Variable Step Size Adams Methods for BSDEs
For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to and 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.
In 1973, Bismut  introduced the linear BSDEs. Until 1990, the well-posedness result of nonlinear BSDEs was rigorously proved by Pardoux and Peng [2–4]. After boomingly developed for three past decades, BSDEs become a vital tool to formulate many problems such as mathematical finance , partial differential equations , actuarial and financial , risk measures , and finance . However, the theory of nonlinear BSDEs indicates that a majority of nonlinear BSDEs do not have analytical solutions . Thus, the main purpose of this paper is to design a new numerical scheme to solve the following BSDE:where denotes a fixed terminal time and is a -dimensional Brownian motion defined on a filtered complete probability space ; is a given terminal condition of BSDE, and is the generator function. In addition, they satisfy the following.
Assumption 1. , and where are the set of all -times continuously differential functions with all partial derivatives bounded.
The papers with respect to numerical solutions of BSDEs are unlikely to list exhaustively because there is a vast literature. Therefore, we recommend milestone papers to readers with respect to time-discretization of BSDEs. The paper  was the first work of designing efficient algorithms for BSDEs. After that, a modified and implementable numerical scheme was adopted to calculate BSDEs in . In the meantime, the Malliavin calculus and Monte Carlo methods were utilized by  to discretize BSDEs. The empirical regression method was constructed by  for BSDEs. The papers [14, 15] presented the -scheme to discretize BSDEs. The forward Picard iterations method was designed by . The cubature method was used to solve BSDEs in [14, 17]. In , authors proposed the BCOS method based on the Fourier cosine series expansions to approximate the solutions of BSDEs. The stochastic grid binding method  was introduced to solve BSDEs. The authors in  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]. This 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. The papers [22, 23] improved the deep learning method via solving the fixed point problem.
If the Euler scheme (explicit, implicit, or generalized) is utilized to discretize BSDEs, the order of discretization error is (see [11, 13, 18, 24]). The -schemes [14, 15] are adopted to discretize BSDEs, and the corresponding rate of convergence is 2. To obtain higher-order schemes, the multistep schemes [16, 25–27] are developed to solve BSDEs.
From the above review, the time-discretization of BSDEs can adopt low-order schemes or high-order schemes. Notice that the Euler schemes, the -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. This implies that the theory of implementable numerical methods of BSDEs is booming. The 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. Thus, 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 . We provide conditions of local truncation errors with respect to and reaching high order (see Lemmas 3 and 4). Moreover, a sufficient and necessary condition for the stability of our schemes is derived (see Theorem 1). Finally, we derive the convergence of our schemes (see Theorem 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.
The 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) The 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) 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 scheme.
An outline of this paper is as follows. In Section 2, we present two lemmas that can be used in the following sections. The 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.
For readers’ convenience, here we present two lemmas which will be utilized in the sequel part.
Lemma 1 (see [3, 4]). Assume that functions and are uniformly Lipschitz with respect to and -Hölder continuous with respect to t. In addition, assume is of class for some . Then, the solution of the BSDE in (4) can be represented aswhere satisfies the parabolic PDE as follows:with the terminal condition where .
For readers’ convenience, we introduce some symbols before providing the lemma. For a multi-index with finite length , let be the length of a multi-index of ; is the set of all functions for which is well defined and continuous; denotes the subset of all functions such that the function is bounded; for positive integer , is the set of functions such that for all .
Lemma 2 (see Proposition 2.2 in ). Let . Then, for a function , where ; , .
3. 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 . A discretization of the time interval is defined with step size ; then, we can restate the BSDE (1) as follows:
Taking conditional expectations on both sides of (4), we get the result as follows:where . It is straightforward that the integrand in the above equation is a deterministic function of time . Naturally, we can replace the function in (5) by using multistep methods through the support points where is a given positive integer and satisfies , namely,where coefficients depend on for and will be given soon; . Inserting (6) into (5), we obtain
Hence, the time-discretization of is, for ,
In what follows, we demonstrate the expression with respect to . Multiplying (4) by and then taking conditional expectation on the derived equation, we obtainwhere . Analogously, we approximate the two integral terms on right-hand side of (9) by the manner as calculating , namely,where and . Plugging (10) and (11) into (9), we deducewhere . Hence, the time-discretization of is, for ,
Thus, from the two equations (8) and (13), we propose the explicit Adams schemes to solve BSDE (1) as follows.
Giving the initial values , we solve random variables for fromwhere for ; coefficients , , and depend on for .
Remark 1. If , the scheme with respect to is explicit and for . If , the scheme with respect to is implicit and for .
