Journal of Nonlinear Mathematical Physics

In Press, Corrected Proof, Available Online: 10 December 2020

Symmetries of Kolmogorov Backward Equation

Authors
Roman Kozlov*
Department of Finance and Management Science, Norwegian School of Economics, Helleveien 30, Bergen 5045, Norway
Corresponding Author
Roman Kozlov
Received 26 June 2020, Accepted 25 October 2020, Available Online 10 December 2020.
DOI
https://doi.org/10.2991/jnmp.k.201104.002How to use a DOI?
Keywords
Lie symmetry analysis, stochastic differential equations, Kolmogorov backward equation
Abstract

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.

Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

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 [14]. Later there were considered general point transformations in the space of the independent and dependent variables [9,10,3537]. For them the transformation of the Brownian motion is induced by the random time change. We refer to a review paper [12] for symmetry development and to a chapter [24] 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 [30]. First, this symmetry relation was treated in Gaeta and Quintero [14] for fiber-preserving symmetries. Later, it was considered for symmetries in the space of the independent and dependent variables in Ünal [36]. In Kozlov [20] (see also [21]) 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 [20]. There are many papers devoted to symmetries of particular Fokker–Planck equations [3,4,8,3134].

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. [5] 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 [6] 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. [5]. 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

dxi=fi(t,x)dt+giα(t,x)dWα(t),i=1,...,n,α=1,...,m, (2.1)
where fi(t, x) is a drift vector, g(t, x) is a diffusion matrix and Wα(t) is a vector Wiener process (vector Brownian motion) [2,7,15,17,29]. We assume summation over repeated indexes and use notation x = (x1, ..., xn). Let us remark that Wα(t), α = 1, ..., m are independent one-­dimensional Brownian motions.

2.1. Itô Formula

The transformation of the dependent variables in stochastic calculus is given by Itô formula (see, for example, [29]). For SDEs (2.1) we perform variable change xy = y(t, x) according to

dyi=yitdt+yixjdxj+122yixjxkdxjdxk,i=1,...,n, (2.2)
where dxjdxk are found with the help of the substitution rules
dtdt=0, (2.3a)
dtdWα=dWαdt=0, (2.3b)
dWαdWβ=δαβdt. (2.3c)

Thus, we obtain the formula for differentials in stochastic calculus

dF(t,x)=D0(F)dt+Dα(F)dWα(t), (2.4)
where
D0=t+fjxj+12gjαgkα2xjxk,Dα=gjαxj. (2.5)

2.2. First Integrals

Stochastic differential equations can possess first integrals.

Definition 2.1.

A quantity I(t, x) is a first integral of a system of SDEs (2.1) if it remains constant on the solutions of the SDEs.

Application of the Itô differential formula (2.4) to a first integral

dI(t,x)=D0(I)dt+Dα(I)dWα(t)=0
leads to a system of partial differential equations
D0(I)=0, (2.6a)
Dα(I)=0. (2.6b)

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

t¯=t¯(t,x,a)t+τ(t,x)a,x¯i=x¯i(t,x,a)xi+ξi(t,x)a, (2.7)
which leave Eq. (2.1) and framework of Itô calculus invariant. Such transformations can be represented by generating operators of the form
X=τ(t,x)t+ξi(t,x)xi. (2.8)

The determining equations for Lie point transformations (2.7) of Itô SDEs (2.1) were derived in Ünal [36]. 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

D0(ξi)X(fi)fiD0(τ)=0, (2.9a)
Dα(ξi)X(giα)12giαD0(τ)=0, (2.9b)
Dα(τ)=0. (2.9c)

In Kozlov [23] 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 [20].

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

ut=fi(t,x)uxi+12giα(t,x)gjα(t,x)2uxixj. (3.1)

For symmetry analysis we rewrite it as

ut+Aijuxixj+Bkuxk=0, (3.2)
where
Aij=12giαgjα,Bk=fk.

In what follows we will assume that Aij are not all zero.

3.1. Determining Equations

Let us find Lie point symmetries

XKB=τ(t,x,u)t+ξi(t,x,u)xi+η(t,x,u)u (3.3)
which are admitted by the KBE. For our purpose we need the prolonged symmetry vector field
pr(2)XKB=τt+ξixi+ηu+ζtut+ζiuxi+ζijuxixj, (3.4)
where the coefficients are computed according to the standard prolongation formulas
ζt=Dt(η)utDt(τ)uxjDt(ξj),ζi=Di(η)utDi(τ)uxjDi(ξj),ζij=Di(ζj)utxjDi(τ)uxkxjDi(ξk).

