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Euler scheme and tempered distributions. (English) Zbl 1106.60053
Consider smooth \(\mathbb R^d\)-valued diffusion processes \((X_t^x)_{t \in [0,1]}\) and its approximation by equidistant Euler method \((Y_{t,n}^x)_{n \geq 1}\) started at point \(x \in \mathbb R^d\) on \([0,1]\). If \(X\) is uniformly elliptic, the author proves that there is a constant \(C_1(f(x))\) such that \[ \mathbb E [ f(X_1^x) ] - \mathbb E [ f(Y_{1,n}^x) ] = \frac{C_1(f(x))}{n} + O \biggl(\frac{1}{n^2}\biggr) \] for all tempered distributions \(f\), i.p. for measurable functions \(f\) with polynomial or exponential growth, Dirac or any derivative of Dirac mass. A motivation with applications to option pricing, hedging, rates of prices, deltas and gammas in finance is given.

MSC:
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C05 Monte Carlo methods
65C20 Probabilistic models, generic numerical methods in probability and statistics
65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
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