School of Science, Tianjin University of Technology, Tianjin 300384, China
Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250199, China
Author to whom correspondence should be addressed.
Received: 22 July 2019 / Accepted: 5 September 2019 / Published: 11 September 2019
In this paper, we introduce the Lie-point symmetry method into backward stochastic differential equation and forward–backward stochastic differential equations, and get the corresponding deterministic equations.
In general, almost all differential equations are difficult to solve explicitly. Numerical methods are frequently used for obtaining approximate solutions. However, exact solutions are important, because with their help, one can analyze the properties of the equations. The symmetric method is one of the methods used for finding exact solutions.
Symmetric method for differential equations was once a popular method in the study of ordinary differential equations (ODEs) and partial differential equations (PDEs). A general survey of this method can be found in [1,2,3,4], etc.
Simply speaking, symmetric method means a group of transformations that maps a solution of a given system of equations to another solution of the same system. This method was first initiated by Sophus Lie in the early 19th century. He found that the problem of finding the group of transformations (known as Lie group), reduced to solving a system of determining equations for its infinitesimal generators. This method helps to reduce the order of differential equations, and find invariant solutions, etc.
In 1990s, there appeared applications of Lie group theory to stochastic differential equations, for example [5,6,7,8], etc. Since then many papers contribute to this area. The symmetries includes fiber preserving symmetry, generalized Lie-point symmetry, W-symmetry ([9,10]), and random Lie symmetry , etc. They are mostly applied in Itô SDE driven by Brownian motion. There are also papers applying this method in SDE driven by both Brownian motion and Poisson processes, or even general semimartingales .
In this paper, we will introduce the symmetric method into multidimensional backward stochastic differential equations (BSDEs).
In 1990, Pardoux-Peng  generalized the linear backward stochastic differential equations that appears in  when Bismut was solving a stochastic control problem in 1973, and got a generalized and usually nonlinear BSDE. They proved the existence and uniqueness under certain conditions. The BSDE with parameters (simply denoted by BSDE ) is as follows
where T denotes the terminal time, Y the terminal condition, g the generator. The solution of the BSDE is a pair of processes that satisfies the above equation. Since then, BSDE theory, as a new field, gains rapid development both in theory and practical applications, for example, finance, stochastic control, etc. However, to get the exact solution of BSDE is generally difficult work, such that hardly any paper contributes to it.
Lie-point symmetry method, because of its property, can be commonly used in BSDE under any conditions if the solution exists.
When we study Lie-point symmetry of BSDE, it is natural to consider the infinitesimals of spatial y and temporal t. However, z, as a special process that appears in the solutions of BSDE, raises a natural problem, i.e., should z be included in the consideration for the transformation? The answer is no. In fact, since is a process accompanying the , we can get the transformation for z spontaneously as we get the transformation for y. Furthermore, we carefully studied the properties of the derived transformation for z.
In this paper, we consider two cases. In the first case, transformation does not include time change. Therefore the terminal time T is always left invariant. In the second case, time change is considered. However, the transformation of t relies on t only, i.e., we only consider projectable transformation. We have also deduced the transformation for z in both cases, and get the determining equations for the corresponding symmetry.
As with the conserved quantity (or first integral) that appears in SDE, we define the ‘martingale transformation function’, a function that transforms the solutions of BSDEs into martingales. This kind of functions leads to a new method for solving BSDEs.
Finally, we use symmetric method to study the solutions of forward–backward stochastic differential equations (FBSDEs). How to deal with the terminal condition of FBSDE becomes a key point.
This paper is organized as follows. In Section 2, we introduce the basic knowledge about BSDE and the application of symmetry method in SDEs. In Section 3, we study the spatial symmetry for BSDE; then in Section 4, similarly to the notion of conserved quantity, we introduce the so-called martingale transformation function, which turns a BSDE into a martingale and get the conditions, under which multiplied by martingale transformation function, the symmetry is still a symmetry; in Section 5, adding time change, which leads to the transformation of Brownian motion, we get the generalized symmetry. Finally, in Section 6, we study the symmetry transformation on forward–backward stochastic differential equations (FBSDEs).
2. Some Facts about BSDE and Lie-Point Symmetries
In this section, we introduce the basic knowledge about BSDEs and Lie-point symmetries.
We denote by the n-dimensional Euclidean space, equipped with the standard inner product and the Euclidean norm . We also denote by the collection of all real matrices, and for matrix , we denote and , where denotes the trace, represents the transposition of z. We simply denote by without causing any confusion. For f being a function of x, represents the derivative of f with respect to x. It may be a real number if both f and x are -valued; a vector if f is -valued while x is -valued; or a Jacobian matrix if x is -valued and f is -valued function.
We consider the probability space . A d-dimensional Brownian motion w is defined on this space, and denotes the filtration generated by w and satisfies the usual conditions.
For a -field defined on this probability space, : Y is a -valued -measurable random variable satisfying }.
