Variable Step Size Adams Methods for BSDEs

. For backward stochastic diﬀerential equations (BSDEs), we construct variable step size Adams methods by means of Itˆo–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coeﬃcients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and suﬃcient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.


Introduction
In 1973, Bismut [1] introduced the linear BSDEs. Until 1990, the well-posedness result of nonlinear BSDEs was rigorously proved by Pardoux and Peng [2][3][4]. After boomingly developed for three past decades, BSDEs become a vital tool to formulate many problems such as mathematical finance [5], partial differential equations [4], actuarial and financial [6], risk measures [7], and finance [8]. However, the theory of nonlinear BSDEs indicates that a majority of nonlinear BSDEs do not have analytical solutions [9]. us, the main purpose of this paper is to design a new numerical scheme to solve the following BSDE: where T > 0 denotes a fixed terminal time and W is a d-dimensional Brownian motion defined on a filtered complete probability space (Ω, F, (F t ) 0≤t≤T , P); Φ(W T ): R d ⟶ R m is a given terminal condition of BSDE, and f(t, y, z): [0, T] × R m × R m×d ⟶ R m is the generator function. In addition, they satisfy the following. where C k b are the set of all k-times continuously differential functions with all partial derivatives bounded. e papers with respect to numerical solutions of BSDEs are unlikely to list exhaustively because there is a vast literature. erefore, we recommend milestone papers to readers with respect to time-discretization of BSDEs. e paper [10] was the first work of designing efficient algorithms for BSDEs. After that, a modified and implementable numerical scheme was adopted to calculate BSDEs in [11]. In the meantime, the Malliavin calculus and Monte Carlo methods were utilized by [12] to discretize BSDEs. e empirical regression method was constructed by [13] for BSDEs. e papers [14,15] presented the θ-scheme to discretize BSDEs. e forward Picard iterations method was designed by [16]. e cubature method was used to solve BSDEs in [14,17]. In [15], authors proposed the BCOS method based on the Fourier cosine series expansions to approximate the solutions of BSDEs. e stochastic grid binding method [18] was introduced to solve BSDEs. e authors in [19] proposed the branching diffusion method for BSDEs, and the branching techniques do not suffer from the curse of dimensionality. A deep learning method was constructed to solve BSDEs in [20,21].
is method could also overcome the curse of dimensionality and deal with the numerical solutions of BSDEs via the Euler scheme under the condition of minimizing the global loss function. e papers [22,23] improved the deep learning method via solving the fixed point problem.
From the above review, the time-discretization of BSDEs can adopt low-order schemes or high-order schemes. Notice that the Euler schemes, the θ-schemes, and the multistep schemes are constant variable step size. And there are a large number of documents about the constant variable step size schemes.
is implies that the theory of implementable numerical methods of BSDEs is booming. e variable step size numerical methods play a vital role in the field of numerical methods of stochastic differential equations (see [28,29]) while they are not seen in the field of numerical theory of BSDEs. us, for this motivation, this paper is to provide novel high-order nonlinear discretization schemes called variable step size Adams scheme (14) by utilizing Itô-Taylor expansion. Note that our high-order nonlinear scheme is always explicit with respect to Z. We provide conditions of local truncation errors with respect to Y and Z reaching high order (see Lemmas 3 and 4). Moreover, a sufficient and necessary condition for the stability of our schemes is derived (see eorem 1). Finally, we derive the convergence of our schemes (see eorem 3). To the best of our knowledge, this is the first attempt to come up with a variable step size numerical scheme for BSDEs. Note that the developed schemes can be also applied to solve decoupled forward-backward stochastic differential equations, and the forward stochastic differential equation can be approximated by using an appropriate scheme. e main contributions are as follows. (i) We derive the variable step size Adams scheme for BSDEs by means of Itô-Taylor expansion. And this scheme is a novel high-order nonlinear time-discretization scheme. (ii) e stability and high-order discretization property of our schemes are rigorously proved. Note that we present a sufficient and necessary condition for the stability of our schemes. (iii) e constant variable one-step size schemes [11,15,18,24] and the constant variable multistep size schemes [16,26,27] are the particular cases of our variable step size Adams scheme.
An outline of this paper is as follows. In Section 2, we present two lemmas that can be used in the following sections. e variable step size Adams schemes of BSDEs are demonstrated in Section 3. Section 4 shows the stability and convergence of the variable step size Adams scheme. In Section 5, numerical experiments are carried out to illustrate the theoretical consequences. In the end, Section 6 is devoted to the conclusion of this paper.

