Second-Order Weak Approximations of CKLS and CEV Processes by Discrete Random Variables

: In this paper, we construct second-order weak split-step approximations of the CKLS and CEV processes that use generation of a three-valued random variable at each discretization step without switching to another scheme near zero, unlike other known schemes (Alfonsi, 2010; Mackeviˇcius, 2011). To the best of our knowledge, no second-order weak approximations for the CKLS processes were constructed before. The accuracy of constructed approximations is illustrated by several simulation examples with comparison with schemes of Alfonsi in the particular case of the CIR process and our ﬁrst-order approximations of the CKLS processes (Lileika– Mackeviˇcius, 2020). 65C30


Introduction
We are interested in weak second-order approximations for the Chan-Karolyi-Longstaff-Sanders (CKLS) model [1] dX t = (θ − β X t ) dt + σX γ t dB t , X 0 = x ≥ 0, with parameters (θ, β, σ, γ) ∈ R + × R × R + × [1/2, 1), where R + := (0, ∞) and R + := [0, ∞). In particular, when θ = 0 and β < 0, we have the constant elasticity of variance (CEV) model [2], and when γ = 1/2 and β ≥ 0, we have the well-known Cox-Ingersoll-Ross (CIR) model [3]. The main problem in developing numerical methods for such a diffusion equation/model is that the diffusion coefficient has unbounded derivatives near zero, and therefore standard methods (see, e.g., Milstein and Tretyakov [4]) are not applicable: discretization schemes that (explicitly or implicitly) involve the derivatives of the coefficients usually lose their accuracy near zero, especially for large σ. This problem for the CIR processes was solved by modifying the scheme considered by switching near zero to another scheme, which is (1) sufficiently regular and (2) sufficiently accurate near zero; we refer, for example, to [5][6][7] and references therein. Typically, a second-order approximation near zero is constructed by discrete random variables matching three or four moments with those of the true solution.
The main result of this paper is the construction of second-order weak split-step approximations of the CKLS and CEV processes by discrete random variables. To the best of our knowledge, no second-order weak approximations of the CKLS process have been constructed before, except for the particular case of the CIR process (Alfonsi [5], Mackevičius [7]). Our construction method is significantly different from that of the firstorder approximation in our previous paper [8]. Another novel feature of our result is that in our schemes, no switching between schemes near zero is used, in contrast to [5,7]. This simplifies the implementation of approximations. We illustrate the accuracy of our

Preliminaries
In this section, we give some definitions for the general one-dimensional stochastic differential equation We assume that the equation has a unique weak solution X x t such that P(X x t ∈ D, t ≥ 0) = 1 for all x ∈ D. For example, for Equation (1), we can take D = R + .
Having a fixed time interval [0, T], consider an equidistant time discretization ∆ h = {ih, i = 0, 1, . . . , [T/h], h ∈ (0, T]}, where [a] is the integer part of a. By a discretization scheme of Equation (2) we mean a family of discrete-time homogeneous Markov chainŝ X h = {X h (x, t), x ∈ D, t ∈ ∆ h } with initial valuesX h (x, 0) = x and one-step transition probabilities p h (x, dz), x ∈ D, in D. For convenience, we only consider steps h = T/n, n ∈ N. For brevity, we writeX x t orX(x, t) instead ofX h (x, t). Note that because of the Markovity, a one-step approximationX x h of the scheme completely defines the distribution of the whole discretization schemeX x t , so that we only need to construct the former. We denote by for some sequence (C n , k n ) ∈ R + × N 0 . Following [5], we say that such a sequence {(C n , k n ), n ∈ N 0 } is a good sequence for f . We will write g( If, in particular, the function g is expressed in terms of another function f ∈ C ∞ pol (R) and the constants C, k, and h 0 only depend on a good sequence for f , then we will write, instead, g(x, h) = O(h n ).

Definition 1.
A discretization schemeX h is a weak νth-order approximation for the solution (X x t , t ∈ [0, T]) of Equation (2) if for every f ∈ C ∞ 0 (D), there exists C > 0 such that Definition 2. Let L f = b f + 1 2 σ 2 f be the generator of the solution of Equation (2).
A discretization schemeX x t is a local νth-order weak approximation of Equation (2) if for all f ∈ C ∞ pol (D) and x ∈ D.
which motivates Definition 2: If L ν+1 f behaves "well" (e.g., b, σ 2 , f ∈ C ∞ 0 (D), and EL ν+1 f is bounded), then for the "one-step" νth-order weak approximation schemeX x h , we have We may expect that in "good" cases, a local νth-order weak discretization scheme is a νth-order (global) approximation. Rigorous statements require certain uniformity of (4) with respect to f and regularity of L.

