Weak approximations of Wright–Fisher equation

We construct weak approximations of the Wright-Fisher model and illustrate their accuracy by simulation examples.


Introduction
We consider Wright-Fisher process defined by the stochastic differential equation where B is a standard Brownian motion, 0 ⩽ a ⩽ b, σ > 0, and x ∈ [0, 1].
The Wright-Fisher model (Fisher 1930;Wright 1931) takes the values in the interval [0, 1] and explicitly accounts for the effects of various evolutionary forces -random genetic drift, mutation, selection -on allele frequencies over time. This model can also accommodate the effect of demographic forces such as variation in population size through time and/or migration connecting populations [5].
In this note, we present a simple first-order weak approximation of the solution of Eq. (1) by discrete random variables that take two values at each approximation step. Recall the definition of such an approximation. By a discretization scheme with time step h > 0 we mean any time-homogeneous Markov chain X h = { X h kh , k = 0, 1, . . . }. We say that a family of discretization schemes X h , h > 0, is a first-order weak approximation of the solution X x of (1) in the interval for a "sufficiently wide" class of functions f : [0, 1] → R and some constants C and h 0 > 0 (depending on the function f ), where N ∈ N. Note that because of the Markovity, the one-step approximation X h h completely defines (in distribution) a weak approximation X h kh , k = 0, 1, . . . . Thus, with some ambiguity, we also call it an approximation and denote it by X x h , with x indicating its starting point. In our context, we introduce the following "sufficiently wide" function class of infinitely differentiable functions with "not too fast" growing derivatives: We easily see that all functions from this class can be expanded by the Taylor series in the interval [0, 1] around arbitrary x 0 ∈ [0, 1] (which, in fact, converges on the whole real line R) and contain, for example, all polynomials and exponential functions.

Approximation
Let us first construct an approximation for the "stochastic" part of Wright-Fisher equation, that is, the solution S x t of Eq. (1) with a = b = 0. Similarly to [4] (see also [3]), we look for an approximationŜ x h as a two-valued discrete random variable taking values By solving the equation system (3)-(4) with respect to x 1 , x 2 , p 1 , p 2 , we get the solution with p 1,2 = x 2x1,2 . It also satisfies conditions (5)-(6). However, for the values of x near 1, the values of x 2 a slightly greater than 1, which is unacceptable. We overcome this problem by using the symmetry of the solution of the stochastic part with respect to the point 1 2 ; to be precise, . Therefore, in the interval [0, 1/2], we can use the values x 1,2 defined by (7)-(8), whereas in the interval (1/2, 1], we use the values corresponding to the process 1 −Ŝ 1−x t , that is,

Weak approximations of Wright-Fisher equation
Now for the initial equation (1), we obtain an approximation X x h by a simple "splitstep" procedure (again, see, e.g., [4] or [3]): Now we can state the following: Theorem 1. LetX x t be the discretization scheme defined by one-step approximation (10). ThenX x t is a first-order weak approximation of equation (1) for functions f ∈ C ∞ * [0, 1].

Backward Kolmogorov equation
The constructed approximation is in fact a so-called potential first-order weak approximation of Eq. (1) (for a definition, see, e.g., Alfonsi [1], Section 2.3.1). The proof that, indeed, it is a first-order weak approximation, is based on the following: In particular, Such theorem is stated for f ∈ C ∞ [0, 1] in [1, Thm. 6.1.12], based on the results of [2]. Our class of functions f is slightly narrower, but our proof of the theorem is significantly simpler and is based on the estimates of the moments of X x t , which show that they grow slower than factorials. The recurrent relations of the moments E[(X x t ) k ] show that they are infinitely differentiable with respect to t and x, which allows us to infinitely differentiate the series

Simulation examples
We illustrate our approximation for f (x) = x 4 and f (x) = exp{−x}. Since we do not explicitly know the moments E exp{−X x t }, we use the approximate equality exp{−x} ≈ 1 − x + x 2 2 − x 3 6 + x 4 24 . In Figs. 1 and 2, we compare the moments Ef ( X x t ) and Ef (X x t ) as functions of t (left plots, h = 0.001) and as functions of discretization step h (right plots, t = 1). As expected, the approximations agree with exact values pretty well.