Estimates of the numerical density for stochastic differential equations with multiplicative noise

We investigate the estimates of the density for the traditional Euler-Maruyama discretization of stochastic differential equations (SDEs) with multiplicative noise. Our estimates focus on two key aspects: (1) the $L^p$-upper bounds for derivatives of the logarithmic numerical density, (2) the sharp error order of the Euler scheme under the relative entropy (or Kullback-Leibler divergence). For the first aspect, we present estimates for the first-order and second-order derivatives of the logarithmic numerical density. The key technique is to adopt the Malliavin calculus to derive expressions of the derivatives of the logarithmic Green's function and to obtain an estimate for the inverse Malliavin matrix. Moreover, for the relative entropy error, we obtain a bound that is second order in time step, which then naturally leads to first-order error bounds under the total variation distance and Wasserstein distances. Compared with the usual weak error estimate for SDEs, such estimate can give an error bound for the worst case of a family of test functions instead of one test function.