Yassine Nachit - Densities for solutions of SDEs driven by non-Gaussian processes

In this talk, we explore the existence of probability density functions for solutions to stochastic differential equations driven by multidimensional non-Gaussian processes. Namely, we consider the following stochastic differential equation on R^d:

Xtk=x0k+∫0tbk(Xs) ds+∑l=1d∫0tσk,l(Xs) dFsl,t∈[0,1],X_t^k = x_0^k + \int_0^t b^k(X_s) \, d s + \sum_{l=1}^d \int_0^t \sigma^{k, l}(X_s) \, d F_s^l, \quad t \in [0, 1],Xtk​=x0k​+∫0t​bk(Xs​)ds+∑l=1d​∫0t​σk,l(Xs​)dFsl​,t∈[0,1],

for k=1,…,dk = 1, \ldots, dk=1,…,d, where:

• x0∈Rdx_0 \in \mathbb{R}^dx0​∈Rd,
• b=(bk)1≤k≤db = (b_k)_{1 \leq k \leq d}b=(bk​)1≤k≤d​,
• σ=(σk,l)1≤k,l≤d\sigma = (\sigma_{k, l})_{1 \leq k, l \leq d}σ=(σk,l​)1≤k,l≤d​, and
• F={F(s);s∈[0,T]}F = \{ F(s); s \in [0, T] \}F={F(s);s∈[0,T]} is a ddd-dimensional non-Gaussian process.

Using Malliavin Calculus, we establish that the solution XtX_tXt​ admits a density for times t∈(0,1]t \in (0, 1]t∈(0,1], provided that:

1. bbb and σ\sigmaσ are smooth enough,
2. σ\sigmaσ satisfies a Hörmander’s condition, and
3. the non-Gaussian driving process FFF verifies certain specific regularity and non-degeneracy criteria.