Here Dt and Di are total differentiation operators with respect to t na xi.

Infinitesimal invariance criteria [16,27,28] states that the application of the second prolongation of the operator XKB to the second order PDE (3.2) should be zero on the solutions of this PDE:

pr(2)XKB(ut+Aijuxixj+Bkuxk)|(3.2)=0. (3.5)

We review briefly the derivation of the determining equations for symmetries of the KBE. It is convenient to use notations

F,t=Ft,F,u=FuandF,i=Fxi
(the last notation uses indexes different from t and u).

Equation (3.5) splits for different spatial derivatives of u. We obtain

τ,u(AijApquxpxqxjuxi)=0
for products of third derivatives with first derivatives that leads to
τ,u=0
and
(Aijτ,i)(Apquxpxqxj)=0
for third derivatives that gives
Aijτ,i=0. (3.6)

Then, for products of second derivatives with first derivatives we get the equations

ξk,u(Aijuxkxiuxj)=0,
which give
ξi,u=0,
and as coefficients for the second derivatives we obtain equations
τAij,t+ξkAij,k+Aij(τ,t+Bpτ,p+Apqτ,pq)Aikξj,kAkjξi,k=0. (3.7)

For products of first derivatives we obtain

η,uu(Aijuxiuxj)=0.

Therefore,

η=φ(t,x)u+ψ(t,x).

Substituting it into the rest of Eq. (3.5), we get

ξi,t+Bpξi,p+Apqξi,pqτBi,tξpBi,p2Aijφ,jBi(τ,t+Bpτ,p+Apqτ,pq)=0, (3.8)
φ,t+Bkφ,k+Aijφ,ij=0 (3.9)
and
ψ,t+Bkψ,k+Aijψ,ij=0 (3.10)
for the terms with the first derivatives, the terms with u and the rest, respectively.

We can summarize the obtained results using the operators D0 and Dα, which were given in Eq. (2.5).

Theorem 3.1.

Lie point symmetries of KBE (3.1) are given by

  1. 1.

    vector fields of the form

    XKB=τ(t,x)t+ξi(t,x)xi+φ(t,x)uu (3.11)
    with coefficients satisfying equations
    giαDα(τ)=0, (3.12a)
    giα(Dα(ξj)X(gjα)12gjαD0(τ))+gjα(Dα(ξi)X(giα)12giαD0(τ))=0, (3.12b)
    D0(ξi)X(fi)fiD0(τ)=giαDα(φ), (3.12c)
    D0(φ)=0 (3.12d)
    and

  2. 2.

    trivial symmetries

    XKB*=ψ(t,x)u, (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).

Remark 3.2.

We see that the determining equations (3.12a)(3.12d) always have a particular solution

τ=0,ξ1=...=ξn=0,φ=const.

It provides us with symmetry

X0=uu, (3.14)
corresponding to linearity of the KBE.

3.2. Symmetries of KBE and Symmetries of SDEs

Now we can relate symmetries of the SDEs to the symmetries of the associated KBE.

Theorem 3.3.

Let operator X of the form (2.8) be a symmetry of the SDEs (2.1), then X is also a symmetry 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.

Theorem 3.4.

Let SDEs (2.1) possess a first integral I(t, x), then the associated KBE admits symmetry

Y=I(t,x)uu. (3.15)

Proof. It follows from Eqs. (2.6a) and (2.6b) that the determining equations (3.12) for symmetries of the KBE are satisfied.

It is possible to state the converse results.

Theorem 3.5.

If KBE (3.1), which corresponds to SDEs (2.1), admits a symmetry X of the form (2.8) with coefficients satisfying equations (2.9b) and (2.9c), then the symmetry X is admitted by the SDEs.

Theorem 3.6.

If KBE (3.1), which correspond to SDEs (2.1), admits a symmetry of the form (3.15) and function I(t, x) satisfies the Eq. (2.6b), then I(t, x) is a first integral of the SDEs.

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 Aij=12giαgjα .

Finally, we summarize the results of this point by presenting four types of Lie point symmetries of the KBE. They are:

  1. 1.

    symmetries (3.13) and (3.14) corresponding to linearity of the KBE

  2. 2.

    symmetries (2.8) which are related to the symmetries of the SDEs

  3. 3.

    symmetries (3.15) which are related to the first integrals of the SDEs

  4. 4.

    the other symmetries, which are not related to the SDEs

3.3. Symmetries of KBE and Symmetries of KFE

For SDEs (2.1) the corresponding KFE, which is also called Fokker–Planck equation [30], takes the form

ut=xi(fi(t,x)u)+122xixj(giα(t,x)gjα(t,x)u). (3.16)

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 [20]. They were based on the following description of the symmetries of the KFE.

