Statistics

Mathematical analysis

Economics

Monte Carlo method

Volterra integral equation

Integral equation

Convergence (economics)

Applied mathematics

Mathematics

Economic growth

Discrete time and continuous time

Infimum and supremum

Stochastic Processes and their Applications

We study discrete-time simulation schemes for stochastic Volterra equations, namely the Euler and Milstein schemes, and the corresponding Multilevel Monte-Carlo method. By using and adapting some results from Zhang (2008), together with the Garsia–Rodemich–Rumsey lemma, we obtain the convergence rates of the Euler scheme and Milstein scheme under the supremum norm. We then apply these schemes to approximate the expectation of functionals of such Volterra equations by the (Multilevel) Monte-Carlo method, and compute their complexity. We finally provide some numerical simulation results.

Discrete-time simulation of Stochastic Volterra equations