We apply Stein's method to obtain a non-uniform exponential bound for normal approximations for certain types of random variables. In particular, we establish the bound for a random variable such that its exchangeable pair coupling exists and the distance between the pair is bounded. Using our result, we obtain better bounds for a wide range of applications, such as sums of bounded independent random variables, the combinatorial central limit theorem, and the numbers of descents and inversions of a permutation.
Several financial instruments have been thoroughly calculated via the price of an underlying asset, which can be regarded as a solution of a stochastic differential equation (SDE), for example the moment swap and its exotic types that encourage investors in markets to trade volatility on payoff and are especially beneficial for hedging on volatility risk. In the past few decades, numerous studies about conditional moments from various SDEs have been conducted. However, some existing results are not in closed forms, which are more difficult to apply than simply using Monte Carlo (MC) simulations. To overcome this issue, this paper presents an efficient closed-form formula to price generalized swaps for discrete sampling times under the inhomogeneous Heston model, which is the Heston model with time-parameter functions. The obtained formulas are based on the infinitesimal generator and solving a recurrence relation. These formulas are expressed in an explicit and general form. An investigation of the essential properties was carried out for the inhomogeneous Heston model, including conditional moments, central moments, variance, and skewness. Moreover, the closed-form formula obtained was numerically validated through MC simulations. Under this approach, the computational burden was significantly reduced.
In this paper, we obtain a nonuniform Berry–Esseen bound for a normal approximation via the Stein method and the exchangeable-pair coupling technique where the boundedness condition of the difference between the exchangeable pair is not required. As applications of the result, we obtain nonuniform bounds for the normal approximations in two well-known applications that are the independence test and the quadratic form. Our results suggest that the obtained bounds for the two applications are sharper than other existing bounds.
We study a new inequality arising from the principle of inclusion and exclusion by mixing the idea from Hlawka's inequality and Tverberg's combinatorial sum. We obtain sharp lower bounds for the sum when the number of variables is small.
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