Theorem 3.7.

Lie point symmetries of KFE (3.16) are given by

  1. 1.

    vector fields of the form

    XKF=τ(t,x)t+ξi(t,x)xi+χ(t,x)uu (3.17)

    with coefficients satisfying equations

    giαDα(τ)=0, (3.18a)
    giα(Dα(ξj)X(gjα)12gjαD0(τ))+gjα(Dα(ξi)X(giα)12giαD0(τ))=0, (3.18b)
    Q=χ+ξi,i+τ,tD0(τ), (3.18c)
    where function Q(t, x) is a solution of equations
    D0(ξi)X(fi)fiD0(τ)=giαDα(Q), (3.19a)
    D0(Q)=0, (3.19b)
    and

  2. 2.

    trivial symmetries

    XKF*=ψ(t,x)u, (3.20)

    where the coefficient is an arbitrary solution of the KFE, corresponding to the linear superposition principle.

By direct comparison of the determining equations given in Theorems 3.1 and 3.7 we can establish the following result.

Theorem 3.8.

Let us consider KBE (3.1) and KFE (3.16) corresponding to the same SDEs (2.1). The KBE admits symmetry (3.11) if and only if the KFE admits symmetry (3.17) and

φ+χ=D0(τ)τ,tξi,i.

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.

Corollary 3.9.

Let us consider KBE (3.1) and KFE (3.16) corresponding to the same SDEs (2.1). The KBE admits symmetry (3.15) if and only if the KFE admits the same symmetry.

4. SCALAR SDES AND (1 + 1)-DIMENSIONAL KOLMOGOROV EQUATIONS

Let us illustrate how one can use symmetries of the scalar SDEs

dx=f(t,x)dt+g(t,x)dW(t),g(t,x)0 (4.1)
to find symmetries of the KBE
ut=f(t,x)ux+12G(t,x)2ux2,G(t,x)=g2(t,x)0,0. (4.2)

Lie point symmetries of the KBE (4.2) are described by Theorem 3.1. They are symmetries

XKB=τ(t,x)t+ξ(t,x)x+φ(t,x)uu (4.3)
with coefficients satisfying equation
DW(τ)=0, (4.4a)
DW(ξ)X(g)12gD0(τ)=0, (4.4b)
D0(ξ)X(f)fD0(τ)=gDW(φ), (4.4c)
D0(φ)=0, (4.4d)
where
D0=t+fx+12g22x2,DW=gx, (4.5)
and trivial symmetries (3.13). Note that from (4.4a) we get τ = τ(t).

In the general case the KBE (4.2) has only symmetries related to its linearity, namely

XKB*=ψ(t,x)uandX0=uu, (4.6)
where ψ(t, x) is an arbitrary solution of the KBE. For particular cases f(t, x) and G(t, x) there can be additional symmetries.

4.1. Symmetries of KBE via Symmetries of SDEs

Lie group classification of the scalar SDE (4.1) was carried out in Kozlov [18] 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 [19].

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

dx=f(t,x)dt+dW(t) (4.7)
because we can perform the variable change
xdxg(t,x).

The corresponding KBE is also simplified. It takes the form

ut=f(t,x)ux+122ux2. (4.8)

4.1.1. SDE with one symmetry

The equivalence class of the SDEs admitting only one symmetry

X1=t (4.9)
can be represented by the equation
dx=f(x)dt+dW(t). (4.10)

The corresponding KBE

ut=f(x)ux+12uxx (4.11)
admits symmetries (4.6) and (4.9).

4.1.2. SDE with two symmetries

For the SDEs admitting two symmetries

X1=t,X2=2tt+xx (4.12)
one can chose the following representative equation
dx=Axdt+dW(t),A0. (4.13)

For the KBE

Y1=2t2t+2txx+(x2(1+2A)t)uu. (4.14)
we get two subcases

1. A ≠ 1

In addition to symmetries (4.6) and (4.12) the KBE admits symmetry

ut=Axux+12uxx (4.15)

2. A = 1

In this particular case the KBE possesses symmetries (4.6), (4.12), (4.15) and

Y2=tx+(xtx)uu,Y3=xuxu. (4.16)

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

dx=dW(t), (4.17)
which admits symmetries
X1=t,X2=2tt+xx,X3=x. (4.18)

In this case we get the KBE

ut=12uxx (4.19)
admits symmetries (4.6), (4.18) and
Y1=tx+xuu,Y2=2t2t+2txx+(x2t)uu. (4.20)

Remark 4.1.