4. 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 satisfy the relation as follows:Then,
Proof. By Assumption 1, the integrand , is a continuous function with respect to (see Theorem 2.2.1 of ). Then, by taking derivative with respect to onwe obtain the following reference ordinary differential equation:Thus, from the definition of , we haveSubstituting (18) into (19) and utilizing Itô–Taylor expansion at , we havewhere . The conclusion is obvious with the help of equation (20). The proof is completed.
Lemma 4. Under Assumption 1, assume that the parameters satisfyThen, .
Proof. By Assumption 1, the two integrands and , , are continuous function of (see Theorem 2.2.1 of ). Taking derivative with respect to , we have the ordinary differential equation as follows:Thus,where the last equality can be verified via relation (2) and integration by parts. Substituting (22) into (23) and utilizing Itô–Taylor expansion at , we haveThe conclusion is obvious with the help of equation (24). The proof is completed.
In what follows, we list the numerical expressions with respect to and by utilizing Lemmas 3 and 4.(1)If , the numerical schemes of for with respect to time are(2)If , the numerical schemes of for with respect to time are(3)The numerical expressions with respect to are provided for , namely,
Lemma 5. If constraints (15) and (21) and are satisfied, then the coefficients , , and in scheme (14) are bounded.
Proof. From Lemma 3, we know that the coefficients are composed of products and sums of for . Under the condition , it is clear that the coefficients are bounded. Analogously, we derive that coefficients and are bounded.
In what follows, two definitions are introduced to serve the stability of scheme (14).
Definition 1. The characteristic polynomials of (14) are given byand Equation (14) is said to fulfil Dahlquist’s root condition if(i)The roots of and lie on or within the unit circle(ii)The roots on the unit circle are simple
Definition 2. Let , be the time-discretization approximate solution given by (14) and be the solution of its perturbed form (see (31)). Then, scheme (14) is said to be -stable ifwhere denotes a constant which changes from line to line; satisfies a perturbed form of (14) for :where ; sequences and which belong to are random variables.
The following theorem is devoted to analyze the stability of scheme (14).
Theorem 1. Suppose Assumption 1 and the condition of Lemma 5 hold. Then, the variable step size explicit Adams methods (that is, the coefficient ) is numerically stable if and only if its characteristic polynomial (29) satisfies Dahlquist’s root condition.
Proof (sufficiency). Let for . We complete the proof of the theorem in three steps. Step 1. From (14) and (31) with respect to , one obtains Furthermore, where denotes the Lipschitz constant. Squaring equation (33), then from the inequalities and , , one deduces Summing over the above inequality from to , we have Step 2. Subtracting (31) from (14) with respect to , we obtain By (2) and (3), one can verify that Plugging (37) into (36), we have We rearrange the -step recursion to a one-step recursion as follows: where To ensure the stability of the -step scheme, the norm of the matrix in equation (39) is not more than 1 (see Chapter III.4, Lemma 4.4 in ). This can be satisfied if the eigenvalues of the matrix make and the eigenvalues are simple if . In addition, the eigenvalues of satisfy the root condition by Definition 1. By Dahlquist’s root condition, it is possible that there exists a nonsingular matrix such that where denotes the spectral matrix norm induced by Euclidian vector norm in . Hence, we can choose a scalar product for as . And we have with the induced vector norm on . Let be the induced matrix norm. Owing to the norm equivalence, we know that there exist positive constants such that where for . Applying to equation (39), we have Squaring equation (42), then from the inequalities and , , one deduces By the Lipschitz condition of with respect to , (43) can be restated as By the Cauchy–Schwarz inequality, we have the following estimates: Summing over the above inequality from to , we have Step 3. Adding (35) to (46), we obtainFrom the discrete Gronwall inequality, we haveMoreover,Necessity. The proof is analogous to ordinary differential equations (see Theorem 6.3.3 in ). So, we omit it.
Theorem 2. Suppose Assumption 1 and the condition of Lemma 5 hold. Then, the variable step size implicit Adams methods (that is, the coefficient ) is numerically stable if and only if its characteristic polynomial (29) satisfies Dahlquist’s root condition.
Proof. The proof of this theorem is analogous to that of Theorem 1. Thus, we omit it here.
Next, the convergence property of scheme (14) is given in the theorem as below.
Theorem 3. Suppose Assumption 1 and the condition of Lemma 5 hold. Let and be solutions of the BSDE in (1) and solutions of the variable step size Adams methods (14), respectively. The terminal values satisfy . Then, as is small enough,where is a constant changing from