Definition 3.
A discretization schemeX x t is a potential νth-order weak approximation for Equation (2) if for every f ∈ C ∞ pol (D), We say that a potential νth-order weak approximation is a strongly potential νth-order weak approximation if it has uniformly bounded moments.

Remark 2.
Typically, a strongly potential νth-order discretization is a νth-order weak approximation in the sense of Definition 1. At least, we do not know any counterexample. A rigorous proof for the CIR equation is given by Alfonsi [9] (see also [10]).
We split Equation (1) into the deterministic part and the stochastic part The solution of the deterministic part is positive for all (x, t) ∈ R + × (0, T], namely: The solution of the stochastic part is not explicitly known. The following theorem allows us to reduce the construction of a weak second-order approximation to that of the stochastic part. LetŜ x t =Ŝ(x, t) be a discretization scheme for the stochastic part (5).
. LetŜ x t be a potential second-order weak approximation of the stochastic part (5) of Equation (1). Then the (split-step) composition defines a potential second-order weak approximation of Equation (1).

Corollary 1.
IfŜ x t is a strongly potential second-order weak approximation of the stochastic part (5) of Equation (1), then composition (6) is a strongly potential second-order weak approximation of Equation (1).
The theorem and corollary allow us to restrict ourselves, without loss of generality, on the (strongly) potential second-order weak approximations of the stochastic part dS x t = σ(S x t ) γ dB t of Equation (1).

A Strongly Potential Second-Order Approximation of the Stochastic Part of the CKLS and CEV Equations
LetŜ x h be any discretization scheme. Denote a := σ 2 . Using Taylor's formula for f ∈ C 6 (R), we get It is worth noting that further technical calculations were mainly made by using MAPLE software.
Since the first and second power of the generator of stochastic part (5) are (see Definition (3)) and For brevity, we denote z := ah = σ 2 h. By the above expression of the remainder Initially, for constructing our approximations, instead of (12), we will require a slightly weaker condition Later, we will see that, actually, all our approximations satisfy the required stronger condition (12).
We easily convert conditions (7)-(11) and (13) for the central moments ofŜ x h into conditions for the noncentral moments: where the "moments"

A Strongly Potential Second-Order Approximation of the CIR Equation
In this section, we construct a strongly potential second-order approximation for the CIR Equation (γ = 1/2) using a three-valued random variable at each generation step without switching to another scheme in a neighborhood of zero. The "moments" (15) in conditions (14) for the central moments E(Ŝ x h ) i in this case become as follows (recall that z := ah = σ 2 h):m We therefore look for approximationsŜ x h taking three positive values x 1 , x 2 , and x 3 with probabilities p 1 , p 2 , and p 3 such that where x ≥ 0, h > 0, together with obvious requirement We have (see [8], Appendix) Solving the system with respect to unknowns x 1 , x 2 , and x 3 , we get: We can get analogous expressions from the last three equations of system (17) (with m 4 , m 5 , m 6 instead of m 1 , m 2 , m 3 ). However, trying to directly solve the obtained six equations with respect to all unknowns x 1 , x 2 , x 3 , p 1 , p 2 , p 3 gave no satisfactory results. In view of the form of approximations presented by Alfonsi [5] and Mackevičius [7] for the CIR equation and of our first-order approximations for the CKLS equations [8], after a number of experiments, we arrived at the following conclusions: • the values of the discretization schemeŜ x h may be chosen of the form (20), together with ensuring the nonnegativity of the solution {x 1 , x 2 , x 3 , p 1 , p 2 , p 3 }, still is a rather technical and long task, even with the help of MAPLE. Note that the right-hand sides O(h 3 ) in conditions (17) give us certain flexibility in finding relatively simple expressions of solutions.
This way we get a family of second-order discretization schemesŜ x h depending on the parameter A ∈ [3/4, 3/2]: Theorem 2. LetX x t be the discretization scheme defined by composition (6), whereŜ x h takes the values x 1 , x 2 , and x 3 defined in (21) with probabilities p 1 , p 2 , and p 3 defined in (19) (Ŝ 0 h = 0). ThenX x t is a strongly potential second-order discretization scheme for the CIR equation.
Proof. Let us first check that for all x ≥ 0 and z > 0. This is equivalent to which in turn is equivalent to This implies that x 1 ≥ 0 for all x, z ≥ 0, provided that A ≥ 3/4. Obviously, x 2 , x 3 ≥ x 1 ≥ 0. Now let us check the nonnegativity of p 1 , p 2 , and p 3 . For p 1 , we have where x ≥ 0, z > 0. We have already checked the nonnegativity of The positivity of (4A + 3) 2 z + 48x)z − 3z is obvious, and 4A 2 − 5A + 3 > 0 for all A ∈ R. Thus, clearly, p 1 ≥ 0 if A ≥ 1. Now let A < 1. Then p 1 ≥ 0 if and only if which clearly holds for all x ≥ 0 and z > 0 if A ∈ [3/4, 3/2]. Thus, p 1 ≥ 0 for x ≥ 0 and z > 0 if A ∈ [3/4, 3/2]. For p 2 , we obviously have Finally, for p 3 , we have for x ≥ 0 and z > 0. The numerator is obviously positive, and the nonnegativity of the denominator follows similarly to that of p 1 .
Let us check that, indeed, the central moments ofŜ x h satisfy conditions (7)-(12) (with γ = 1/2). The first three are obvious, since the moments of the random variableŜ x h exactly match the three first moments of S x h , so they also match the first three central moments: Conditions (10), (11), and (13) are satisfied, since, respectively, Finally, by the last relation and the expression of the maximal value x 3 ofŜ x h , condition (12) is satisfied for every f ∈ C ∞ pol (D) (suppose | f (6) (x)| ≤ C 6 (1 + x k 6 )): It remains to check that the discretization schemeŜ x h has uniformly bounded moments, that is, that there exists h 0 > 0 such that By elementary but tedious calculations, we arrive at the following expression for the moments: where the constant C > 0 depends on p and σ, from which the boundedness of the moments of the approximation follows in a standard way (see [5] [Prop. 1.5]).

