SDEs with random and irregular coefficients (Q2135424)

In this article, the author studies \(\mathbb{R}^n\)-valued stochastic differential equations (SDEs) with random and irregular coefficients of the form \[ X_t(\omega) = X_0(\omega) + \int_0^t \sigma_s(X_s,\omega) \, \mathrm{d} W_s(\omega) + \int_0^t b_s(X_s,\omega) \, \mathrm{d}s, \] where \(\{ W_t \}_{t \in [0,1]}\) is a \(d\)-dimensional Brownian motion, and \(\sigma : \mathbb{R}^n \times [0,1] \times \Omega \to \mathbb{R}^n \otimes \mathbb{R}^d\) and \(b : \mathbb{R}^n \times [0,1] \times \Omega \to \mathbb{R}^n\) are measurable coefficients. The main result is Theorem 1.2, where sufficient conditions on \(\sigma\) and \(b\) for existence and uniqueness of solutions to the SDE are presented. In order to prepare the proof, the author provides Schauder estimates for random Banach-valued PDEs in Section 2 (see, in particular Theorem 2.3) and a Schauder estimate for a backward SPDE in Section 3 (see, in particular Theorem 3.5). Afterwards, the proof of Theorem 1.2 is given in Section 4. The main ideas are solving backward stochastic Kolmogorov equations and utilizing a modified Zvonkin type transformation.