We assume that we observe $N$ independent copies of a diffusion process on a time-interval $[0,2T]$. For a given time $t$, we estimate the transition density $p_t(x,y)$, namely the conditional density of $X_{t + s}$ given $X_s = x$, under conditions on the diffusion coefficients ensuring that this quantity exists. We use a least squares projection method on a product of finite dimensional spaces, prove risk bounds for the estimator and propose an anisotropic model selection method, relying on several reference norms. A simulation study illustrates the theoretical part for Ornstein-Uhlenbeck or square-root (Cox-Ingersoll-Ross) processes.

翻译：假设我们观测到扩散过程在时间区间 $[0,2T]$ 上的 $N$ 个独立样本。对于给定时间 $t$，我们在扩散系数满足转移密度存在性的条件下，估计转移密度函数 $p_t(x,y)$，即给定 $X_s = x$ 时 $X_{t + s}$ 的条件密度。我们采用有限维空间乘积上的最小二乘投影方法，证明了估计量的风险界，并提出一种基于多重参考范数的各向异性模型选择方法。针对 Ornstein-Uhlenbeck 过程或平方根（Cox-Ingersoll-Ross）过程的模拟研究验证了理论部分的结果。