Joint characteristic function and simultaneous simulation of iterated Itô integrals for multiple independent Brownian motions
The author considers a system of stochastic differential equations \[ dX(t) = b(X(t),t) dt + \sigma(X(t),t) dW(t), \tag{1} \] where \(X(t)\) is a \(d\)-dimensional vector and \(W(t)\) is an \(m\)-dimensional vector of independent Brownian motions. In the case where the dispersion matrix \(\sigma\) does not satisfy the commutativity condition (which is \(L^i \sigma_{kj}= L^j \sigma_{ki}\), \(i,j = 1,\ldots,m\), \(k=1,\ldots, d\) with \(L^i = \sum_{l=1}^d \sigma_{li} \frac{\partial}{\partial x_l}\)), numerical schemes do not converge in the mean square sense with order larger than \(\frac{1}{2}\). Order \(1\) can be achieved when the iterated stochastic Itô integrals \( I_{i,j}(t_n,t_n+h) = \int_{t_n}^{t_n+h} \int_{t_n}^s dW_i(u) dW_j(s)\) are included (e.g. in the Milstein scheme). An expression for the conditional joint characteristic function of the iterated Itô integrals given the Brownian increments is derived. Moreover, the author suggests a method to approximate \( I_{i,j}\) which can be used in the case \(m>2\) [for \(m=2\) see \textit{J. G. Gaines} and \textit{T. J. Lyons}, SIAM J. Appl. Math. 54, 1132-1146 (1994; Zbl 0805.60052) and \textit{T. Rydén} and the author, Stochastic Processes Appl. 91, 151-168 (2001)], based on the method proposed by \textit{P. E. Kloeden, E. Platen} and \textit{I. W. Wright} [Stochastic Anal. Appl. 10, 431-441 (1992; Zbl 0761.60048)]. The algorithm is essentially a truncation of an infinite series representation of the iterated Itô integral. The mean-square error of the original method is of order \(h^2/n\), where \(n\) is the number of terms in the truncated sum. By a modification of the method, which amounts to adding a term approximating the tail-sum of the discarded terms of the series, the author achieves a mean-square error of order \(h^2/n^2\).
multidimensional stochastic differential equation
scientific article; zbMATH DE number 1906244
scientific article