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The $T_1$ inequality, which bounds the 1-Wasserstein distance in terms of the relative entropy, is known to characterize Gaussian concentration. To extend the $T_1$ inequality to laws of discrete-time processes while preserving their temporal structure, we investigate the adapted $T_1$ inequality which relates the 1-adapted Wasserstein distance to the relative entropy. We prove that $\mu$ satisfies the adapted $T_1$ inequality if and only if $\mu$ is subgaussian.
Estimating a $d$-dimensional distribution $\mu$ by the empirical measure $\hat{\mu}_n$ of its samples is an important task in probability theory, statistics and machine learning. It is well known that $\mathbb{E}[\mathcal{W}_p(\hat{\mu}_n, \mu)]\lesssim n^{-1/d}$ for $d>2p$, where $\mathcal{W}_p$ denotes the $p$-Wasserstein metric. An effective tool to combat this curse of dimensionality is the smooth Wasserstein distance $\mathcal{W}^{(\sigma)}_p$, which measures the distance between two probability measures after having convolved them with isotropic Gaussian noise $\mathcal{N}(0,\sigma^2\text{I})$. We show that the smooth adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p^{(\sigma)}$ achieves the fast rate of convergence $\mathbb{E}[\mathcal{A}\mathcal{W}_p^{(\sigma)}(\hat{\mu}_n, \mu)]\lesssim n^{-1/2}$, if $\mu$ is subgaussian. This result follows from the surprising fact, that any subgaussian measure $\mu$ convolved with a Gaussian distribution has locally Lipschitz kernels.
While the topological differences between the adapted Wasserstein distance $\mathcal{A}\mathcal{W}_p$ and the Wasserstein distance $\mathcal{W}_p$ are well understood, their differences as metrics remain largely unexplored beyond the trivial bound $\mathcal{W}_p\lesssim \mathcal{A}\mathcal{W}_p$. This paper closes this gap by providing upper bounds of $\mathcal{A}\mathcal{W}_p$ in terms of $\mathcal{W}_p$ through investigation of the smooth adapted Wasserstein distance. Our upper bounds are explicit and are given by a sum of $\mathcal{W}_p$, Eder’s modulus of continuity and a term characterizing the tail behavior of measures. As a consequence, upper bounds on $\mathcal{W}_p$ automatically hold for $\mathcal{AW}_p$ under mild regularity assumptions on the measures considered. A particular instance of our findings is the inequality $\mathcal{A}\mathcal{W}_1\le C\sqrt{\mathcal{W}_1}$ on the set of measures that have Lipschitz kernels.
Our work also reveals how smoothing of measures affects the adapted weak topology. In fact, we find that the topology induced by the smooth adapted Wasserstein distance exhibits a non-trivial interpolation property, which we characterize explicitly: it lies in between the adapted weak topology and the weak topology, and the inclusion is governed by the decay of the smoothing parameter.
We establish concentration inequalities for the optimal transport cost under radial cost functions with superpolynomial growth. Compared with the classical Fournier-Guillin rate for the Wasserstein distance $\mathcal{W}_p$, our new estimates yield sharper rates.
We consider asian options in local volatility models. We present asymptotics for prices and deltas when the maturity $T$ is small. The decay of OTM and ITM options are known to be characterized by the large deviation principle. This paper derives asymptotic formulas for ATM options.