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An analytic approximation of solutions of stochastic differential equations. (English) Zbl 1075.60067
In general, it is not possible to determine explicitly the solution of the equation \[ dX(t)=a(t,X(t)) dt +b(t,X(t)) dW(t), \;t \in [0,1], \quad X(0)=\eta. \] To obtain an approximate solution, some applicable analytic or numerical methods are usually used. An analytic approximation for the solution of the equation above is given, namely the drift and diffusion coefficients are Taylor approximations of the functions \(a\) and \(b\). If \(0=t_0<t_1<t_2<\ldots<t_n=1,\) \(\delta_n=\max_{0\leq k \leq n-1}(t_{k+1}-t_k)\) is an arbitrary partition of the interval \([0,1]\), then the approximation is the form \[ \begin{split} X_t^n =X_{t_k}^n+\int_{t_k}^t \sum_{i=1}^{m_1} \frac{a_x^{(i)}(s, X_{t_k}^n)}{i!} (X_s^n-X_{t_k}^n)^{i} \,ds+\\ \int_{t_k}^t \sum_{i=1}^{m_2} \frac{b_x^{(i)}(s, X_{t_k}^n)}{i!} (X_s^n-X_{t_k}^n)^i \,dw_s, \quad 0 \leq k\leq{n-1},\end{split} \] where \(X_{0}^n=\eta\) a.s. The authors compare in the \(L^p\)-norm, \(p\geq 2\), the solution of the first equation to its approximation. They prove that the sequence of the approximate solutions tends with probability one to the solution of first equation.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
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