Now we explain BSDE (Equation (1)) in detail. The terminal time , the terminal condition Y is usually assumed to be in the space , the generator . Please note that g is usually assumed to be dependent on in the literature on BSDE. However, in this paper, similarly to most applications of symmetric method in SDE, we assume g being deterministic. Moreover, the integral with respect to w denotes the Itô’s integral. The solution of BSDE (1) is a pair of processes that satisfies the Equation (1), where is continuous in t, a.s., and is -adapted, is -predictable process and usually satisfies .
The existence and uniqueness conditions of BSDE are diverse. For example, the existence and uniqueness conditions:
g is Lipschitz continuous in y and z, i.e., , see for example ;
g is of quadratic growth in z, see for example, [15,16,17,18];
g is uniformly continuous and of at most linear growth in , see ;
the existence conditions: g is continuous in and of at most linear growth, etc., see . Different assumptions of g may correspond to different conditions of . Since the symmetric method only requires the existence of the solutions of BSDEs, we will not specify the conditions.
2.2. Symmetries on SDEs
Consider n-dimensional SDE
where and are smooth functions.
We introduce a projectable vector field X on . It is given by
where and () are functions smooth enough. Here (resp. ) denotes the i-th component of (resp. x) and we use Einstein summation convention. ‘projectable’ means that depends on t alone.
Every vector field (3) generates a flow acting on which we denote by ,
for . The relation between X and is given by
Therefore forms a one-parameter continuous transformation group. For this notion, the readers can refer to . X is also called the infinitesimal generator of .
Here is the definition for symmetry of SDE.
The vector field X given in (3) is a symmetry of SDE (2) if the action given by keeps the solutions of (2) invariant.
Since acts on the variable t according to (5), if , this action, hence, transforms the Brownian motion w to a new process. It is proved that this new process, which we denote by , is again a Brownian motion (see [21,22] for more details).
Generally for variables and the transformation , we denote simply the transformed variables by and , i.e., . Let be a solution of (2). Then Definition 1 means that the vector field X is a symmetry of (2) if and only if the transformed process satisfies the following equation
Please note that and have different initial values. The corresponding initial values and also satisfy the transformation .
2.3. Conserved Quantity
Now we introduce the notion of conserved quantity (i.e., first integral) of SDE.
A function is a conserved quantity (or first integral) of a system of SDEs (2) if it remains constant on the solutions of SDEs.
For this notion, the readers can refer to [5,6,7,8,23] etc.
A system of SDEs (2) with a non-zero diffusion matrix admits two linearly connected symmetries
if an only if function is a conserved quantity (first integral) of the system.
3. Symmetry Transformation I
In this section, we start to introduce the Lie symmetry method into the BSDE theory. Let us consider the m-dimensional BSDE without terminal conditions and terminal time,
We suppose that the generator is smooth enough with respect to .
First, we consider a near-identity change of coordinates, passing from y to via
Using Itô’s formula, we have
where on the right-hand side, the default variables are . We let . Please note that at first order in , we have
We simply denote by and . For g, we have the following deduction
We can deduce that
Thus, we have the following theorem.
The Lie-point spatial symmetries of BSDE (1) are given by with ξ satisfying the determining equations, i.e., for each ,
with summation over and .
According to (4) and (5), X corresponds to the continuous transformation group . We also introduce the corresponding transformation on z generated by X or equivalently . That is
with summation over , for . We will get the following relationships.
It holds that for and generated by (4) and (9) respectively.
We consider the system of linear differential equations with variable coefficients
Please note that is an -dimensional Jacobian matrix. We denote the transition matrix by . Then , the ith column of , is a solution of Equation (10) with initial value , of which the i-th component is 1, and the others are all 0.
For a general vector , the solution of Equation (10) with initial value can be represented as . Therefore the solution of (9) can be represented by .
We only need to prove that , i.e., , the i-th column of , is a solution of (10) with initial value .
In fact, since each is a smooth function of y, is also a smooth function of y and r. Therefore, we have the following deduction
Moreover, . So is a solution of Equation (10) with initial value . Thus, the result follows. The proof is complete. □
With this transformation, the filtration is still generated by the same Brownian motion w and the terminal time does not change. However, the terminal condition changes. Therefore a solution of BSDE is transformed into the solution of BSDE .
Here are several examples of symmetries.
Let , . We get
Rearranging the equation, we get
Letting , the above formula is equivalent to the following equations
Solving the above equations, we get , for any . Thus, the symmetries are as follows
Therefore , . corresponds to transformation functions
Therefore , and . Both and keep the solutions of BSDE (6) invariant. More specifically, transforms the solutions of BSDE to those of BSDE , and transforms solutions of BSDE to those of BSDE .
For , with , Equation (8) is equivalent to the following equations
Solving the above equations, we get that ξ is a constant independent of , i.e., the only symmetry is . The corresponding transformation is .
For , , Equation (8) is equivalent to the following equations
Solving the above equations, we get for any . Therefore, we get two symmetries , . In addition, the corresponding transformations are , and , .
