Application of B-theory for numerical method of functional differential equations in the analysis of fair value in financial accounting

Let the derivative y′(x_n) of function y(x) at point x_n be replaced by a two-point expression, i.e., (2) y'(xn)≈y(xn+1)−y(xn)h {y^\prime }({x_n}) \approx {{y({x_{n + 1}}) - y({x_n})} \over h} Then use y_n to replace y(x_n) approximately, then the initial value problem in Eq. (2) becomes (3) {yn+1=yn+hf(xn,yn)y0=y(x0),n=0,1,2,⋯ \left\{ {\matrix{{{y_{n + 1}} = {y_n} + hf({x_n},{y_n})} \hfill \cr {{y_0} = y({x_0}),n = 0,1,2, \cdots } \hfill \cr } } \right. Equation (2) is the famous B-theoretical formula. Figure 1 shows the B-theoretical image representation.

Fig. 1

The B-theoretical method has a simple form and low accuracy. In order to improve the accuracy, the two ends of the equation y′ = f (x,y) are integrated over the interval [x_n,x_n+1]. (4) y(xn+1)=y(xn)+∫xnxn+1f[x,y(x)]dx y({x_{n + 1}}) = y({x_n}) + \int_{{x_n}}^{{x_{n + 1}}} f[x,y(x)]dx Use the trapezoid method to calculate its integral term, i.e., (5) ∫xnxn+1f[x,y(x)]dx≈xn+1−xn2[f(xn,y(xn))+f(xn+1,y(xn+1))] \int_{{x_n}}^{{x_{n + 1}}} f[x,y(x)]dx \approx {{{x_{n + 1}} - {x_n}} \over 2}\left[ {f({x_n},y({x_n})) + f({x_{n + 1}},y({x_{n + 1}}))} \right] Substituting into Eq. (3) and replacing y_n in the equation with y(x_n), we obtain the following trapezoidal formula; (6) yn+1=yn+h2[f(xn,yn)+f(xn+1,yn+1)] {y_{n + 1}} = {y_n} + {h \over 2}\left[ {f({x_n},{y_n}) + f({x_{n + 1}},{y_{n + 1}})} \right] Because the trapezoidal formula of numerical integration is more accurate than the rectangular formula, the trapezoidal formula in Eq. (6), which is a numerical method, is more accurate than the B-theory (3). The right end of Eq. (6) contains the unknown factor y_n+1, which is a function equation about y_n+1. This type of method is called an implicit method. Figure 2 shows a schematic diagram of a trapezoidal formula [7].

Fig. 2

The trapezoidal formula requires multiple iterations in actual calculation, so the calculation amount is large. Practically, the trapezoidal formula in Eq. (6) is only iterated once, i.e., the estimated value B of y_n+1 is first calculated using y¯n+1 {\overline y _{n + 1}} ,-theory, and then using the trapezoidal formula Eq. (6) to perform an iteration to obtain the correction value y_n+1, i.e., (7) {y¯n+1=yn+hf(xn,yn)yn+1=yn+h2[f(xn,yn)+f(xn+1,y¯n+1)] \left\{ {\matrix{{{{\overline y }_{n + 1}} = {y_n} + hf({x_n},{y_n})} \hfill \cr {{y_{n + 1}} = {y_n} + {h \over 2}\left[ {f({x_n},{y_n}) + f({x_{n + 1}},{{\overline y }_{n + 1}})} \right]} \hfill\cr } } \right.

A major criterion for measuring the quality of a solving formula is the accuracy of the solving formula, so the concepts of local truncation error and order are introduced.

For the B-theory, assuming y_n = y(x_n), then (8) yn+1=y(xn)+h[f(xn,y(xn))]=y(xn)+hy'(xn) {y_{n + 1}} = y({x_n}) + h\left[ {f({x_n},y({x_n}))} \right] = y({x_n}) + h{y^\prime }({x_n}) And the true solution y(x) according to the second-order Taylor expansion at x_n, has (9) yn+1=y(xn)+hy'(xn)+h22!y‴(ξ), ξ∈(xn,xn+1) {y_{n + 1}} = y({x_n}) + h{y^\prime }({x_n}) + {{{h^2}} \over {2!}}{y^{'''}}(\xi ),\quad \xi \in ({x_n},{x_{n + 1}}) So, there is (10) y(xn+1)−yn+1=h22!y″(ξ) y({x_{n + 1}}) - {y_{n + 1}} = {{{h^2}} \over {2!}}{y^{''}}(\xi ) If the local truncation error of the numerical method is O(h^p+1), then the order of this numerical method is called P. The smaller the step size (h <1), the higher is the P, the smaller is the local truncation error, and the higher is the calculation accuracy [8].

