Symmetries of Kolmogorov Backward Equation
- https://doi.org/10.2991/jnmp.k.201104.002How to use a DOI?
- Lie symmetry analysis, stochastic differential equations, Kolmogorov backward equation
The note provides the relation between symmetries and first integrals of Itô stochastic differential equations and symmetries of the associated Kolmogorov Backward Equation (KBE). Relation between the symmetries of the KBE and the symmetries of the Kolmogorov forward equation is also given.
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Lie group theory of differential equations is well developed [16,27,28]. It studies transformations which take solutions of differential equations into other solutions of the same equations. This theory became a powerful tool for finding analytical solutions of differential equations.
Successful applications of Lie group theory to differential equations motivated its development for other equation types. Here we consider Stochastic Differential Equations (SDEs). First attempts were devoted to transformations which change only the dependent variables, i.e. transformations which do not change time [1,25,26]. After them fiber-preserving transformations were approached . Later there were considered general point transformations in the space of the independent and dependent variables [9,10,35–37]. For them the transformation of the Brownian motion is induced by the random time change. We refer to a review paper  for symmetry development and to a chapter  for symmetry applications. More general framework includes transformations which depend on the Brownian motion [11,13,22,23].
One of the applications of the symmetries of SDEs was their relation to symmetries of the associated Kolmogorov Forward Equation (KFE), which is also known as the Fokker–Planck equation in physics . First, this symmetry relation was treated in Gaeta and Quintero  for fiber-preserving symmetries. Later, it was considered for symmetries in the space of the independent and dependent variables in Ünal . In Kozlov  (see also ) a more precise formulation of the symmetry relation was provided. There is also a relation between first integrals of the SDEs and symmetries of the KFE . There are many papers devoted to symmetries of particular Fokker–Planck equations [3,4,8,31–34].
Symmetries of the Kolmogorov Backward Equation (KBE) received much less attention than symmetries of the KFE. The KBE is useful when one is interested whether at some future time the system will be in a target set, i.e. in a specified subset of states. In De Vecchi et al.  the authors considered symmetries of a KBE with diffusion matrix [matrix Aij in Eq. (3.2)] of full rank. For such equations corresponding to autonomous SDEs and time changes restricted to scalings it was shown that symmetries of the SDEs are also symmetries of the KBE. The paper  examines more general stochastic transformations able to change the underlying probability measure. In this framework the weak extended symmetries of SDEs are more general than the Lie point symmetries of the KBE.
In the present note we consider Lie point symmetries of the KBE and examine how these symmetries can be related to the strong symmetries of the underlying SDEs without the restrictions which were imposed in De Vecchi et al. . We also consider the Lie point symmetries of the KBE corresponding to first integrals of the underlying SDEs and show how the symmetries of KBEs are related to the symmetries of KFEs corresponding to the same underlying SDEs.
The paper is organized as follows. In the next section we recall basic results on Itô SDEs and their symmetries. In Section 3 we examine Lie point symmetries of the KBE and find out how they can be related to the symmetries of the SDEs and to the symmetries of the KFE. Finally, in Section 4 we consider scalar SDEs and (1 + 1)-dimensional Kolmogorov equations to illustrate the theoretical results of this paper. The last section also illustrates the theory on an example of geometric Brownian motion.
2. SDES AND LIE POINT SYMMETRIES
Let us consider a system of stochastic differential equations in Itô form
2.1. Itô Formula
Thus, we obtain the formula for differentials in stochastic calculus
2.2. First Integrals
Stochastic differential equations can possess first integrals.
2.3. Determining Equations
We will be interested in infinitesimal group transformations (near identity changes of variables) in the space of the independent and dependent variables
The determining equations for Lie point transformations (2.7) of Itô SDEs (2.1) were derived in Ünal . It is convenient to present them with the help of the operators D0 and Dα given in Eq. (2.5). The determining equations take a compact form
In Kozlov  it was shown that one can also obtain these determining equations by restriction of more general transformations which involve Brownian motion. The Lie point symmetries (2.8) of Itô SDEs, which are given by the determining equations (2.9a)–(2.9c), form a Lie algebra .
3. SYMMETRIES OF KOLMOGOROV BACKWARD EQUATION
In this section we derive the determining equations for Lie point symmetries of the KBE and find out how these symmetries can be related to the symmetries and first integrals of SDEs. Later we show how these symmetries can be related to the symmetries of the KFE.
For SDEs (2.1) the associated KBE has the form
For symmetry analysis we rewrite it as
In what follows we will assume that Aij are not all zero.
3.1. Determining Equations
Let us find Lie point symmetries
Here Dt and Di are total differentiation operators with respect to t na xi.
We review briefly the derivation of the determining equations for symmetries of the KBE. It is convenient to use notations
Equation (3.5) splits for different spatial derivatives of u. We obtain
Then, for products of second derivatives with first derivatives we get the equations
For products of first derivatives we obtain
Substituting it into the rest of Eq. (3.5), we get
We can summarize the obtained results using the operators D0 and Dα, which were given in Eq. (2.5).
Lie point symmetries of KBE (3.1) are given by
vector fields of the form(3.11)with coefficients satisfying equations(3.12a)(3.12b)(3.12c)(3.12d)and
trivial symmetries(3.13)where the coefficient is an arbitrary solution of the KBE, corresponding to the linear superposition principle.