Preliminaries
For readers' convenience, here we present two lemmas which will be utilized in the sequel part.
Lemma 1 (see [3,4]). Assume that functions f and Φ are uniformly Lipschitz with respect to (y, z) and 1/2-Hölder continuous with respect to t. In addition, assume Φ is of class for some κ ∈ (0, 1). en, the solution (Y t , Z t ) of the BSDE in (4) can be represented as where u ∈ C 1,2 b ([0, T] × R d ) satisfies the parabolic PDE as follows: For readers' convenience, we introduce some symbols before providing the lemma. For a multi-index with finite length α, let ℓ(α) be the length of a multi-index of α; A α is the set of all functions v:

Variable Step Size Adams Methods
In this part, we introduce the variable step size Adams schemes of BSDEs in detail. Now, we deduce the variable step size Adams schemes of BSDEs with respect to Y. A discretization π � t 0 , t 1 , . . . , t N of the time interval [0, T] is defined with step size h i � t i+1 − t i and h � max 0≤i≤N− 1 h i ; then, we can restate the BSDE (1) as follows: Taking conditional expectations on both sides of (4), we get the result as follows: where coefficients Γ j,i depend on h i for i � N − 1, . . . , 0 and will be given soon; In what follows, we demonstrate the expression with respect to Z. Multiplying (4) by ΔW i,n : � W t i+n − W t i , n ∈ N + and then taking conditional expectation on the derived equation, we obtain where ΔW i,s � W s − W t i . Analogously, we approximate the two integral terms on right-hand side of (9) by the manner as calculating Y t i , namely, where (10) and (11) into (9), we deduce where us, from the two equations (8) and (13), we propose the explicit Adams schemes to solve BSDE (1) as follows. Giving

Theoretical Analysis
Before showing the stability and convergence analysis of the variable step size Adams scheme (14), we first provide a few lemmas.

Lemma 3.
Under Assumption 1, assume that the parameters Γ j,i 0≤j≤k satisfy the relation as follows: en, is a continuous function with respect to s (see eorem 2.2.1 of [31]). en, by taking derivative with respect to s on we obtain the following reference ordinary differential equation: Substituting (18) into (19) and utilizing Itô-Taylor expansion at (t i , W t i ), we have where u t i � u(t i , W t i ). e conclusion is obvious with the help of equation (20). e proof is completed. □ Lemma 4. Under Assumption 1, assume that the parameters Γ j,i 0≤j≤k satisfy are continuous function of s (see eorem 2.2.1 of [31]). Taking derivative with respect to s, we have the ordinary differential equation as follows: 4 Journal of Mathematics us, where the last equality can be verified via relation (2) and integration by parts. Substituting (22) into (23) and utilizing Itô-Taylor expansion at (t i , W t i ), we have e conclusion is obvious with the help of equation (24). e proof is completed. In what follows, we list the numerical expressions with respect to Y and Z by utilizing Lemmas 3 and 4.
(1) If Γ 0,i � 0, the numerical schemes of Y for k � 1, 2, 3 with respect to time are (25) Journal of Mathematics 5 (2) If Γ 0,i ≠ 0, the numerical schemes of Y for k � 0, 1, 2 with respect to time are (3) e numerical expressions with respect to Z are provided for k � 0, 1, 2, namely, Lemma 5. If constraints (15) and (21)   e characteristic polynomials of (14) are given by and Equation (14) is said to fulfil Dahlquist's root condition if (i) e roots of P y (ζ) and P z (ζ) lie on or within the unit circle (ii) e roots on the unit circle are simple . . , N − k, be the timediscretization approximate solution given by (14) and (Y π i , Z π i ) be the solution of its perturbed form (see (31)). en, scheme (14) (14) for e following theorem is devoted to analyze the stability of scheme (14).

