We consider a Backward Stochastic Differential Equation (BSDE for short) in a Markovian framework for the pair of processes (Y, Z), with generator with quadratic growth with respect to Z. The forward equation is an evolution equation in an abstract Banach space. We prove an analogue of the Bismut–Elworty formula when the diffusion operator has a pseudo-inverse not necessarily bounded and when the generator has quadratic growth with respect to Z. In particular, our model covers the case of the heat equation in space dimension greater than or equal to 2. We apply these results to solve semilinear Kolmogorov equations in Banach spaces for the unknown v, with nonlinear term with quadratic growth with respect to \(\nabla v\) and final condition only bounded and continuous, and to solve stochastic optimal control problems with quadratic growth.

我们考虑马尔可夫框架中的一对过程（Y，  Z）的向后随机微分方程（BSDE ），其中生成器相对于Z具有二次增长。前向方程是抽象Banach空间中的演化方程。当扩散算子具有不一定要有界的伪逆数并且当生成器关于Z的平方增长时，我们证明了Bismut-Elworty公式的类似物。特别是，我们的模型涵盖了空间维数大于或等于2的热方程的情况。我们将这些结果用于求解未知v的Banach空间中的半线性Kolmogorov方程，其中非线性项相对于二次增长\（\ nabla v \）和最终条件仅是有界和连续的，并且可以解决具有二次增长的随机最优控制问题。