On the convergence of the Lie-Trotter formula for stochastic differential equations
The authors consider the one-dimensional stochastic differential equation \[ dX_t = \sigma (t,X_t) dB_t - b(t,X_t) dt, \quad X_0 = x, \] on \([0,T]\) together with the corresponding Lie-Trotter formula that defines approximate solutions \(X^n_t\) and \(Y_t^n\) for \(t\) belonging to an interval \([ih, ih + h]\) of an \(n\)-point equidistant grid on \([0,T]\) by \[ X^n_t = Y^n_{ih} - \int^t_{ih} b(s, X^n_s) ds, \quad Y^n_t = X^n_t + \int^t_{ih} \sigma (s, Y_s^n) dB_s. \] It is proved that if \(b\) and \(\sigma\) are continuous and bounded and a pathwise unique solution \(X\) exists, then both \(X^n\) and \(Y^n\) converge towards \(X\) in the norm of the \(L_2\)-space of \(C[0,T]\)-valued random variables, uniformly for all initial values \(x\) in a compact.
scientific article; zbMATH DE number 784552
scientific article