Suppose that g is independent of z. We get equation
The equation is equivalent to
(1) In this paper ξ is assumed to be independent of z. That is because if ξ is a function of z, when we take derivative of ξ with respect z, we will get , which usually does not exist. (2) Equation (8) is a system of equations with independent variables . However, ξ is just a vector function of . Therefore, Equation (8) may be unsolvable because of z.
4. Transformation to Martingale
Inspired by the notion of conserved quantity, we try to find the conserved quantity of BSDE (6). For a function , applying Itô’s formula to , we have
summation over . Therefore, by the definition of conserved quantity, we should let I satisfy the following two equations
Since z can be arbitrary, we have that , for each , and . Therefore, the conserved quantity degenerates to a constant. Thus, there is no nontrivial conserved quantity (or first integral) for BSDE.
However, for BSDE, we can consider another important kind of functions , satisfying that forms a martingale. Thus, I is the solution of the equation . Now we introduce the following definition
A smooth function is called martingale transformation function of a process if it turns a process into a martingale, i.e., is a martingale.
We have the following theorem.
A function forms a martingale transformation function for the solution of BSDE (6), if and only if it satisfies the following equation
(summation over ).
Suppose and that is a martingale transformation function of BSDE (1). If moreover, is invertible about y for each t, we can use to solve BSDE (1). In fact, for terminal value Y and terminal time T, we can get a process . Since I is invertible in y for each fixed t, we can directly get . In fact, this method was already used by Bahlali, Eddahbi, Ouknine .
Inspired by Lemma 1, for BSDE (1), we suppose that is a martingale transformation function of the solution of BSDE (1). We consider the operator . By Theorem 1, we can deduce that for each ,
(summation over ). Therefore, we get the following theorem.
Suppose is a symmetry of BSDE (6) and is a martingale transformation of the BSDE. If moreover, for each i, holds for any , then forms a new symmetry of BSDE (6).
5. Symmetry Transformation II
Following the results in , we consider again BSDE (6) and introduce a time change
Here is a smooth function of t. We have the following substitution:
The transformations of the Wiener process into is specifically discussed in Appendix A in . For more details about this transformation, the readers can also refer to [11,21,22], etc.
According to Equation (7), the last item in the right-hand side of the third equation can be transformed as follows
Thus, considering the first order, we can introduce with summation over j.
with summation over . Then we get the following theorem.
The projectable vector field is a symmetry generator for BSDE (6) if and only if τ and ξ satisfy the full determining equations for projectable symmetries of BSDE:
(summation over ), for each .
By symmetry X, we can define and , and moreover,
(1) Please note that in this case, , because of the transformation of w. (2) For any fixed ε, generates a new filtration . In addition, if a transformation transforms t to , then . (3) The transformation generated by X transforms the terminal time and terminal condition from to , and from the solution to .
Let and , then we have
Letting , the above equation is equivalent to
We get solutions
where is an arbitrary function that is smooth enough and are arbitrary constants. So, the symmetries of the BSDE are as follows:
Therefore and generated by and respectively, are the same as those in Example 1, while . Similarly, for , is the solution of the following equations
Suppose and g is independent of z. Equation (11) leads to the following equations
6. Symmetry Transformation on FBSDE
In this section, we consider the following FBSDE
where , , , are all smooth functions. In addition, we suppose that the solution of the FBSDE exists.
We introduce the following near-identity change of coordinates,
for , . For ease of notation, we denote , , and (resp. ) the partial derivative of f with respect to (resp. ), while (resp. ) the partial derivative of f with respect to (resp. ), the partial derivative of f with respect to . Then we have
(summation over the repeated index and ).
By , at first order in , we have
Let . We have
Therefore, we have
Also, we have
Using Taylor’s series expansion at point gives
We also introduce . For terminal condition , We have the following result.
For any fixed (It can be similarly deduced for ), we define and, without loss of generality, we suppose that . Suppose that transforms any solution of FBSDE (13) into .
For all , and any solution of FBSDE (13), the terminal condition holds, if and only if for each and ,
Since , we consider with initial values , where . By the definition of , we know that for ,
(with summation over i).
holds for each and . Then
Taking integral from 0 to for both sides, we get
Since , we get . Therefore, Equation (16) can be easily assured by condition (14) which is a rather weaker condition than (15). The sufficiency follows.
Necessity. Since for any , , taking derivative, we get
Therefore, Since can take any values in , and can take any values in , we have (14). The proof is complete. □
Therefore, we get the following theorem.
The projectable vector field is a symmetry generator of FBSDE (13) allowing time change if and only if satisfy the following determining equations:
If , we get symmetries that leave the terminal time invariant.
Conceptualization, methodology, formal analysis, writing—original draft preparation, writing—review and editing: N.Z. and G.J.
Supported by the National Natural Science Foundation of China grant number 11601387 and 11171186, National Key R&D Program of China grant number 2018YFA0703900.
We thank referees for useful comments.
Conflicts of Interest
The authors declare no conflict of interest.
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