B-theory can be rewritten as follows: (11) {yn+1=yn+hK1K1=f(xn,yn) \left\{ {\matrix{{{y_{n + 1}} = {y_n} + h{K_1}} \hfill \cr {{K_1} = f({x_n},{y_n})} \hfill\cr } } \right. Then the expression of y_n+1 is exactly the same as the first two terms of Taylor's expansion of y(x_n+1), i.e., the local truncation error is O(h²). The improved B-theory can be rewritten as follows: (12) {yn+1=yn+h2(K1+K2)K1=f(xn,yn)K2=f(xn+1,yn+hk1) \left\{ {\matrix{{{y_{n + 1}} = {y_n} + {h \over 2}({K_1} + {K_2})} \hfill \cr {{K_1} = f({x_n},{y_n})} \hfill\cr {{K_2} = f({x_{n + 1}},{y_n} + h{k_1})} \hfill\cr } } \right. The above two sets of formulas have one thing in common in their form: they both use the linear combination of the values of f (x,y) at some points to obtain the approximate value y_n+1 of y(x_n+1), and increase the number of calculations of f (x,y), which can improve the order of truncation error. In the B-theory, the value of f (x,y) is calculated once per step, which is a first-order method. The improved B-theory needs to calculate the value of f (x,y) twice, which is a second-order method. Its local truncation error is O(h³).

Generally, Runge-Kutta method sets the approximate formula as (13) {yn+1=yn+h∑i=1pciKiK1=f(xn,yn)Ki=f(xn+aih,yn+h∑j=1i−1bijKj)(i=2,3,⋯,p) \left\{ {\matrix{{{y_{n + 1}} = {y_n} + h\sum\limits_{i = 1}^p {c_i}{K_i}} \hfill\cr {{K_1} = f({x_n},{y_n})} \hfill\cr {{K_i} = f({x_n} + {a_i}h,{y_n} + h\sum\limits_{j = 1}^{i - 1} {b_{ij}}{K_j})} \hfill\cr } } \right.\left( {i = 2,3, \cdots ,p} \right) Among them, a_i, b_ij, and c_i are parameters. The principle to determine them is to make the Taylor expansion of the approximate formula at y_m and the Taylor expansion of y(x) at x_n coincide as much as possible, so that the approximate formula has as much high accuracy as possible. With this, we can get the commonly used third-and fourth-order Runge-Kutta formulas through a complicated calculation process, as shown in Figure 3. (14) {yn+1=yn+h6(K1+4K2+K3)K1=f(xn,yn)K2=f(xn+h2,yn+h2K1)K3=f(xn+h,yn−hK1+2hK2) \left\{ {\matrix{{{y_{n + 1}} = {y_n} + {h \over 6}({K_1} + 4{K_2} + {K_3})} \hfill\cr {{K_1} = f({x_n},{y_n})}\hfill \cr {{K_2} = f({x_n} + {h \over 2},{y_n} + {h \over 2}{K_1})} \hfill\cr {{K_3} = f({x_n} + h,{y_n} - h{K_1} + 2h{K_2})} \hfill\cr } } \right. With (15) {yn+1=yn+h6(K1+2K2+2K3+K4)K1=f(xn,yn)K2=f(xn+h2,yn+h2K1)K3=f(xn+h2,yn+h2K2)K4=f(xn+h,yn+hK3) \left\{ {\matrix{{{y_{n + 1}} = {y_n} + {h \over 6}({K_1} + 2{K_2} + 2{K_3} + {K_4})} \hfill\cr {{K_1} = f({x_n},{y_n})} \hfill\cr {{K_2} = f({x_n} + {h \over 2},{y_n} + {h \over 2}{K_1})} \hfill\cr {{K_3} = f({x_n} + {h \over 2},{y_n} + {h \over 2}{K_2})} \hfill\cr {{K_4} = f({x_n} + h,{y_n} + h{K_3})} \hfill\cr } } \right. Equation (15) is called the classic Runge-Kutta method.

