Blog

Although my resarch focuses on PDEs (optimal transport, these days), I am currently trying to learn stochastic calculus and integration. I am just beginning in this topics, but I was wondering: why can we only integrate with respect to martingales? Typically $ $ Y_t=\int_0^s X_s \,dH_s $ $ seems to require that $ H_s$ beRead more

Let $ (\Omega,\mathcal A,\operatorname P)$ be a complete probability space $ (\mathcal F_t)_{t\ge 0}$ be a complete and right-continuous filtration of $ \mathcal A$ $ H$ be a separable $ \mathbb R$ -Hilbert space $ (e_n)_{n\in\mathbb N}$ be an orthonormal basis of $ H$ $ \mathfrak L_1(H)$ denote the space of nuclear operators on $Read more

Let $ (M_n)_{n \geq 1}$ be a uniformly bounded martingale over a probability space $ (\Omega,\mathcal{F},\mathbb{P})$ . Define the probability measure $ \mu$ on $ \mathbb{R}^\mathbb{N}$ to be the law of $ (M_n)_{n \geq 1}$ , and for each $ n$ define the probability measure $ \mu_n$ on $ \mathbb{R}^n$ to be the law ofRead more