arXiv:0707.4466v1 [math.NA] 30 Jul 2007 Convergence in the Prokhorov Metric of Weak Methods for Stochastic Differential Equations BENOIT CHARBONNEAU,YURIY SVYRYDOV , P. F. TUPPER Department of Mathematics and Statistics, McGill University, Montr´ eal QC, H3A 2K6 Canada. March 4, 2019 Abstract There are two important classes of numerical methods for stochastic differential equations (SDEs): strong meth- ods and weak methods. Strong methods construct numerical approximations to trajectories of the SDEs directly, using the Brownian motions driving the SDEs. Weak methods compute a numerical trajectory using a sequence of random variables independent of the Brownian motions. The convergence of weak methods is usually expressed indirectly in terms of the convergence of expected values of test functions of the trajectories. Here we present an alternative formulation of convergence for weak methods in terms of the well-known Prokhorov metric on spaces of random variables. For a general class of weak methods, we establish bounds on the rates of convergence in terms of the Prokhorov metric. In doing so, we revisit the original proofs of convergence for weak methods and show explic- itly how the bounds on the error depend on the smoothness of the test functions. As an application of our result, we use the Strassen–Dudley theorem to show that the true solution to the system of SDEs and the approximation from the weak method can be embedded in the same probability space in such a way that values generated by the weak method converge there in a strong sense. We conclude with a review of the existing results for pathwise convergence of weak methods and the corresponding strong results available under embedding. stochastic differential equations, convergence in distribution, weak convergence, Prokhorov metric, Strassen–Dudley Theorem 1 Introduction Consider the following system of Ito stochastic differential equations (SDEs) dX = a(X )dt + q ∑ r=1 σ r (X )dW r (t ), X (0)= x 0 , (1.1) for X (t ) ∈ R n , where the W r (t ) are independent Brownian motions. The simplest numerical method for obtaining approximate solutions to this system is the Euler–Maruyama method: for k ≥ 0, timestep ∆t , and ∆ k W r = W r ((k + 1)∆t ) − W r (k∆t ), X k+1 = X k + a(X k )∆t + q ∑ r=1 σ r (X k )∆ k W r , X 0 = x 0 . (1.2) For each k, X k is an approximation to X (k∆t ). Euler–Maruyama is a strong method because for each realization of the W r (t ), the method gives an approximation to the exact solution of the SDE with that same realization. In particular, as shown in [5, p. 342 ], (E(X (T ) − X T /∆t ) 2 ) 1/2 ≤ C∆t 1/2 , (1.3) under certain assumptions on the coefficients a and σ . In order for such a result to be possible X (T ) and X T /∆t must be defined on the same probability space. Another class of methods is the weak methods. These methods do not use the original Brownian processes W r in the construction of the approximate solution. An example of such a method for solving Equation (1.1) is obtained