The proof follows from the previous discussion of the equations for symmetry coefficients. In particular, Eqs. (3.12a)–(3.12d) for coefficients of the symmetry (3.11) represent Eqs. (3.6)–(3.9), which are rewritten with the help of the operators D0 and Dα. The coefficient of the symmetry (3.13) satisfies Eq. (3.10).
3.2. Symmetries of KBE and Symmetries of SDEs
Now we can relate symmetries of the SDEs to the symmetries of the associated KBE.
Proof. From the determining equations (2.9b) and (2.9c) it follows that Eqs. (3.12a) and (3.12b) hold. Choosing φ ≡ 0, which is always a solution of Eq. (3.12c) [if Eq. (2.9a) hold] and (3.12d), we get X as a symmetry of the KBE.
We can also relate some symmetries of the KBE to first integrals of the SDEs.
Let SDEs (2.1) possess a first integral I(t, x), then the associated KBE admits symmetry
It is possible to state the converse results.
The additional requirements of Theorems 3.5 and 3.6 are not surprising. They specify the particular SDEs: the same KBE can correspond to different SDEs, which have the same drift coefficients fi and diffusion matrix .
Finally, we summarize the results of this point by presenting four types of Lie point symmetries of the KBE. They are:
3.3. Symmetries of KBE and Symmetries of KFE
The relation of symmetries of KFE and the symmetries of the underlying SDEs was considered in several papers [14,20,36]. The most general results were established in Kozlov . They were based on the following description of the symmetries of the KFE.
Lie point symmetries of KFE (3.16) are given by
vector fields of the form(3.17)
with coefficients satisfying equations(3.18a)(3.18b)(3.18c)where function Q(t, x) is a solution of equations(3.19a)(3.19b)and
where the coefficient is an arbitrary solution of the KFE, corresponding to the linear superposition principle.
Proof. The result follows from the observation that the sets of variables (τ, ξ1, ..., ξn, φ) and (τ, ξ1, ..., ξn, −Q) satisfy the same equations.
4. SCALAR SDES AND (1 + 1)-DIMENSIONAL KOLMOGOROV EQUATIONS
Let us illustrate how one can use symmetries of the scalar SDEs
In the general case the KBE (4.2) has only symmetries related to its linearity, namely
4.1. Symmetries of KBE via Symmetries of SDEs
Lie group classification of the scalar SDE (4.1) was carried out in Kozlov  by direct method. Alternatively, one can obtain this Lie group classification with the help of real Lie algebra realizations by vector fields. It was done in Kozlov .
In the general case the SDE (4.1) has no symmetries. Therefore, the KBE (4.2) admits only symmetries (4.6) corresponding to its linearity. We shall go through the cases of the Lie group classification of the scalar SDEs (4.1) and find the symmetries of the corresponding KBEs. It should be noted that we can always choose a representative SDE for each equivalence symmetry class in the form
The corresponding KBE is also simplified. It takes the form
4.1.1. SDE with one symmetry
The equivalence class of the SDEs admitting only one symmetry
The corresponding KBE
4.1.2. SDE with two symmetries
For the SDEs admitting two symmetries
For the KBE
1. A ≠ 1
2. A = 1
4.1.3. SDE with three symmetries
Scalar SDEs can admit at most three symmetries. The equivalence class for SDEs with three symmetries can be represented by the equation
In this case we get the KBE
The KBE (4.14) with A = 1, namely the equation
We cannot expect that Lie group classification of SDEs will provide us with Lie group classification of the associated KBE. Indeed, it gives only partial results on the symmetries of specified KBEs as we will see in the next point.
4.2. Lie Group Classification of (1 + 1)-Dimensional KBE
Lie group classification of the (1 + 1)-dimensional KFE
In addition to the symmetries
Lie group classification of the KBE (obtained with the help of Lie group classification of the KFE) can be compared with results of the previous point. We find out the following.
However, we do not obtain the correct description of the equivalence class for the KBEs admitting three additional symmetries. It is easy to see from the next theorem.
Using Theorem 3.8, which relates symmetries of the KBE and KFE, we state a similar result for the KBE.
Therefore, using the Lie group classification of the scalar SDEs, we get partial results of the Lie group classification of the (1 + 1)-dimensional KBE. The same was observed for the (1 + 1)-dimensional KFE in Kozlov .
4.3. The Geometric Brownian Motion Equation
Let us examine the geometric Brownian motion 
The associated KBE (4.2) takes the form
It admits the trivial symmetries (4.6) because of its linearity. Theorem 3.3 states that the symmetries (4.27) are also admitted by this KBE. Direct computation of the symmetries of the form (2.7) provides us with the two additional symmetries
The KFE (4.21) for the geometric Brownian motion equation takes the form
CONFLICTS OF INTEREST
The author declares no conflicts of interest.
Cite this article
TY - JOUR AU - Roman Kozlov PY - 2020 DA - 2020/12 TI - Symmetries of Kolmogorov Backward Equation JO - Journal of Nonlinear Mathematical Physics SN - 1776-0852 UR - https://doi.org/10.2991/jnmp.k.201104.002 DO - https://doi.org/10.2991/jnmp.k.201104.002 ID - Kozlov2020 ER -