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The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we provedTV⁢(XTε,X¯Tε,Mil,(n))≤C⁢ε2/nd_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{Mil},(n)})\leq C\varepsilon^{2}/nanddTV⁢(XTε,X¯Tε,EM,(n))≤C⁢ε/nd_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{EM},(n)})\leq C\varepsilon/n, wheredTVd_{\mathrm{TV}}is the total variation distance,XεX^{\varepsilon}is a solution of a stochastic differential equation with a small parameter , andX¯ε,Mil,(n)\bar{X}^{\varepsilon,\mathrm{Mil},(n)}andX¯ε,EM,(n)\bar{X}^{\varepsilon,\mathrm{EM},(n)}are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.

Total variation bound for Milstein scheme without iterated integrals