The KBE (4.14) with A = 1, namely the equation

ut=1xux+12uxx,
can be transformed into the KBE (4.19) by the change of the dependent variable
u=1xu¯.

However, the SDE (4.13) with A = 1 cannot be transformed into the SDE (4.17).

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

ut=x(f(t,x)u)+122x2(G(t,x)u) (4.21)
is knows [34]. It can be used to obtain Lie group classification of the (1 + 1)-dimensional KBE with the help of Theorem 3.8, which relates symmetries of the KBE and KFE.

In addition to the symmetries

XKF*=ψ(t,x)uandX0=uu, (4.22)
where ψ(t, x) is an arbitrary solution of the KFE, the KBE (4.21) can admit 0, 1, 3 or 5 symmetries. Due to Theorem 3.8 we get the same results for the KBE (4.2). It can admit 0, 1, 3 or 5 symmetries in addition to symmetries (4.6).

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.

  • Using Lie group classification of the scalar SDEs, we obtain correct description of the equivalence classes for KBEs admitting 0, 1 and 5 symmetries in addition to the symmetries (4.6). These equivalence classes are represented by Eqs. (4.8), (4.11) and (4.19), respectively.

  • 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.

Theorem 4.2

([34]). KFEs (4.21) admitting three symmetries in addition to the linearity symmetries (4.22) can be transformed into the equation

ut=(2k(x)u)x+uxx, (4.23)
where k(x) is a solution of the equation
k(k)2=λx2,λ0. (4.24)

In addition to the symmetries (4.22) Eq. (4.23) admits operators

X1=t,X2=2tt+xxxk(x)uu,X3=4t2t+4txx(x2+4t+4txk(x))uu,
where k(x) is a solution of Eq. (4.24).

Using Theorem 3.8, which relates symmetries of the KBE and KFE, we state a similar result for the KBE.

Corollary 4.3.

KBEs (4.2) admitting three symmetries in addition to the linearity symmetries (4.6) can be transformed into the equation

ut=2k(x)ux+uxx, (4.25)
where k(x) is a solution of Eq. (4.24).

In addition to the symmetries (4.6) Eq. (4.25) admits operators

X1=t,X2=2tt+xx+xk(x)uu,X3=4t2t+4txx+(x2+4txk(x))uu,
where k(x) is a solution of Eq. (4.24). For
k(x)Ax
symmetry X2 (up to factorization by X0) is beyond the framework based on the symmetries of the underlying SDE. Thus, with the help of the scalar SDE classification we get only a subcase of the class of KBEs admitting three additional symmetries.

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 [21].

4.3. The Geometric Brownian Motion Equation

Let us examine the geometric Brownian motion [29]

dx=αxdt+σxdW(t),σ0 (4.26)
as a theory application. This SDE is an important model for stochastic prices in economics.

The SDE (4.26) admits three symmetries of the form (2.8), namely

X1=t,X2=2tt+((ασ22)tx+xlnx)x,X3=xx (4.27)
and has no first integrals.

The associated KBE (4.2) takes the form

ut=αxux+12σ2x22ux2. (4.28)

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

Y1=txx1σ2((ασ22)tlnx)uu, (4.29a)
Y2=2t2t+2txlnxx+(1σ2((ασ22)tlnx)2t)uu. (4.29b)

The KFE (4.21) for the geometric Brownian motion equation takes the form

ut=αx(xu)+12σ22x2(x2u). (4.30)

It is invariant with respect to symmetries (4.22). The other symmetries can be obtained with the help of Theorem 3.8 and the symmetries of the KBE (4.28). We find

X1=t,X2=2tt+((ασ22)tx+xlnx)x((ασ22)t+lnx)uu,X3=xx,Y1=txx+(1σ2((ασ22)tlnx)t)uu,Y2=2t2t+2txlnxx(1σ2((ασ22)tlnx)2+2tlnx+t)uu.

Remark 4.4.

Let us note that symmetries of SDEs can be used to find symmetries of the associated KFE [20]. A symmetry (2.8) of the SDEs (2.1) provides with the symmetry

X¯=X+(D0(τ)τ,tξi,i)uu (4.31)
admitted by the associated KFE. This results can be used to find the symmetries X1, X2 and X3 for the KFE (4.30).

CONFLICTS OF INTEREST

The author declares no conflicts of interest.

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Journal
Journal of Nonlinear Mathematical Physics
Publication Date
2020/12
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.2991/jnmp.k.201104.002How to use a DOI?
Copyright
© 2020 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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  -