Appears in collection : A Random Walk in the Land of Stochastic Analysis and Numerical Probability / Une marche aléatoire dans l'analyse stochastique et les probabilités numériques

After a few reminders on the convex order $\leq _{cv}$ between two random vectors $U$ and $V$ defined by $U\leq _{cv}V$ if $\mathbb{E}f(U)\leq \mathbb{E}f(V)$ for every convex function $f:\mathbb{R}^{d}\rightarrow \mathbb{R}$, (with some variants like monotonic convex order) and their first applications in finance, we will explain how to extend this order in a functional way to stochastic processes, in particular to diffusions (Brownian, with jumps, McKean Vlasov type), even to non-Markovian processes, such as the solutions of Volterra equations with singular kernels like those appearing in rough volatility modeling in Finance. We systematically establish our comparison results by an approximation procedure of Euler scheme type, generally simulable. Thus, among other virtues, this approach makes it possible in finance to ensure that the prices of derivative products computed by simulation cannot give rise to arbitrages by lack of convexity. As a by-product we will also establish the convexity of functionals $x\to \mathbb{E}F(X^{x})$ of such stochastic processes $X^{x}$ when $F$ is convex and $x$ is the starting value of $X^{x}$. (includes some joint works with B. Jourdain and Y. Liu).