaced by a grid still counting $N$ points but refined near the origin of times.

##### - Strong convergence properties of the Ninomiya Victoir scheme and multilevel Monte-Carlo estimators.

With their former PhD student, A. Al Gerbi, E. Clément and B. Jourdain [1] have proved strong convergence with order $1/2$ of the Ninomiya-Victoir scheme which is known to exhibit order 2 of weak convergence [82]. This study was aimed at analysing the use of this scheme either at each level or only at the finest level of a multilevel Monte Carlo estimator : indeed, the variance of a multilevel Monte Carlo estimator is related to the strong error between the two schemes used in the coarse and fine grids at each level. In [38], they proved that the order of strong convergence of the crude Ninomiya Victoir scheme is improved to 1 when the vector fields corresponding to each Brownian coordinate in the SDE commute, and in [39], they studied the error introduced by discretizing the ordinary differential equations involved in the Ninomiya-Victoir scheme.

##### - Non-asymptotic error bounds for the multilevel Monte Carlo Euler method.

A. Kebaier and B. Jourdain are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, they dealt in [20] with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. They obtained Gaussian-type concentration. To do so, they used the Clark-Ocone representation formula and derived bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives. The estimation of such differences is much more complicated than the one of a single Euler scheme contribution and explains why they suppose the diffusion coefficient to be constant. This assumption ensures boundedness of the Malliavin derivatives of both the SDE and its Euler scheme.

##### - Computation of sensibilities of integrals with respect to the invariant measure.

In [48], R. Assaraf, B. Jourdain, T. Lelièvre and R. Roux considered the solution to a stochastic differential equation with constant diffusion coefficient and with a drift function which depends smoothly on some real parameter $\lambda$, and admitting a unique invariant measure for any value of $\lambda$ around $\lambda =0$. Their aim was to compute the derivative with respect to $\lambda$ of averages with respect to the invariant measure, at $\lambda =0$. They analyzed a numerical method which consists in simulating the process at $\lambda =0$ together with its derivative with respect to $\lambda$ on a long time horizon. They gave sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to $\lambda$ of the mean of an observable through Monte Carlo simulations.

##### - Parametrix methods.

V. Bally and A. Kohatsu-Higa have recently proposed an unbiased estimator based on the parametrix method to compute expectations of functions of a given SDE ( [54]). This method is very general, and A. Alfonsi, A. Kohastu-Higa and M. Hayashi  [42] have applied it to the case of one-dimensional reflected diffusions. In this case, the estimator can be obtained explicitly by using the scheme of Lépingle  [79] and is quite simple to implement. It is compared to other simulation methods for reflected SDEs.

#### Estimation of the parameters of a Wishart process

A. Alfonsi, A. Kebaier and C. Rey  [43] have computed the Maximum Likelihood Estimator for the Wishart process and studied its convergence in the ergodic and in some non ergodic cases. In the ergodic case, which is the most relevant for applications, they obtain the standard square-root convergence. In the non ergodic case, the analysis rely on refined results for the Laplace transform of Wishart processes, which are of independent interest.

#### Optimal stopping and American options

In joint work with A. Bouselmi, D. Lamberton studied the asymptotic behavior of the exercise boundary near maturity for American put options in exponential Lévy models. In [7], they deal with jump-diffusion models, and establish that, in some cases, the behavior differs from the classical Black and Scholes setting. D. Lamberton has also worked on the binomial approximation of the American put. The conjectured rate of convergence is $O\left(1/n\right)$ where $n$ is the number of time periods. He was able to derive a $O\left({\left(lnn\right)}^{\alpha }/n\right)$ bound, where the exponent $\alpha$ is related to the asymptotic behavior of the exercise boundary near maturity.