Remark 3.
(Third-order approximation for the stochastic part of the CIR equation) By a similar procedure, we can obtain a strongly potential third-order weak approximation of the stochastic part (5) of the CIR Equation (1) (γ = 1/2). Although composition (6) then theoretically gives only second-order approximation, numerical simulations show that, practically, it gives a slightly better accuracy of approximation than with second-order approximation of the stochastic part.
We look at a discretization schemeŜ x h taking four values x 1 , x 2 , x 3 , x 4 with probabilities p 1 , p 2 , p 3 , p 4 such that and Its solution with respect to x 1 , x 2 , x 3 , and x 4 is as follows: , , , Again, after a number of experiments, we chose to look for a solution of (22) and (23), together with ∑ i p i = 1 and p i ≥ 0, in the form with parameters A 1 , A 2 , B 1 , B 2 , C 1 , C 2 > 0 and probabilities p 1 , p 2 , p 3 , p 4 defined in (24). The main difficulty was obtaining a nonnegative solution {x 1 , The final result is a strongly potential third-order weak approximationŜ x h of the stochastic part (5) of the CIR equation taking the four values with the corresponding probabilities p i , i = 1, 2, 3, 4, given by (24).

A Strongly Potential Second-Order Approximation of the CKLS Equations
In this section, we apply to the CKLS equations the method of constructing secondorder approximations used in the previous section in the CIR case. As an example, we present strongly potential second-order approximations in the cases γ = 3/4 and γ = 5/6, where the results look relatively simple.
Let γ = 3/4 in the CKLS Equation (1). Then for the stochastic part dS x t = σ(S x t ) 3/4 dB t , S x 0 = x ≥ 0, we have (see [8], Appendix) In [8], we have constructed a strongly potential first-order two-valued approximation of the stochastic part with (26) In particular, for γ = 3/4, This motivated us to look for the second-order approximations with values of the following form: with probabilities (19). Using the same method as in the CIR case, after tedious and rather complex calculations, we arrived at the scheme with values x 2 = x + 11 8 x 1/2 z + 15 64 z 2 , and probabilities p 1 , p 2 , and p 3 defined in (19). Similarly, in the case γ = 5/6, we have The corresponding approximation takes the values with probabilities p 1 , p 2 , and p 3 defined in (19).

Simulation Examples
We indicate a particular γ of the stochastic part (5) by the left subscript γ as in γ S x t . We first give a short algorithm for calculatingX (i+1)h givenX ih = x at each simulation step i:

2.
Draw a uniform random number U in the interval [0, 1].
In the legends of figures, we use the following notation.

Conclusions
We have constructed second-order weak split-step approximations of the Chan-Karolyi-Longstaff-Sanders (CKLS) and constant elasticity of variance (CEV) processes. The approximations use generation of a three-valued random variable at each discretization step. To illustrate the accuracy of constructed approximations, we performed several simulations with different parameters and test functions. Our method can be applied to constructing second-order weak approximations for other stochastic differential equations. It would be interesting to construct third-order weak approximations for the CKLS equations, as we did for the CIR equation.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: A function of polynomial growth with respect to h n , i.e., we write g(x, h) = O(h n ) if for some C > 0, k ∈ N, and h 0 > 0, |g(x, h)| ≤ C(1 + |x| k )h n , A function of polynomial growth with respect to h n when the function g is expressed in terms of another function f ∈ C ∞ pol (D) and the constants C, h 0 , and k depend on a good sequence for f only