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Quantitative Finance · Glossary

What is Strong and weak orders of convergence, Milstein scheme?

Also known as: strong order of convergence · weak order of convergence · Milstein scheme

Definition 26.11 Quantitative Methods · Chapter 26 — Monte Carlo

A scheme with step hh has strong order of convergence γ\gamma if E∣XTh−XT∣≤Chγ\E|X_T^h - X_T| \le Ch^\gamma, and weak order of convergence β\beta if ∣Eg(XTh)−Eg(XT)∣≤Chβ|\E g(X_T^h) - \E g(X_T)| \le Ch^\beta for smooth gg. The Milstein scheme for dX=a(X) dt+b(X) dWdX = a(X)\,dt + b(X)\,dW adds to the Euler–Maruyama step the term 12b(X)b′(X)((ΔW)2−h)\frac12b(X)b'(X)((\Delta W)^2 - h).

Strong error |X_Th - X_T| of the Euler and Milstein schemes (20 000 paths of a geometric Brownian motion, = 5\%, = 50\%, X_0 = 100, one year) and the exact weak error of the mean, against the time step. Data: the chapter’s tutorial, seeded.
Figure 26.4. Strong error E∣XTh−XT∣\E|X_T^h - X_T| of the Euler and Milstein schemes (20 000 paths of a geometric Brownian motion, μ=5%\mu = 5\%, σ=50%\sigma = 50\%, X0=100X_0 = 100, one year) and the exact weak error of the mean, against the time step. Data: the chapter’s tutorial, seeded.
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