Weak approximations of Wright–Fisher equation

Wright–Fisher lygties silpnosios aproksimacijos

We consider Wright–Fisher process defined by the stochastic differential equation

where $\mathit{\text{B}}$ is a standard Brownian motion, $0⩽a⩽b,\sigma >0,$ and $x\in [0,1].$

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 $\widehat{{\text{X}}^h}=\{\widehat{{\text{X}}_{kh}^h},k=0,1,...\}.$

We say that a family of discretization schemes $\widehat{{\text{X}}^h},h>0,$ is a first-order weak approximation of the solution ${\text{X}}^x$ of (1) in the interval $[0,\mathit{T}]$ if

for a “sufficiently wide” class of functions $f:[0,1]\to R$ and some constants $\mathit{\text{C}}$ and $h_0>0$ (depending on the function $f$), where $\text{N}\text{∈}\text{N}.$ Note that because of the Markovity, the one-step approximation $\stackrel{ˆ}{{\mathit{\text{X}}}_h^h}$ completely defines (in distribution) a weak approximation $\stackrel{ˆ}{{\mathit{\text{X}}}_{kh}^h},k=0,1,....$ Thus, with some ambiguity, we also call it an approximation and denote it by $\stackrel{ˆ}{{\mathit{\text{X}}}_h^x},$ 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 $x_0\in [0,1]$ (which, in fact, converges on the whole real line $\mathit{\text{R}}$) and contain, for example, all polynomials and exponential functions.

Let us first construct an approximation for the “stochastic” part of Wright–Fisher equation, that is, the solution ${\mathit{\text{S}}}_t^x$ of Eq. (1) with $a=b=0.$ Similarly to [4] (see also [3]), we look for an approximation $\stackrel{ˆ}{{\mathit{\text{X}}}_h^x}$ as a two-valued discrete random variable taking values $x_{1,2}\mathrm{\in }[0,1]$ with probabilities $p_{1,2}$ such that

By solving the equation system (3)–(4) with respect to $x_1,x_2,p_1,p_2,$ we get the solution

with $p_{1,2}=\frac{x}{2_{x_{1,2}}}.$ 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 $\frac{1}{2};$ to be precise, ${\mathit{\text{S}}}_t^x\stackrel{\mathit{d}}{=}1-{\mathit{\text{S}}}_t^{1-x}.$ 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 $1-\stackrel{ˆ}{{\mathit{\text{S}}}_t^{1-x}}$, that is,

with probabilities

with probabilities $\widehat{p_{1,2}}=\frac{1-x}{2_{x_{1,2}}(1-x,h)}.$ Thus we obtain a correct (i.e., with values in $[0,1]$ ) approximation $\stackrel{ˆ}{{\mathit{\text{S}}}_h^x}$ taking the values

Now for the initial equation (1), we obtain an approximation $\widehat{X_h^x}$ by a simple "splitstep" procedure (again, see, e.g., [4] or [3]):

Now we can state the following:

Theorem 1.Let $\widehat{X_t^x}$ be the discretization scheme defined by one-step approximation (10). Then $\widehat{X_t^x}$ is a first-order weak approximation of equation (1) for functions $\textcolor{red}{}f\in C_{*}^{\infty }[0,1].$

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:

Theorem 2. Let $f\in C_{*}^{\mathrm{\infty }}[0,1].\mathit{The}u(t,x):=Ef\left(X_{\iota }^x\right)isa{C}^{\mathrm{\infty }}$ function on $[0,1]\mathrm{x}\mathrm{R}$ that solves the backward Kolmogorov equation

In particular,

Such theorem is stated for $\textcolor{red}{}f\in C^{\mathrm{\infty }}[0,1]$ in $\textcolor{red}{}[1,\mathrm{Thm}.6.1.12],$ 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 $X_t^x,$ which show that they grow slower than factorials. The recurrent relations of the moments $E\left[{\left(X_t^x\right)}^k\right]$ show that they are infinitely differentiable with respect to $t$ and $x,$ which allows us infinitely differentiate the series

termwise with respect to $t$ and $x,$ where $f\left(x\right)={\sum }_{k=0}^{\infty }{c}_{{kx}^k}$ is the Taylor expansion of $f.$

We illustrate our approximation for $f\left(x\right)=x^4$ and $f\left(x\right)=\exp \{-x\}.$ Since we do not explicitly know the moments $E\exp \{-X_t^x\},$we use the approximate equality $\exp \{-x\}\approx 1-x+\frac{x^2}{2}-\frac{x^3}{6}+\frac{x^4}{24}.$ In Figs. 1 and 2, we compare the moments $Ef\left(\widehat{X_t^x}\right)$ and $Ef\left(x_t^x\right)$ as functions of $t$ $(\mathrm{left\; plots},h=0.001)$ and as functions of discretization step $h$ $(\mathrm{right\; plots},t=1).$ As expected, the approximations agree with exact values pretty well.