Fig. 3

Differential equation: (20) dxdy=f(x,y),x∈[a,b] {{dx} \over {dy}} = f(x,y),x \in [a,b] Integrating both ends from x_n to x_n+1, we get (21) y(xn+1)=y(xi)+∫xnxn+1f(x,y(x))dx y({x_{n + 1}}) = y({x_i}) + \int_{{x_n}}^{{x_{n + 1}}} f(x,y(x))dx We replace the integrand on the right with an interpolation polynomial, as shown in Figure 4.

Fig. 4

Adams extrapolation selects k points x_n, x_n−1,=, ⋯, x_n−k+1 The k – 1 order polynomial in Adams extrapolation method used as the interpolation base point to construct f (x,y) is calculated as follows [9]: (22) yn+1=yn+h∑j=0k−1aj∇jfm {y_{n + 1}} = {y_n} + h\sum\limits_{j = 0}^{k - 1} {a_j}{\nabla ^j}{f_m} where a_j satisfies the following algebraic recursion: (23) aj+12aj−1+13aj−2+⋯+1j+1a0=1 j=0,1,⋯ {{\rm{a}}_{\rm{j}}} + {1 \over 2}{a_{j - 1}} + {1 \over 3}{a_{j - 2}} + \cdots + {1 \over {j + 1}}{a_0} = 1\quad j = 0,1, \cdots According to this recursive formula, a_j (j = 0,1, ⋯) can be calculated one by one. Table 1 gives some values of a_j.

According to the interpolation theory, the choice of the interpolation node directly affects the accuracy of the interpolation formula, and the accuracy of the interpolation formula of the same degree is higher than that of the extrapolation formula. The Adams interpolation formula for the initial value problem in Eq. (1) can be derived as follows: (24) y(xn+1)=y(xn)+∫xnxn+1Ln,k−1(p)(x)dx y({x_{n + 1}}) = y({x_n}) + \int\limits_{{x_n}}^{{x_{n + 1}}} L_{n,k - 1}^{(p)}(x)dx The above formula degenerates into the interpolation formula at p = 0, as shown in Figure 5.

Fig. 5

In addition to the known value of f at x_n−k+1, ⋯, x_n, the interpolation formula in Eq. (24) also contains the unknown value at point x_n, ⋯, x_n+p, so the interpolation Eq. (9) only gives the relationship of the unknown quantity y_n, ⋯, y_n+p. Actual calculation is still necessary to solve the simultaneous equations. The interpolation formula of p = 1 is the most applicable. Ln,k−1(1)(x) L_{n,k - 1}^{(1)}(x) uses the Newton backward interpolation formula to obtain the Adams interpolation formula. (25) yn+1=yn+h∑j=0ka*j∇jfm {y_{n + 1}} = {y_n} + h\sum\limits_{j = 0}^k {a^*}_j{\nabla ^j}{f_m} where the coefficient a^*_j is defined as (26) a*j=(−1)j∫−10(−τj)dτ j=0,1,⋯ {a^*}_j = ( - {1)^j}\int_{ - 1}^0 \left( {\matrix{{ - \tau } \cr j \cr } } \right)d\tau \quad j = 0,1, \cdots Here, a^*_j satisfies the following algebraic recursion: (27) a*j+12a*j−1+13a*j−2+⋯+1j+1a*0={1,j=00,j>0 {a^*}_j + {1 \over 2}{a^*}_{j - 1} + {1 \over 3}{a^*}_{j - 2} + \cdots + {1 \over {j + 1}}{a^*}_0 = \left\{ {\matrix{{\matrix{{1,j = 0} \cr {0,j > 0} \cr } } \cr } } \right.

According to this recursive formula, a^*_j (j = 0,1, ⋯) can be calculated one by one. Table 2 gives some values of a^*_j: