The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs measure using appropriate stochastic dynamics of Langevin type. Rigorous ergodicity results as well as elements on the estimation of numerical errors are provided. The last part is devoted to the computation of transport coefficients such as the mobility or autodiffusion in fluids, relying either on integrated equilibrium correlations à la Green-Kubo, or on the linear response of nonequilibrium dynamics in their steady-states.[-]

Backward stochastic differential equations have been a very successful and active tool for stochastic finance and insurance for some decades. More generally they serve as a central method in applications of control theory in many areas. We introduce BSDE by looking at a simple utility optimization problem in financial stochastics. We shall derive an important class of BSDE by applying the martingale optimality principle to solve an optimal investment problem for a financial agent whose income is partly affected by market external risk. We then present the basics of existence and uniqueness theory for solutions to BSDE the coefficients of which satisfy global Lipschitz conditions.[-]

In this lecture, we shall discuss the key steps involved in the use of least squares regression for approximating the solution to BSDEs. This includes how to obtain explicit error estimates, and how these error estimates can be used to tune the parameters of the numerical scheme based on complexity considerations.
The algorithms are based on a two stage approximation process. Firstly, a suitable discrete time process is chosen to approximate the of the continuous time solution of the BSDE. The nodes of the discrete time processes can be expressed as conditional expectations. As we shall demonstrate, the choice of discrete time process is very important, as its properties will impact the performance of the overall numerical scheme. In the second stage, the conditional expectation is approximated in functional form using least squares regression on synthetically generated data – Monte Carlo simulations drawn from a suitable probability distribution. A key feature of the regression step is that the explanatory variables are built on a user chosen finite dimensional linear space of functions, which the user specifies by setting basis functions. The choice of basis functions is made on the hypothesis that it contains the solution, so regularity and boundedness assumptions are used in its construction. The impact of the choice of the basis functions is exposed in error estimates.
In addition to the choice of discrete time approximation and the basis functions, the Markovian structure of the problem gives significant additional freedom with regards to the Monte Carlo simulations. We demonstrate how to use this additional freedom to develop generic stratified sampling approaches that are independent of the underlying transition density function. Moreover, we demonstrate how to leverage the stratification method to develop a HPC algorithm for implementation on GPUs.
Thanks to the Feynmann-Kac relation between the the solution of a BSDE and its associated semilinear PDE, the approximation of the BSDE can be directly used to approximate the solution of the PDE. Moreover, the smoothness properties of the PDE play a crucial role in the selection of the hypothesis space of regressions functions, so this relationship is vitally important for the numerical scheme.
We conclude with some draw backs of the regression approach, notably the curse of dimensionality.[-]

We will consider the discretization of the stochastic differential equation$$X_t=X_0+W_t+\int_0^t b\left(s, X_s\right) d s, t \in[0, T]$$where the drift coefficient $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is measurable and satisfies the integrability condition : $\|b\|_{L^q\left([0, T], L^\rho\left(\mathbb{R}^d\right)\right)}<\infty$ for some $\rho, q \in(0,+\infty]$ such that$$\rho \geq 2 \text { and } \frac{d}{\rho}+\frac{2}{q}<1 .$$Krylov and Röckner [3] established strong existence and uniqueness under this condition.Let $n \in \mathbb{N}^*, h=\frac{T}{n}$ and $t_k=k h$ for $k \in \left [ \left [0,n \right ] \right ]$. Since there is no smoothing effect in the time variable, we introduce a sequence $\left(U_k\right)_{k \in \left [ \left [0,n-1 \right ] \right ]}$ independent from $\left(X_0,\left(W_t\right)_{t \geq 0}\right)$ of independent random variables which are respectively distributed according to the uniform law on $[k h,(k+1) h]$. The resulting scheme Euler is initialized by $X_0^h=X_0$ and evolves inductively on the regular time-grid $\left(t_k=k h\right)_{k \in \left [ \left [0,n \right ] \right ]}$ by:$$X_{t_{k+1}}^h=X_{t_k}^h+W_{t_{k+1}}-W_{t_k}+b_h\left(U_k, X_{t_k}^h\right) h$$where $b_h$ is some truncation of the drift function $b$. When $b$ is bounded, one of course chooses $b_h=b$. Then the order of weak convergence in total variation distance is $1 / 2$, as proved in [1]. It improves to 1 up to some logarithmic correction under some additional uniform in time bound on the spatial divergence of the drift coefficient. In the general case (1), we will see that for suitable truncations $b_h$, the difference between the transition densities of the stochastic differential equation and its Euler scheme is bounded from above by $C h^{\frac{1}{2}\left(1-\left(\frac{d}{\rho}+\frac{2}{q}\right)\right)}$ multiplied by some centered Gaussian density, as proved in [2].[-]

We will consider the discretization of the stochastic differential equation$$X_t=X_0+W_t+\int_0^t b\left(s, X_s\right) d s, t \in[0, T]$$where the drift coefficient $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is measurable and satisfies the integrability condition : $\|b\|_{L^q\left([0, T], L^\rho\left(\mathbb{R}^d\right)\right)}<\infty$ for some $\rho, q \in(0,+\infty]$ such that$$\rho \geq 2 \text { and } \frac{d}{\rho}+\frac{2}{q}<1 .$$Krylov and Röckner established strong existence and uniqueness under this condition.Let $n \in \mathbb{N}^*, h=\frac{T}{n}$ and $t_k=k h$ for $k \in \left [ \left [0,n \right ] \right ]$. Since there is no smoothing effect in the time variable, we introduce a sequence $\left(U_k\right)_{k \in \left [ \left [0,n-1 \right ] \right ]}$ independent from $\left(X_0,\left(W_t\right)_{t \geq 0}\right)$ of independent random variables which are respectively distributed according to the uniform law on $[k h,(k+1) h]$. The resulting scheme Euler is initialized by $X_0^h=X_0$ and evolves inductively on the regular time-grid $\left(t_k=k h\right)_{k \in \left [ \left [0,n \right ] \right ]}$ by:$$X_{t_{k+1}}^h=X_{t_k}^h+W_{t_{k+1}}-W_{t_k}+b_h\left(U_k, X_{t_k}^h\right) h$$where $b_h$ is some truncation of the drift function $b$. When $b$ is bounded, one of course chooses $b_h=b$. Then the order of weak convergence in total variation distance is $1 / 2$, as proved in [1]. It improves to 1 up to some logarithmic correction under some additional uniform in time bound on the spatial divergence of the drift coefficient. In the general case (1), we will see that for suitable truncations $b_h$, the difference between the transition densities of the stochastic differential equation and its Euler scheme is bounded from above by $C h^{\frac{1}{2}\left(1-\left(\frac{d}{\rho}+\frac{2}{q}\right)\right)}$ multiplied by some centered Gaussian density, as proved in [2].[-]

We introduce a new strategy for the solution of Mean Field Games in the presence of major and minor players. This approach is based on a formulation of the fixed point step in spaces of controls. We use it to highlight the differences between open and closed loop problems. We illustrate the implementation of this approach for linear quadratic and finite state space games, and we provide numerical results motivated by applications in biology and cyber-security.[-]

The mathematical framework of variational inequalities is a powerful tool to model problems arising in mechanics such as elasto-plasticity where the physical laws change when some state variables reach a certain threshold [1]. Somehow, it is not surprising that the models used in the literature for the hysteresis effect of non-linear elasto-plastic oscillators submitted to random vibrations [2] are equivalent to (finite dimensional) stochastic variational inequalities (SVIs) [3]. This presentation concerns (a) cycle properties of a SVI modeling an elasto-perfectly-plastic oscillator excited by a white noise together with an application to the risk of failure [4,5]. (b) a set of Backward Kolmogorov equations for computing means, moments and correlation [6]. (c) free boundary value problems and HJB equations for the control of SVIs. For engineering applications, it is related to the problem of critical excitation [7]. This point concerns what we are doing during the CEMRACS research project. (d) (if time permits) on-going research on the modeling of a moving plate on turbulent convection [8]. This is a mixture of joint works and / or discussions with, amongst others, A. Bensoussan, L. Borsoi, C. Feau, M. Huang, M. Laurière, G. Stadler, J. Wylie, J. Zhang and J.Q. Zhong.[-]