Euler scheme for solutions of a countable system of stochastic differential equations (Q5953977)

The authors consider a countable system of stochastic differential equations NEWLINE\[NEWLINEX(t) = X(0) + \int_0^t a(s,X(s)) ds + \int_0^t b(s,X(s)) dW(s),NEWLINE\]NEWLINE or equivalently NEWLINE\[NEWLINEx_i(t) = x_i(0) + \int_0^t a_i(s,X(s)) ds + \sum_{j=1}^m \int_0^t b_i^j(s,X(s)) dW_i^j(s),NEWLINE\]NEWLINE where for \(S\) a countable set, \(X(t)=(x_i(t))_{i\in S}\) is a stochastic process in \(({\mathbb R}^d)^S\). A suitable function space with a weighted norm is introduced. They define a countable Euler approximation and prove the following results. The countable Euler scheme converges in mean-square (with respect to the weighted norm) to the solution of the countable system of stochastic differential equations. They then define a truncated and thus finite-dimensional Euler scheme and show that this converges to the solution of the countable Euler scheme in mean-square. Finally they prove convergence in the sense of weak approximations of the countable Euler scheme to the solution of the countable system of stochastic differential equations.