Theorem 1. Suppose Assumption 1 and the condition of Lemma 5 hold. en, the variable step size explicit Adams methods (that is, the coefficient Γ 0,i � 0) is numerically stable if and only if its characteristic polynomial (29) satisfies
Dahlquist's root condition.
We complete the proof of the theorem in three steps.
Step 1. From (14) and (31) with respect to Y, one obtains Furthermore, where L f denotes the Lipschitz constant. Squaring equation (33), then from the inequalities Summing over the above inequality from i to N − k, we have

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Step 2. Subtracting (31) from (14) with respect to Z, we obtain By (2) and (3), one can verify that Plugging (37) into (36), we have We rearrange the k-step recursion to a one-step recursion as follows: where To ensure the stability of the k-step scheme, the norm of the matrix A in equation (39) is not more than 1 (see Chapter III.4, Lemma 4.4 in [32]). is can be satisfied if the eigenvalues eig(A) of the matrix A make |eig(A)| ≤ 1 and the eigenvalues are simple if |eig(A)| � 1. In addition, the eigenvalues of A satisfy the root condition by Definition 1. By Dahlquist's root condition, it is possible that there exists a nonsingular matrix D such that ‖D − 1 AD‖ 2 ≤ 1 where ‖ · ‖ 2 denotes the spectral matrix norm induced by Euclidian vector norm in R k×m×d . Hence, we can choose a scalar product for A, A ∈ R k×m×d as 〈A, A〉 * : And we have |A| 2 * : � 〈A, A〉 * with the induced vector norm on R k×m×d . Let ‖A‖ * � ‖D − 1 AD‖ 2 be the induced matrix norm.
Owing to the norm equivalence, we know that there exist positive constants c 1 and c 2 such that where |A| Journal of Mathematics Squaring equation (42), then from the inequalities ( n i�1 a i ) 2 ≤ n n i�1 a 2 i and (a + b) 2 ≤ δ(1 + δ)a 2 + (1 + 1/δ)b 2 , δ > 0, one deduces By the Lipschitz condition of f with respect to (y, z), (43) can be restated as By the Cauchy-Schwarz inequality, we have the following estimates: Summing over the above inequality from i to N − k, we have (46)

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Step 3. Adding (35) to (46), we obtain From the discrete Gronwall inequality, we have Moreover, Necessity. e proof is analogous to ordinary differential equations (see eorem 6.3.3 in [33] Proof. e proof of this theorem is analogous to that of eorem 1. us, we omit it here. Next, the convergence property of scheme (14) is given in the theorem as below. (1) and solutions of the variable step size Adams methods (14), respectively.

Theorem 3. Suppose Assumption 1 and the condition of
e terminal values satisfy where C is a constant changing from line to line.
Proof. e proof is obvious with the help of eorem 1 and Lemmas 3 and 4.

Numerical Experiments
In this section, we demonstrate the theory results of scheme (14) via numerical examples. First, we choose a method to approximate the conditional mathematical expectations numerically. Among the popular methods, we focus on the least squares Monte Carlo (LSMC) method (see [13,27] Take T � 1, d � 5, and M � 100000. Figure 2 presents the relationship of the absolute error between the numerical solution and the exact solution of BSDE (59) with respect to Y at time 0 and the number of time points via the variable step size scheme (53), the variable step size scheme (54), the constant variable step size scheme (55), and the constant variable step size scheme (56). Figure 2 implies that the variable step size schemes possess almost the same convergence rates as the constant variable step size scheme, but the number of steps of the variable step size schemes is smaller than that of the constant variable step size schemes.

Conclusions
We design in this article the variable step size Adams schemes to calculate the solution of BSDEs. To enrich the numerical schemes with high-order convergence, we infer the conditions with respect to the coefficients in the numerical method by means of Itô-Taylor expansion. We then provide a sufficient and necessary condition for the stability of our variable step size Adams schemes via Dahlquist's root condition.
e convergence of the proposed scheme is analyzed too. We illustrate the theory results of our numerical algorithms with two examples. Finally, note that the constant variable one-step size schemes [11,15,18,24] and the constant variable multistep size schemes [16,26,27] are the particular cases of our variable step size Adams schemes.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest.