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Volume 25, Issue 2, March 2018, Pages 262 - 289

Symmetry and integrability for stochastic differential equations

Authors
G. GaetaORCID, C. LuniniORCID
Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy,giuseppe.gaeta@unimi.it,claudia.lunini@studenti.unimi.it
Received 6 October 2017, Accepted 11 December 2017, Available Online 6 January 2021.
DOI
10.1080/14029251.2018.1452673How to use a DOI?
Abstract

We discuss the interrelations between symmetry of an Ito stochastic differential equations (or systems thereof) and its integrability, extending in party results by R. Kozlov [J. Phys. A 43 (2010) & 44 (2011)]. Together with integrability, we also consider the relations between symmetries and reducibility of a system of SDEs to a lower dimensional one. We consider both “deterministic” symmetries and “random” ones, in the sense introduced recently by Gaeta and Spadaro [J. Math. Phys. 58 (2017)].

Copyright
© 2018 The Authors. Published by Atlantis Press and Taylor & Francis
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

It is well known that symmetry methods are among the most powerful tools for the study of nonlinear deterministic differential equations [3,7,25,26,29]. It is thus natural to think they could be useful also in the study of stochastic differential equations (SDEs).

This observation is of course not new, and indeed there is by now a substantial literature devoted to symmetries of SDEs (see [12] for an extended reference list). In a first phase the efforts focused on determining the proper – that is, useful – definition of symmetry for a SDE and hence the determining equations; there is now a consensus on what this suitable definition is (see below). The second and crucial phase is of course to understand how these can be used in the study of SDEs.

The main idea – paralleling the approach for deterministic equations – is to use symmetry-adapted coordinates to implement a reduction of the SDE. It should be stressed that here one should think of the decomposition of a given system into a smaller (lower dimensional) one plus one or more reconstruction equations, as in the case of deterministic Dynamical Systems; we are here going to study systems of Ito equations, which are the stochastic counterpart of first order ODEs, i.e. indeed Dynamical Systems.

A key result in this direction was obtained by Kozlov [18] for scalar SDEs, showing that if such an SDE possesses a symmetry of a certain type (“simple” symmetries, to be defined below), then it can be integrated; the theorem is actually constructive, i.e. the symmetry determines a change of variables which allows for explicit integration of the SDE (see below for details).

In the case of systems of SDE, Kozlov’s approach [19,20] shows that a symmetry of the appropriate type implies that the system can be “partially integrated” (in a sense to be made precise below).

Kozlov approach is based – like analogous results for deterministic equations – on changes of variables to use favorable properties of symmetry-adapted coordinates; it is essential that symmetries are preserved under changes of coordinates. In a companion paper [13] we have argued that while this fact is entirely obvious for deterministic equations, preservation of symmetries for Ito stochastic equations cannot be given for granted. The reason is that Ito equations are not geometrical objects, and indeed do not transform in the “usual” way (i.e. under the familiar chain rule) under changes of coordinates: in fact, they transform according to the Ito rule. However it turns out that there is a rather ample class of symmetries which are preserved under changes of coordinates, and these include the kind of symmetries relevant for Kozlov’s theory; the reader is referred to our work [13] for a discussion, while the results relevant to our present subject are recalled – after discussing in Section 2 some general features of symmetries of SDEs – in Section 3.

Equipped with these preliminary results, we pass to study the relation between symmetry and integration – or at least reduction – for (Ito) SDEs. For what concerns a single symmetry, the canonical result in the literature, already recalled above, is due to Kozlov [18] and says that the existence of a symmetry of a given type is a sufficient condition for the integrability of an Ito scalar SDE. We will show that the condition is not only sufficient but also necessary, see Section 4.

We will then pass to consider the case of multi-dimensional systems of SDEs with a single symmetry and discuss the reduction which is possible in this case (see Section 5); this will also give an idea of the kind of results which can be obtained in the general case.

Finally (in Section 6) we will consider the case of a n-dimensional system of Ito SDEs with a q-dimensional solvable symmetry algebra (the restriction to considering solvable symmetry algebras descends from well known results in the study of deterministic equations and their symmetry reduction [3,7,25,26,29]), and discuss how the symmetry allows for a multiple-stages reduction of the system; in general this is actually setting the system in a “partially triangular” form, with an m-dimensional (with m < n) system and a number (q = n − m) of “reconstruction equations”. Here again we will compare our results with those obtained by Kozlov [19,20].

So far we have considered maps acting by transformation of the dependent variables, possibly depending on time; we will also show (in Section 7, with examples in Section 8) that the results discussed so far can be extended to the case of random maps [4,5,15], i.e. transformations depending on the Wiener processes entering in the equations under study. In this case the transformation is subject to certain restrictions to guarantee we remain within the category of Ito equations, and a number of other details should be controlled. For this setting, we will consider only scalar equations, postponing the study of systems to a future contribution. It will turn out that it is possible to obtain a nearly full extension – at least for scalar equations – of Kozlov theory to the setting of random maps and random symmetries; the only difference being that now an additional compatibility condition (7.18) involving the coefficients of the Ito equation and of the symmetry vector field should be satisfied.

We also provide two Appendices; the first shows that albeit one could apparently extend Kozlov’s theory to other types of integrable SDEs (separable ones), this framework is already contained in the “standard” case; in the second we discuss when one has infinitely many (simple) random symmetries for a given scalar Ito equation.

Finally, it should be mentioned that the literature also considers (systems of) higher order SDEs and their symmetries [12,28]; we will not consider these, neither other possible generalizations of the setting (i.e., that of systems of Ito equations) we have chosen to study. Similarly, when dealing with deterministic equations one also considers more general classes of symmetries than Lie-point ones [3,7,25,26,29]; but we will not consider these here.

2. Symmetry of stochastic differential equations

In this note, by “differential equation” (DEs) we will always mean possibly a vector one, i.e. a system of differential equations, unless the contrary is explicitly stated (in this case we speak of a scalar equation).

Given a deterministic DE Δ of order n in the basis manifold M (this include dependent and independent variables), it is well known that this is identified with its solution manifold SΔ in the Jet bundle Jn M. A Lie-point symmetry (generator) is then a vector field X on M which, once prolonged to the vector field X(n) on JnM, is tangent to SΔ ⊂ JnM [3,7,25,26,29].

Both submanifolds and vector fields are geometrical object; they are defined independently of any coordinate system, and it is thus entirely obvious that tangency relations – hence in particular X being a symmetry for Δ – hold independently of coordinates. This also means that symmetries are preserved under changes of coordinates.

Let us consider an (Ito) stochastic differential equation (SDE) [4,9,10,16,17,24]

dxi=fi(x,t)dt+σji(x,t)dwj; (2.1)
in (2.1) and below, t ∈ R, x ∈ Rn (the xi can be thought as local coordinates on a smooth manifold, as we take care of distinguishing covariant and contravariant indices albeit never introducing explicitly a metric), f and σ are smooth vector and matrix functions of their arguments, and wj(j = 1,…,m) are independent standard Wiener processes.

For Ito equations, a geometrical interpretation of the equation is missing, and one is forced to resort to a purely algebraic notion of symmetry. Actually, not only a geometrical interpretation is missing, but this is not possible: in fact Ito equations and more generally Ito differentials do not transform in the “usual” way, i.e. according to the familiar chain rule, under changes of coordinates; they transform instead according to the Ito rule, and this is inherent to their very nature [4,9,10,16,17,24,30] (see [27], chapter 5, for a Physics point of view).

Remark 1.

The situation is different for Stratonovich SDEs; actually the main motivation for their introduction was precisely to have SDEs transforming “nicely” under changes of coordinates. For these a geometrical description is indeed possible, and hence symmetries are trivially preserved under any change of coordinates (see also the detailed discussion in [13]). On the other hand, Stratonovich equations present several problems from the probabilistic point of view; see e.g. [4,9,10,16,17,24,30]. It should be stressed that albeit there is a correspondence between Ito and Stratonovich equations (which is not entirely trivial, see e.g. the discussion in [30]), there is not an identity between general symmetries – even of deterministic type [32] – of an Ito equation and those of the corresponding Stratonovich one. It is instead known that such an identity holds for (both deterministic [32] and random [13]) simple symmetries (see again [13] for discussion and examples). Note that previous work by one of the present authors [12,15] contains wrong statements in this respect (originating in a trivial mistake in the definition of the Ito Laplacian, as discussed in [13]).

Remark 2.

It should be stressed that, in view of the non-covariance of Ito equations, and of the purely algebraic definition of symmetries for them (see e.g. the review [12] for details) it is not at all guaranteed a priori that symmetries will be preserved under changes of coordinates. An analysis of this problem was provided recently [13], and it turned out that the symmetries which are relevant for integrability and/or reduction are preserved. Such results will be recalled below, and are based on the relation with symmetries of the corresponding Stratonovich equation, see Remark 1.

One could, in principles, consider general mappings and vector fields in the (x, t, w) space albeit with some restrictions if these have to make physical sense, see e.g. [12,15]. That is, we could generally speaking consider maps

(x,t;w)→(x˜(x,t,w), t˜(x,t,w); w˜(x,t,w)).

Within these, we will denote by simple the maps (and vector fields) which act only on the spatial variables, leaving the time t (and the Wiener processes) unchanged. We will also denote as deterministic the maps (and vector fields) which only depend on x (and possibly t), as opposed to the random ones (to be considered in subsection 2.2 below), in which dependence on the Wiener processes w is also present.

Within our present context, we are interested only in simple maps. The reason for this is that only these are relevant in Kozlov theory, as discussed in his works [18–20] and as will be recalled below. This simplifies considerably our task.

2.1. Simple deterministic maps & symmetries

Let us first consider a (simple) smootha map

(x,t)→(x˜,t),x˜=x˜(x,t); (2.2)
this induces a map dx→dx˜ and hence a map on SDEs (2.1); in particular if (2.2) is (locally) inverted to give
x=Φ(x˜,t), (2.3)
then by Ito formula
dxi=(∂Φi∂x˜j)dx˜j+(∂Φ∂t)dt+12(∂2φi∂x˜j∂x˜k) σ˜𝓁jσ˜𝓁kdt; (2.4)
similarly the functions fi(x, t) and σji(x,t) are mapped into functions f˜i(x˜,t) and σ˜ji(x˜,t) . In this way, (2.1) is mapped into a new Ito equation
dx˜i=f^i(x˜,t)dt+σ^ji(x˜,t)dwj; (2.5)
note that here the f^ and σ^ take into account not only the change of their variables, but also the contribution arising from dx expressed in the new variables via the Ito formula.

We say that (2.2) is a symmetry for (2.1) if (2.5) is identical to (2.1), i.e. if the (n + m ⋅ n) conditions

fi^(x,t)=fi(x,t),σ^ji(x,t)=σji(x,t) (2.6)
are satisfied identically in (x, t).

We are specially interested in the case where the map (2.2) is a near-identity one, x˜i=xi+ε(δx)i and can hence be seen as the infinitesimal action of a vector field X,

X=φi(x,t)∂∂xi≡φi(x,t)∂i. (2.7)
(Note that X has no component along the t variable; this corresponds to the restriction made above, see eq. (2.2), for simple symmetries.) In this case we speak of a Lie-point (simple) symmetry.

We obtain then easily (the reader is referred to [14] for details and explicit computations) the determining equations for (simple, deterministic) Lie-point symmetries of SDEs:

φti+fj(∂jφi)-φj(∂jfi)=-12(Δφi), (2.8)
σkj(∂jφi)-φj(∂jσki)=0. (2.9)

Here and below Δ denotes the Ito Laplacian, which in general (for functions possibly depending also on the wk, see next subsection) is defined as

Δf=δik[(∂2f∂wi∂wk)+σijσkm(∂2f∂xj∂xm)+2σij(∂2f∂xj∂wk)]. (2.10)

2.2. Simple random symmetries

We can consider more general transformations; in particular one can consider maps and vector fields which depend on the Wiener processes realizations wk. This approach (actually in a more general setting) was first considered by Arnold and Imkeller [4,5] in the context of normal forms for SDEs; random symmetries of SDEs are considered e.g. in [15] (see also [12]), to which we refer for explicit computations and details.

For what concerns our discussion here, we consider simple random maps,

(x,t;w)→(x˜,t;w), x˜=x˜(x,t;w); (2.11)
and the corresponding symmetries, i.e. simple random symmetries. These are the vector fields
X=φi(x,t;w)∂i (2.12)
leaving the equation (2.1) invariant

It is shown in [15] that, using the notation, to be used routinely in the following,

∂^k≔∂∂wk, (2.13)
the determining equations for simple random Lie-point symmetries of a SDE (2.1) read
(∂tφi)+fj(∂jφi)-φj(∂jfi)=-12(Δφi), (2.14)
(∂^kφi)+σkj(∂jφi)-φj(∂jσki)=0. (2.15)

3. Preservation of simple symmetries of Ito equations

Ito equations are not geometrical objects. In fact, under changes of coordinates they do not transform in the “usual” way, i.e. under the chain rule, but in their own way, i.e. under the Ito rule.

This means that symmetries are defined in an algebraic rather than geometrical way; and, at difference with deterministic differential equations, it is not obvious that if we consider an Ito equation E, a vector field X which is a symmetry for E, and a change of coordinates (mapping the equation E into an equation E˜ ), the vector field X will also be a symmetry for the transformed equation E˜ . This point was raised and discussed in a recent paper of ours [13].

The situation is however quite different-and similar to the familiar one for deterministic equations – if one considers, as in Kozlov theory, only simple symmetries.

Lemma 1.

Simple (random or deterministic) symmetries of an Ito equation are preserved under changes of coordinates.

Proof. See [13].

4. Deterministic symmetry and integrability for a scalar Ito SDE

We will consider the following result, which in its essential part is due to Kozlov:

Theorem 1.

The SDE

dy=f˜(y,t)dt+σ˜(y,t)dw (4.1)
can be transformed by a deterministic map into
dx=f(t)dt+σ(t)dw, (4.2)
and hence explicitly integrated, if and only if it admits a simple deterministic symmetry.

If the generator of the latter is

X=φ(y,t)∂y, (4.3)
then the change of variables y = F(x, t) transforming (4.1) into (4.2) is the inverse to the map x = Φ(y, t) identified by
Φ(y,t)=∫1φ(y,t)dy. (4.4)

Proof. Consider a (scalar) SDE of the form (4.2). This is obviously and elementarily integrable; moreover it admits the Lie symmetry generator

X=∂x. (4.5)

If we operate a change of variable of the form

x=Φ(y,t) (4.6)
(note we are leaving t and w = w(t) untouched), the Ito formula gives
dx=(∂tΦ)dt+(∂yΦ)dy+12(∂yΦ)2σ2(t)dt. (4.7)

Putting together this and (4.2) we get

Φydy=[f(t)-Φt-12Φyy σ˜2(t)]dt+σ(t)dw. (4.8)

Now we just note that Φy ≠ 0, or (4.6) would not be a proper change of variable, so that we can divide by Φy. Thus (4.2) reads, in terms of the new variable, precisely as (4.1), where we have written

f˜(y,t)≔1Φy[f-Φt-12Φy2σ2]; (4.9)
σ˜≔σΦy. (4.10)

On the other hand, in the new coordinates the vector field X reads

X=∂y∂x∂y=∂F(x,t)∂x∂∂y=1Φy∂y. (4.11)

Following the discussion in Sect.3, we are guaranteed by Lemma 1 that (4.11) is a symmetry for (4.1).

In view of the general nature of (4.6), we have thus shown that if the equation (4.1) can be mapped via a simple change of variables of the form

y=F(x,t), (4.12)
to the equation (4.2), then necessarily it enjoys the symmetry (4.11). The required change of variables (4.12) is just the inverse to (4.6), i.e. these satisfy
Φ[F(x,t),t]=x;F[Φ(y,t),t]=y. (4.13)

It just remains to identify F in terms of the coefficient φ(y, t) in (4.3). Comparing the latter and (4.11) we immediately have

φ(y,t)=1Φy(y,t), (4.14)
which shows that (4.4) holds and identifies Φ(y, t) in terms of φ(y, t). As we need the change of variables leading from y to x, we just need the change of variable y = F(x, t) inverse to Φ, as stated in (4.13). This completes the proof in one direction, i.e. shows that indeed symmetry is a necessary condition for the equation (4.1) to be integrable.

In order to show that the condition is also sufficient for the integrability of (4.1), it suffices to perform the change of variables, thus obtaining (4.2) which is obviously integrable.

Remark 3.

The Theorem is due to Kozlov [18], who stated it in one direction only – i.e. identifying the symmetry condition as a sufficient one to guarantee integrability – and gave it in constructive form, i.e. identifying the change of variables which makes the equation explicitly integrable. Our contribution here is to show that this condition is not only sufficient but also necessary.

Remark 4.

One could think of extending the Kozlov theorem to more general situations: in fact, stochastic equations of the form (4.2) are not the only ones which can be integrated, and one could e.g. consider also those of the form dx = β(x)[f(t)d t + σ(t)dw]. This can be done, but actually gives no new result. The discussion of this fact is given in Appendix A.

Remark 5.

The most relevant feature of Kozlov theorem is of course that it is constructive: once we have determined the simple symmetry (if any) of the equation, it suffices to invert 4.14), which of course yields (4.4), to explicitly get the required change of variables and explicitly integrate the equation.

Remark 6.

The function Φ(y, t) is determined by (4.4) up to an “integration constant” which is actually an arbitrary function of t; it is clear from (4.9), (4.10) that the only effect of this would be to add a function of t alone to F(y, t) – or, seen from the other end of the procedure, to f(t) – while leaving S(y, t) and σ(t) unchanged; thus with no loss of generality for what concerns the integration of the SDE under study. (Similar considerations would also apply for the results in the next section 5, and will not be repeated there.)

Example 1.

The Ito equationb

dy=[e-y-(1/2)e-2y]dt+e-ydw (4.15)
admits the vector field X=e-y∂y as a (Lie-point) symmetry generator. By the change of variables x = exp[y] the vector field reads X=∂x , and the initial equation (4.15) reads
dx=dt+dw.

Example 2.

The (quite involved) equation

dy=e-t(1+y2)28y3(-4y2+et(3y4+2y2-1))dt-(1+y2)22ydw (4.16)
admits the vector field X=-[(1+y2)2/(2y)]∂y as a symmetry. Correspondingly, passing to the variable x=1/(1+y2) , we get X=∂x and the equation (4.16) just reads
dx=e-tdt+dw.

5. Deterministic symmetry and reduction for a system of Ito SDEs

In the previous section we have considered reduction of a scalar Ito equation under a simple Lie-point symmetry. In view of the discussion given there it is not surprising that the same result – with some obvious changes – also holds when we consider a system of Ito SDEs with a simple symmetry. In this section we will discuss in detail such an extension. We will again show that Kozlov’s sufficient conditions are also necessary.

We thus consider a system of n SDEs for x1(t),…,xn(t), which enjoys a symmetry of an appropriate type, and want to reduce this to a system of n − 1 equations for y1(t),…,yn-1(t); the processes yi(t) will be defined in terms of the xi(t). We have the following result.

Theorem 2.

Consider the system (2.1), with i = 1,…,n. The necessary and sufficient condition for the existence of a simple deterministic non-degenerate change of variables (2.2) which brings the system into the form

dyi=gi(y1,…,yn-1,t)dt+ρki(y1,…,yn-1,t)dwk(t) (i=1,…,n) (5.1)
is that (2.1) admits a simple deterministic Lie-point symmetry (2.7).

Proof. We will proceed substantially in the same way as in the proof for Theorem 1. We start from the “reduced” system (5.1), which is clearly invariant under the action of the vector field X=(∂/∂yn) ; we consider a general (obviously, invertible) change of variables, i.e. we bring it to its general form in arbitrary coordinates. So we consider the change of coordinates (y1,…,yn) → (x1,…,xn) defined by

yi=Fi(x,t), (5.2)
with inverse
xi=Φi(y,t). (5.3)

Under this change of variables the system (5.1) becomes (using Ito rule)

dyi=dFi(x,t)=∂Fi∂xkdxk+∂Fi∂tdt+12∂2Fi∂xk∂x𝓁dxkdx𝓁. (5.4)

Let us focus on the last term, and more specifically on the factor dxkdx𝓁 ; using (5.3) and again Ito rule

dxkdx𝓁=∂Φk∂ym∂Φ𝓁∂ysdymdys+o(dt)=∂Φk∂ym∂Φ𝓁∂ysρpmρqsdwpdwq+o(dt)=∂Φk∂ym∂Φ𝓁∂ysρqmρqsdt+o(dt).

Inserting this into (5.4), the latter reads

dyi=∂Fi∂xkdxk+∂Fi∂tdt+12∂2Fi∂xk∂x𝓁(∂Φk∂ym∂Φ𝓁∂ysρqmρqs)dt. (5.5)

These allow to deduce the SDEs satisfied by the process in terms of the new (i.e. the x) variables; in fact, using also (5.1), the above relation (5.5) is rewritten as

∂Fi∂xkdxk=[gi-∂Fi∂t-12∂2Fi∂xk∂xl(∂Φk∂ym∂Φ𝓁∂ysρqmρqs)]dt+ρkidwk(t). (5.6)

In order to obtain a system of the form (2.1), we need to invert the matrix (∂Fi/∂xk) , that is the Jacobian matrix of F. This inverse is of course the Jacobian for the inverse change of coordinates, i.e. the Jacobian for Φ; in fact

δki=∂yi∂yk=∂Fi[Φ(y,t),t]∂xk=∂Fi∂xl∂Φl∂yk. (5.7)

By acting on (5.6) from the left with the matrix (∂F/∂y)=(∂F/∂x)-1 , we get

dxk=∂Φk∂yi[gi-∂Fi∂t-12∂2Fi∂xm∂xl(∂Φk∂ym∂Φ𝓁∂ysρqmρqs)]dt+∂Φk∂yiρ𝓁idw𝓁    ≔fk(x,t)dt+σ𝓁k(x,t)dw𝓁. (5.8)

In the last step we have simply defined

fi(x,t)≔gi-∂Fi∂t-12∂2Fi∂xm∂xl(∂Φk∂ym∂Φ𝓁∂ysρqmρqs), (5.9)
σki(x,t)≔∂Φi∂yjρkj; (5.10)
here all the quantities which are function of y should be thought as function of x through (5.2).

Let us now turn our attention to symmetries. The original system (5.1) is by construction invariant under X=(∂/∂yn) . In the x variables this vector field is described by

X=(∂xi∂yn)∂∂xi=∂Φi∂yn∂∂xi≔φi(x,t)∂i. (5.11)

It should be stressed that when considering (possibly time-dependent) vector fields acting in Rn, we are dealing with the xi, yi as coordinates in Rn, not with the stochastic processes xi(t), yi(t) defined by SDEs (5.1), (2.1); thus we do not have to use the Ito formula.

As discussed in Section 3, it is not guaranteed apriori that symmetries present in one system of coordinates will still be present in another one; however as we deal with simple symmetries we are guaranteed by Lemma 1 we still have X – now given as (5.11) – as a symmetry.

We have thus determined the most general SDE (2.1) which can be obtained from (5.1) by a change of variables; these are identified by (5.9), (5.10). Reversing our point of view, we have shown that all the SDEs (2.1) which can be reduced to the form (5.1) by a change of variables are characterized by f, σ as given in (5.9), (5.10). Moreover these admit by construction the Lie-point symmetry (5.11). This shows that indeed the presence of such a symmetry is a necessary condition for the equation (2.1) to be reducible to the form (5.1).

Moreover, our discussion shows that there is a simple relation between the symmetry (5.11) and the change of variables needed to take (2.1) into the reduced form (5.1); this is given by

φi=∂Φi∂yn. (5.12)

In order to show that the presence of such a symmetry (5.11) for the equation (2.1) characterized by (5.9), (5.10) is also a sufficient condition for its (reducibility to the form (5.1) and hence) integrability, it suffices to note (as in [18]) that the change of variables (5.2) with Φ given by (5.12) produces exactly the required reduction.

Remark 7.

If (y1(t),…,yn − 1(t)) are a solution to the autonomous subsystem consisting of the first n − 1 equations in (5.1), then yn(t) is given by

yn(t)=yn(t0)+∫t0tgn[y1(t),…,yn-1(t);t]dt+∫t0tρkn[y1(t),…,yn-1(t);t]dwk(t), (5.13)
and is therefore known. In the language used in the symmetry analysis of deterministic equations, this can be seen as a “reconstruction equation”, and amounts to a (stochastic) quadrature.

Example 3.

Let us consider the two-dimensional system (we write all indices as subscripts to avoid confusion

dy1=(ey1-12e-2y1)dt+e-y1dw1
dy2=12ey2(2ey1+ey2+e2y1+y2)dt-ey1+y2dw1-ey2dw2. (5.14)

This admits as symmetry the vector field X=-ey2(∂/∂y2) . Actually, passing to the variables x1 = ey1, x2 = e − y2, the initial system (5.14) is rewritten as

dx1=x12dt+dw1
dx2=-x1dt+x1dw1+dw2. (5.15)

In these coordinates, X=(∂/∂x2)

6. Multiple symmetries and reductions for systems of SDEs

It is natural to ask now what happens if the system of SDEs (2.1) admits more than one symmetry, in particular – in view of well known results holding for deterministic equations [3,7,25,26,29] – what happens if it admits a r-parameter solvable group of symmetry. This was stated in [19,20] but without a formal proof. We will now give a formal statement and a detailed proof.

Theorem 3.

Suppose the system (2.1) admits an r-parameter solvable algebra 𝔤 of simple deterministic symmetries, with generators

Xk=∑i=1nφki(x,t)∂∂xi k=1,…,r, (6.1)
acting regularly with r-dimensional orbits.

Then it can be reduced to a system of m = (n − r) equations,

dyi=gi(y1,…,ym;t)dt+σki(y1,…,ym;t)dwk (i,k=1,…,m) (6.2)
and r “reconstruction equations”, the solutions of which can be obtained by quadratures from the solution of the reduced (n − r)-order system.

In particular, if r = n, the general solution of the system can be found by quadratures.

Proof. We want to proceed by induction on the dimension r of the algebra (for r = 1 this theorem obviously reduces to the previous one).

First of all we recall that the hypothesis on the algebraic structure of 𝔤 implies that there exists a chain of algebras 𝔤(k)(k=1,…,r) of dimension k such that 𝔤(0)⊂𝔤(1)⊂⋯⊂𝔤(r+1)=𝔤 , and each algebra 𝔤(k) is a normal subalgebra of 𝔤(k+1) , i.e.

[𝔤(k),𝔤(k+1)]⊂𝔤(k).

Suppose now we have an r = s + 1-parameter solvable group of symmetries G with Lie algebra g. Let Xs + 1 be a generator of 𝔤(s+1) that does not lie in 𝔤(s) , and consider the one-parameter group Gs + 1 generated by it. Thanks to the solvability condition, the system is invariant under the action of Gs + 1, since its action coincides to that of the quotient 𝔤(s+1)/𝔤(s) . Thus we can invoke Theorem 2 and reduce the system via a change of variable x→x˜ , getting a “reduced” system of n − 1 equations for x˜1(t),…, x˜n-1(t); and a last “reconstruction” equation which amounts to

x˜n=x˜n(t0)+∫t0tf¯n(x˜1,…,x˜n-1,t)dt+∫t0tσ¯kn(x˜1,…,x˜n-1,t)dwk(t). (6.3)

The key observation now is that it has been proven (Lemma 1) that the symmetries are maintained under changes of variables; thus we know for sure that the remaining s = r − 1 symmetries of the original system are still symmetries of the reduced one (once expressed in the new variables).

This allows to iterate the previous step and carry on the procedure described above, using the remaining s vector fields one by one. After performing all the allowed r steps we will have a reduced system of n − r equations for (x^1(t),…,x^n-r(t)) , and r “reconstruction” equations generalizing (6.3). Note that if we are able to obtain a solution to the reduced system, solutions to the full system require to solve the reconstruction equation in the “proper” order, i.e. the one dictated by the solvable structure of the symmetry algebra (to obtain solutions to the original equations we will of course also have to invert all the change of coordinates performed at each step in the reduction procedure).

Note also that if r = n, at the last step (after using r − 1 symmetries) the reduced system will amount to a single equation, with a symmetry of the form considered in Theorem 1; using this we will get, as in Theorem 1, a single SDE with coefficients depending only on the variable t, hence an integrable SDE. Solutions to the full systems are then obtained by solving the reconstructions equations.

Remark 8.

We stress that the solvability hypothesis was essential to guarantee that the actions of the quotients G(k + 1) / G(k) (where G(k) is the Lie group generated by 𝔤(k) , and so on) were symmetries of the system, according to the fact that if a normal subgroup H of the group G is a symmetry of a system, then G itself is also a symmetry if and only if the system reduced under H admits the quotient G/H as a symmetry group.

Example 4.

We will just refer the reader to Example 4.2 in Kozlov paper [18]. It is shown there that any equation of the form

dx1=(a1+b11x1+b12x2)dt+s11dw1+s12dw2dx2=(a2+b21x1+b22x2)dt+s21dw1+s22dw2 (6.4)
with ai, bij and sij real constant and satisfying the “full rank condition”, which in this case just reads
Det(s11s12s21s22)≠0,
admits symmetries
X=φ1(t)∂∂x1+φ2(t)∂∂x2,
with φi arbitrary smooth functions. Looking at two vector fields of the form
Xi=Ai(t)∂∂x1+Bi(t)∂∂x2,
with Δ: = A1 B2−A2 B1 ≠ 0 to ensure independence, the corresponding change of variables is
y1=B2(t)x1-A2(t)x2A1(t)B2(t)-A2(t)B1(t);y2=-B1(t)x1+A1(t)x2A1(t)B2(t)-A2(t)B1(t)

With this, the system (6.4) is mapped into

dy1=(a1B2-a2A2A1B2-A2B1)dt+(B2s11-A2s21A1B2-A2B1)dw1+(B2s12-A2s22A1B2-A2B1)dw2dy2=(a2A1-a1B1A1B2-A2B1)dt+(A1s21-B1s11A1B2-A2B1)dw1+(A1s22-B1s12A1B2-A2B1)dw2,
which is indeed of the form
dyi=Fi(t)dt+Ski(t)dwk
and hence integrable. The reader is referred to [18] for full detail.

7. Random maps and Kozlov theorem

We have so far considered changes of variables of the form (2.2); we will now consider a more general class of transformations, i.e. random maps [5].

Lemma 1 does still guarantee that symmetries are preserved under (simple) random maps, but in this case the discussion becomes more involved, and we will limit to consider simple random maps, see (2.11) and (2.12), and the framework of scalar Ito equations, i.e. the analogue of the situation discussed in Section 4 (systems will be discussed in a forthcoming contribution).

The difficulty here lies in that we are not guaranteed a simple random map (2.11) will take an Ito equation into an equation of Ito type. In fact, let us consider a scalar Ito equation (4.1) and assume the existence of a symmetry in the form X=φ(y,t,w)∂y . If we change coordinates passing to

x=Φ(y,t,w)=∫1φ(y,t,w)dy (7.1)
(this integral is defined up to an additive function β(t, w), which is a “constant of integration” in this framework), the vector field X is mapped again into X=∂x , and Lemma 1 guarantees this is a symmetry of the transformed equation, which we write as dx = fdt + σdw. Knowing that X = ∂x is a symmetry for this equation guarantees – in view of the determining equations (2.14), (2.15) – that (∂f/∂x)=0,(∂σ/∂x)=0; but now the coefficients f and σ could depend on w as well, in which case the transformed equation is not even an Ito one.

Remark 9.

On the other hand, if the equation (4.1) can be obtained from an integrable equation, i.e. one with f˜=f˜(t), σ˜=σ˜σ(t) , via a simple change of coordinates y = Θ(x, t, w), then necessarily the map identified by Φ given above – or more precisely by some Φ as above (the one which defines the change of variable inverse to the one defined by Θ), i.e. for some choice of the function β(t, w) playing the role of integration constant – takes (4.1) back to the original one, with no dependence of the coefficients on w.

In this context, we will find useful to consider how a simple random change of variables

x=Φ(y,t;w) (7.2)
acts on the the general (formal and) not necessarily Ito equation
dx=f(t,w)dt+σ(t,w)dw. (7.3)

Lemma 2.

Under the simple random change of variables (7.2), the general formal equation (7.3) is mapped into a (formal) equation of the type

dy=F^(y,t,w)dt+S^(y,t,w)dw (7.4)
with coefficients given by
F^(y,t,w)=f-Φt-(1/2)ΔΦΦy,S^(y,t,w)=σ-ΦwΦy. (7.5)

Proof. This follows by a standard computation. We start from (7.3), and operate with the general change of variables (7.2); note that t and w – and therefore also f(t, w) and σ(t, w) – are not changed in any way, and that to have a proper change of variables we must require (we recall that here Θ denotes the change of coordinates inverse to Θ) Φy≠0∀y and Θx≠0∀x .

Using Ito formula,

dx=Φydy+Φtdt+Φwdw+(1/2)Δ(Φ)dt=Φydy+Φtdt+Φwdw+(1/2)(ΦyyS^2+Φww+2S^Φyw)dt; (7.6)
note that here we have introduced S^ , i.e. the coefficient of the noise term in the equation for y, which we have not yet determined. From (7.6) and (7.3) we have
Φydy=[f-Φt-(1/2)Δ(Φ)]dt+[σ-Φw]dw; (7.7)
recalling that Φy is never vanishing, we divide by Φy and obtain an equation of the form (7.4) with coefficients given precisely by (7.5).

Corollary 1.

The equation (7.4) is obtained from an equation (7.3) by a simple change of coordinates (5.2) if and only if there exist functions f, σ and Φ satisfying the equations

(σ-Φw)=S^Φy, (f-Φt-(1/2)ΔΦ)=F^Φy. (7.8)

Proof. This is a trivial restatement of Lemma 2 above.

Remark 10.

Obviously eq. (7.3) admits X = ∂x as a Lie-point symmetry. By Lemma 1, this will also be a symmetry of the transformed equation; on the other hand, in the new coordinates we have ∂x = (∂y / ∂x) ∂y = (1 / Φy) ∂y. Thus, in the situation considered by Lemma 2, the transformed equation (7.4) admits the vector field

X=[Φy(y,t,w)]-1∂y≔φ(y,t,w)∂y (7.9)
as a (simple) Lie-point symmetry, as follows at once from Lemma 1.

This can also be checked by direct computation using the explicit form of (2.14), (2.15).

We are now interested in the case where the equation (7.4) is actually an Ito equation,

F^(y,t,w)=F(y,t), S^(y,t,w)=S(y,t). (7.10)

Lemma 3.

Let the equation (7.4) be of Ito type, i.e. let (7.10) be satisfied. Then the functions f, σ, Φ of Lemma 2 satisfy the equations

σ=Φw+SΦy; f=Φt+FΦy+(1/2)(Φww+2SΦyw+S2Φyy). (7.11)

Proof. This is exactly (7.8) with S taking the place of S^ and writing explicitly ΔΦ.

Lemma 4.

Let the “target” equation (7.4) be of Ito type, i.e. let (7.10) hold; moreover, let (7.4) be obtained from (7.3) by a simple random map (7.2). Then the “source” equation (7.3) is of Ito type if and only if

Φww+SΦyw=0; Φtw+FΦyw+(1/2)(ΔΦ)w=0. (7.12)

Proof. We are in the framework of Lemma 3. The equation (7.3) is of Ito type if and only if the coefficients f and σ do not depend on w. The requirement that σw = 0 and fw = 0, in view of (7.11), are exactly the first and second equation respectively in (7.12).

Remark 11.

Introducing Γ≔Φw , the equations (7.12) are rewritten as

Γw+SΓy=0; Γt+FΓy+(1/2)(ΔΓ)=0. (7.13)
(To obtain these, recall that now by assumption Sw = 0).

Theorem 4.

Let the Ito equation

dy=F(y,t)dt+S(y,t)dw (7.14)
admit as Lie-point symmetry the simple random vector field
X=φ(y,t,w)∂y. (7.15)

If there is a determination of

Φ(y,t,w)=∫1φ(y,t,w)dy (7.16)
such that the equations (7.12) are satisfied, then the equation is reduced to the explicitly integrable form
dx=f(t)dt+σ(t)dw (7.17)
by passing to the variable x = Φ(y, t, w).

Proof. When we operate with the map x = Φ(y, t, w), the equation for x obtained from (7.14) will be written as dx = fdt + σdw.

With such a map-for any determination of Φ – the vector field will X will just read as X = ∂x in the new coordinates, and we are guaranteed by Lemma 1 this will be a symmetry of the transformed equation. By (2.14), (2.15), this means that fx = 0, σx = 0, i.e. the equation in the x coordinate will be of the form (7.3).

Moreover, by Lemma 4, the coefficients of this satisfy fw = 0 = σw, i.e. we have a proper Ito equation, if and only if Φ satisfies the equations (7.12), in which now F and S are assigned functions, i.e. the coefficients of our equation (7.14). Thus, by choosing such a Φ (among those of the form (7.16)), we are guaranteed the f and σ do not depend on x nor on w, i.e. the equation is precisely of the form (7.17).

Remark 12.

This Theorem is based on the existence of a determination of an integral with certain properties. It is natural to wonder how one goes in order to study the existence of such a determination. In other words, one would like to have a criterion based on the directly available data, i.e. the functions F(y, t), S(y, t) and φ(y, t, w). This is provided by the next Theorem.

Theorem 5.

Let the Ito equation (7.14) admit as Lie-point symmetry the simple random vector field (7.15); define γ(y, t, w): = ∂w(1 / φ).

If the functions F(y, t), S(y, t) and γ(y, t, w) satisfy the relation

Sγt+Stγ=Fγw+(1/2)[Sγww+S2γyw], (7.18)
then the equation (7.14) can be mapped into an integrable Ito equation (7.17) by a simple random change of variables.

Proof. For any given determination Ξ of the integral, i.e. any function satisfying Ξy=(1/φ) , the most general integral is given by

Φ=∫1φdy=Ξ(y,t,w)-α(t,w),
with α an arbitrary function. With this, we write
Γ=Φw=Ξw-αw≔ψ-β.

The equations (7.13) read then

βw=ψw+Sψy, βt=ψt+Fψy+(1/2)[(Δψ)-Δ(β)]. (7.19)

Noting now that Δβ = βww, and that in view of the first of (7.19) we can write βww = ψww + Sψyw, we rewrite the equations as

βw=ψw+Sψy, βt=ψt+Fψy+(1/2)[Sψyw+S2ψyy]. (7.20)

The condition of compatibility βtw = βwt reads therefore

Sψyt+Stψy=Fψyw+(1/2)[Sψyww+S2ψyyw]. (7.21)

This can be slightly simplified by writing γ: = ψy; actually, recalling that ψ=Ξw , and that βy=0 , this definition yields

γ(y,t,w)=ψy=Ξwy=Φwy=∂w(1/φ), (7.22)
i.e. exactly the definition provided in the statement.

Thus the compatibility condition (7.21) can be written entirely in terms of the accessible data, i.e. the coefficients F and S in (7.14) and the coefficient φ of the symmetry vector field. More precisely, plugging γ = ψy into (7.21), we get exactly (7.18).

Remark 13.

Note that for φ independent of w (i.e. deterministic simple symmetries), which entails γ = 0, the equation (7.18) is always trivially satisfied – as it should be.

Remark 14.

Even in the case of deterministic symmetries, it can make sense to consider simple random changes of variables: by enlarging the set of considered transformations, we could obtain a simpler transformed equation (this will be the case in Example 5).

The Theorems 4 and 5 identify the presence of a simple random symmetry (7.15) such that the compatibility condition (7.18) is satisfied as a sufficient condition for integrability. It is quite simple to observe this is also a necessary condition.

Theorem 6.

Let the Ito equation (7.14) be reducible to the integrable form (7.17) by a simple random change of variables (7.2). Then necessarily (7.14) admits (7.9) as a symmetry vector field, and – with γ = ∂w(1 / φ) – (7.18) is satisfied.

Proof. Let y = Θ(x, t, w) be the change of variables inverse to (7.2). By assumption, there is an integrable Ito equation (7.17) which is mapped into (7.14) by Θ. Equation (7.17) admits X=∂x as a symmetry, and by Lemma 1 this will also be a symmetry of (7.14); on the other hand, X is written in the (y, t, w) coordinates exactly in the form (7.9). The fact that (7.18) is satisfied follows at once from the assumption that both (7.17) and (7.14) are Ito equations.

Thus, in the end, even within the framework of (simple) random maps and (simple) random symmetries, the presence of a simple random symmetries-now with the additional requirement that a compatibility condition (7.18) is satisfied-is a necessary and sufficient condition for a given (scalar) Ito equation to be integrable by a (simple random) change of variables.

8. Examples of integration via random symmetries

In this Section we will present several examples of reduction to integrable form via (changes of variables identified by) simple random symmetries. As this kind of reduction is new in the literature, we will discuss these examples in greater detail than the previous ones.

Example 5.

Let us consider the equation

dy=((1/2)t2e-t-y)dt+(t+k)e-tdw; (8.1)
this admits as symmetry X = e−t∂y. Here φw = 0 and the equation (7.18) is trivially satisfied. According to our general result, we should therefore use the change of variable described by
Φ=∫etdy=ety+β(t,w); (8.2)
with this, the equation (8.1) is mapped into
dx=(t22+(1/2)βww+βt)dt+(k+t+βw)dw. (8.3)

As expected, its coefficients are always independent of y.

For this to be an Ito equation, from the coefficient of dw it is needed to have βww = 0, hence β(t, w) = b(t) + b(1)(t)w, and from the coefficient of dt we get that actually we should require b(1)(t) = c. In other words, we have

Φ=cw+ety+b(t), (8.4)
where the constant c and the function b(t) are arbitrary. With the map x = Φ(y, t, w) and this choice of Φ, we get the equation
dx=(t22+b′(t))dt+(k+c+t)dw≔f(t)dt+σ(t)dw; (8.5)
this is always (that is, for any choice of b(t) and c) integrable.

We can get a somewhat simpler equation choosing c = − k and b(t) = 0, which corresponds to Φ(y, t, w) = − k w + ety; now the transformed equation is c

dx=t22dt+tdw. (8.6)

Note that in this case the symmetry is actually deterministic; if we were working in the framework of deterministic simple maps, the only difference would have been that we should have just considered c = 0 and β(t, w) = b(t).

Example 6.

The equation

dy=-(e-y+(1/2)e-2y)dt+e-ydw (8.7)
admits the vector field
X=e-yt+ey-w+1∂y≔φ(y,t,w)∂y (8.8)
as symmetry (this is a genuine random symmetry); condition (7.18) is satisfied. We hence have
Φ=∫1φ(y,t,w)dy=ey(t-w+1)+e2y2+β(t,w). (8.9)

By passing to the variable x = Φ(y, t, w) the equation (8.7) reads

dx=(βt+(1/2)βww+w-t-32)dt+(βw-w+t+1)dw≔fdt+σdw.

Again the coefficients are – as expected – always independent of y.

In order to have σw = 0 we must require βw w = 1, i.e. β(t, w) = b(t) + [b(1)(t)] w + (1 / 2) w2; with this, requiring fw = 0 enforces b(1)(t) = c1−t. The equation reads now

dx=-(1+t+b′(t))dt+(1+c1)dw, (8.10)
which is obviously integrable. Note that with our choice for β we have got
Φ(y,t,w)=w22+(c1-ey-t)w+[ey+e2y2+eyt+b(t)]. (8.11)

By choosing c1 = 0, b(t) = 0 we get slightly simpler expressions: the transformed equation is

dx=-(1+t)dt+dw, (8.12)
and the integrating map is
Φ(y,t,w)=w22-(ey+t)w+[1+ey2+t]ey. (8.13)

Note that (8.7) also admits a simple deterministic symmetry, i.e. X0 = e−y ∂y. d

Example 7.

Consider the Ito equations

dy=yp(t)dt+ydw, (8.14)
for p(u) any smooth function. The determining equation (2.15) reads in this case φ = φw + yφy and hence yields φ(y, t, w) = y ψ(z, t), where z: = ye−w; with this the other determining equation (2.14) reads ψt + z[p(t)−(1 / 2)]ψz = 0 and yields ψ(z, t) = η(ζ), where we have defined
ζ=zexp[R(t)]=yexp[-w+R(t)]; R(t)≔∫0t(12-p(u))du. (8.15)

Thus, in the end, the simple symmetries of (8.14) are identified by

φ(y,t,w)=yη(ζ), (8.16)
with η an arbitrary non-zero function. The compatibility condition (7.18) is always satisfied.

The change of variables identified by such symmetries are

x=Φ(y,t,w)=∫1yη(ζ)dy+β(t,w). (8.17)

In order to apply Lemma 3, and in particular (7.11), we compute partial derivatives of Φ; in doing this we should distinguish the case η = K (with K ≠ 0 a constant) and the generic one.

In fact, if η′(ζ) ≠ 0, we have

Φt=βt-R′(t)e-w+R(t)∫(ηζ/η2)dy=βt-R′(t)∫(ηζ/η2)dζ=βt+R′(t)η,Φw=βw+e-w+R(t)∫(ηζ/η2)dy=βw+∫(ηζ/η2)dζ=βw-1η;
by the same computation it is clear that in the case η = K, and hence ηζ = 0, we get
Φt=βt, Φw=βw.

Obviously we always have Φy = 1 /(yη).

As for the second order partial derivatives, these are (these formulas also hold for η = K, simply setting ηζ = 0 in them

Φyy=-η+ζηζy2η2,Φww=βww-ζηζη2,Φwy=ζηζyη2;

With these – and recalling S = y, F = y p(t), see (8.14) – we always get Δ(Φ) = βww−(1 / η). We can now apply (7.11) to get f and σ of the transformed equation.

(A) (η not constant). For η ≠ K, applying the first of (7.11) we get immediately

σ=βw. (8.18)

Applying the second of (7.11), for η non constant we get f = [−1 + 2p(t) + 2 R′(t) + ηβww + 2ηβt] /(2η); recalling now the expression for R(t), hence R′(t) = (1 / 2)−p(t), this just reduces to

f=βt+12βww. (8.19)

If now we require that f and σ are both independent of w, we get

β(t,w)=b(t)+cw.

With this-recalling (8.18) and (8.19) – we obtain e

dx=[b′(t)]dt+cdw. (8.20)

This shows that we can have several random symmetries – actually, in this case, infinitely many ones, depending on the choice of an arbitrary non-constant smooth function η – leading to the same integrable equation; this situation is discussed in more detail in Appendix B.

(B) (η constant). For η constant, applying the first of (7.11) we get

σ=βw+K-1; (8.21)
applying now also the second of (7.11) we get
f=(bt+12βww)+(p(t)-12)K-1. (8.22)

In order to have sw = 0 and fw = 0 we must again require β = b(t) + cw. Thus for η = K the reduced equation is

dx=1K[(p(t)-12+Kb′(t))dt+(1+Kc)dw]. (8.23)

Note also that if we choose η to be constant, we have a deterministic symmetry; this leads to a different reduction (for the same choice of β(t, w); it is clear that by suitable different choices of β the two reductions can be made equal), as seen by comparing (8.20) and (8.23).

Example 8.

Consider the Ito equation

dy=dt+ydw; (8.24)
the only simple Lie-point symmetry this admits is
X=ew-t/2∂y≔φ(y,t,w)∂y. (8.25)

In fact, in this case eq. (2.15) yields immediately

φ(y,t,z)=yψ(z,t), z=ye-w;
plugging this into (2.14) we obtain
(ψ+zψz)+(ψt-(1/2)zψz)y=0.

Each of the two terms in brackets must vanish, hence ψ(z, t) = η(t) / z and η = k e−t / 2. Going back to the original variables and function, we obtain (8.25).

For this form of φ, we have γ=∂w(1/φ)=et/2-w , and it is easy to check that the compatibility condition (7.18) is not satisfied.

Let us now consider the changes of variables (7.2) such that X=∂x .in the new variables, i.e. yielding equations of the form (7.3); we know these are of the form

Φ(y,t,w)=∫1φdy=(et/2-w)y+β(t,w).

The coefficients f, σ in (7.3) are given in this case by (7.11), and we obtain

σ=βw, f=et/2-w+12βww+βt; (8.26)
requiring σw = 0 would enforce βww = 0, i.e.
β(t,w)=b0(t)+b1(t)w;
with this, we get f=et/2-w+β′0+wb′1 , and the requirement fw = 0 amounts to
b′1(t)=et/2-w;
which does not admit any solution.

Thus in this case the equation admits a simple random symmetry, but it can not be transformed into an integrable Ito equation by a simple random map; this corresponds to the fact that the compatibility condition (7.18) is not satisfied.

9. Conclusions and discussion

We have considered in constructive terms the relation between symmetry and integrability – or at least reducibility – for stochastic differential equations (or systems thereof).

We started by recalling that symmetries of an Ito equation are defined only algebraically, not geometrically; thus – contrary to the deterministic case or even to the Stratonovich one – in this context it is not granted that symmetry are preserved under a change of variables. On the other hand, preservation of simple deterministic symmetries (which are exactly the symmetries relevant in Kozlov theory [18–20]) of Ito equations is guaranteed, as can be shown by considering the relation with symmetries of the associated Stratonovich equations [13,32].

We have then considered Kozlov results [18–20]; our Theorems 1, 2 and 3 reproduce them, but while Kozlov was interested in providing sufficient symmetry conditions for integrability or reducibility, we have also considered necessary ones; actually showing that Kozlov conditions are necessary and sufficient. Moreover, we have provided fully explicit and complete proofs to his results.

On the other hand, one can consider more general sets of maps and hence symmetries; in particular, we have considered simple random maps and symmetries [15]. In this case the preservation of symmetries of an Ito equation is granted by a recent result of ours [13].

We have then tackled, confining our study to the scalar case, the problem of extending Kozlov theory to the framework of (simple) random maps and symmetries. This extension is provided by Theorem 5 and Theorem 6, which are a direct extension of Kozlov theorem [18] and of our completion of it (Theorem 1), except that now we have to ask further conditions (7.12) to ensure we obtain an Ito equation, or equivalently a compatibility condition (7.18) to be sure they can be satisfied.

Some final considerations are in order here, also regarding possible further extensions of Kozlov theory.

  1. (a)

    Our study of random maps and possibly random symmetries was limited to scalar equations; we trust it can be extended to the case of systems, but such study is deferred to a forthcoming contribution.

  2. (b)

    A “naive” extension of Kozlov theory would consider reduction to SDEs which can be integrated albeit not being of the form considered here (and hence not “immediately” integrable); this would be e.g. the case for separable SDEs. As mentioned above, in this case one does not obtain anything new w.r.t. the standard Kozlov theory; this is discussed in detail in Appendix A.

  3. (c)

    Further generalizations can be envisaged, albeit they have not been considered here. The discussion by Kozlov [18–20] suggests that there is no point in considering deterministic symmetries which are not simple, i.e. which also act on t; on the other hand, there is probably a point in considering W-symmetries [11], which will be done elsewhere.

  4. (d)

    Also, here we have mainly investigated the relation between symmetries and integrability, which corresponds to an extremely simplified form of SDE; but other – less radical – simplification of the SDEs under study can also be of interest. One has been considered in Section 5, where we considered systems which are reducible to a system of lower dimension, albeit not integrable, plus some “reconstruction equations”. One could also consider e.g. systems which are separable into two or more subsystems, with no “reconstruction equation”. Such systems can also be characterized, albeit less immediately, in symmetry terms and it is thus possible that systems which can be brought into such form can also be detected in terms of their symmetry properties.

  5. (e)

    In this note we have only considered Ito equations. One could of course consider also more general stochastic differential equations and their symmetries (see e.g. the recent paper [1]; some of its authors also propose an alternative approach to reduction and reconstruction of stochastic differential equations via symmetries [8]). In this framework the approach presented here can be of use to identify such more general equations which can be mapped back to an Ito integrable equation.

  6. (f)

    Finally we note that, as in the deterministic case, one expects that the relation between symmetries and integrability displays special features in the case of stochastic equations with a variational origin. This lies beyond the limit of the present study but several results exist in this direction, see e.g. [2,21,31,33,34].

Acknowledgements

We are most grateful to an unknown Referee of the companion paper [13] who pointed out a missing term in the formula of the Ito Laplacian (2.10), thus avoiding incorrect conclusions. We are equally grateful to the Referee of the present paper for detecting several errors and for several other constructive remarks; and above all for kindly but firmly pushing us to substantially improve the paper. We are also grateful to the Editor for his patience in awaiting the revised version. The first part of this work follows from the M.Sc. Thesis of CL. We thank Francesco Spadaro (EPFL Lausanne) for useful discussions. A substantial part of this work was performed while GG was visiting SMRI; the work of GG is also supported by GNFM-INdAM.

Appendix A.

The “extended” Kozlov theorem

Equations of the form (4.2) are not the only SDEs which can be integrated. Another class is provided by equations of the form

dx=β(x)f(t)dt+β(x)σ(t)dw. (A.1)

Note that these are integrated to provide the solution in implicit form only (in general), i.e. as

B(x)≔∫1β(x)dx=∫f(t)dt+∫σ(t)dw.

Correspondingly we would have an extension of Theorem 1 to consider this case. It turns out, however, that equations of the form (A.1) do not in general admit simple Lie symmetries. This is detailed in Lemma A.1 below.

Lemma A.1.

The equation (A.1) does not admit any simple Lie point symmetry generator unless β(x) = x; in this case the admitted symmetry generator is

X=x∂x. (A.2)

Proof. The equation (2.9) reads in this case

σ(βφx-φβx)=0,
and hence (for σ ≠ 0) has solution
φ(x,t)=A(t)β(x).

Plugging this general expression for φ into (2.8), we get

A′(t)β(x)=-(1/2)[β(x)σ(t)]2A(t)βxx(x).

This in turn implies

A′Aσ=-(1/2)ββxx;
but here the expression on the l.h.s. is a function of t alone, that on the r.h.s. of x alone. Hence we must have
A′Aσ=c0=-(1/2)ββxx,
with c a numerical constant.

The first of these equations yields

A(t)=c1exp[c0∫g2(t)dt
with c1 an arbitrary constant; the second one requires
ββxx=-2c0,
and the solution is given by
β(x)=b0x, (A.3)
with b0 another constant. This also implies c0 = 0, and hence A(t) = c1.

In other words, (A.1) has a simple Lie symmetryf if and only if β(x) is of the form (A.3) (then we can move the constant b0 into the f(t) and g(t) functions, i.e. we can assume b0 = 1 with no loss of generality).

It should be noted that for β(x) of the form (A.3) and the associated (A.1), the determining equations give (A.2). The symmetry group generated by (A.2) is just a scaling one, x → λ x.

As mentioned above, one could in principles follow the approach of Theorem 1 and get a result similar to the one there for equations which can be reduced to the form (A.1). However, we have seen that a characterization in terms of simple symmetries is possible only for β as in (A.3). But for such a case we can come back to the case considered by Kozlov.

Lemma A.2.

The equation (A.1) in the case (A.3) is mapped to an equation of the form (4.2) by the change of variables

x=Φ(y)≔ey.

Proof. By direct computation.

Appendix B.

Multiple simple random symmetries

We have observed, in Examples 7 and 8, that one can have a situation where random symmetries correspond to a certain pattern in which enters an arbitrary function. Here we want to discuss this situation – and the conditions in which it can occur – in some detail. We will again confine ourselves to scalar Ito equations, as in Sections 7 and 8.

First of all, given an Ito equationg (2.1) we introduce the linear operators

L≔∂t+[f(x,t)-12σ(x,t)σx(x,t)]∂x; M≔∂w+σ(x,t)∂x. (B.1)

With these, the determining equations (2.14), (2.15) are written as

L(φ)+12M2(φ)=φ(∂xF),
M(φ)=φ(∂xσ). (B.2)

Suppose now that we have a solution χ(x, t, w) to these equations, and let us look for solutions in the form φ = ψχ. Plugging this into (B.2), we obtain

χL(ψ)+ψL(χ)+12[χM2(ψ)+2M(ψ)M(χ)+ψM2(χ)]=ψχ(∂xF),χM(ψ)+ψM(χ)=ψχ(∂xσ). (B.3)

Recalling now that χ is, by assumption, a solution to (B.2), these reduce to

χL(ψ)+12[2M(χ)M(ψ)+χM2(ψ)]=0, χM(ψ)=0. (B.4)

The second equation requires ψ ∈ Ker(M), and assuming this the first ones reduces to ψ ∈ Ker(L) Thus ψ is any function in

𝒡=Ker(L)∩Ker(M). (B.5)

Looking back at Example 7, in that case f = yp(t), σ = y; hence (with the present notation) L=∂t+[p(t)-1/2]x∂x,M=∂w+x∂x . It is then immediate to obtain that Ker(L) consists of arbitrary functions P[z𝓁,w] , while Ker (M) consists of arbitrary functions Q[zm, t] where

z𝓁=xexp[-∫0t[p(u)-1/2]du], zm=xe-w . 

In this case, 𝒡=Ker(L)∩Ker(M) is non-trivial, and consists of arbitrary functions

η(xexp[-w-∫0t[p(u)-1/2]du]) (B.6)

To be more concrete, e.g. with the choice p(t) = 1 we get 𝒡=η(xe-w-t/2) ,. while for p(t) = t we get 𝒡=η(xe-w+t/2-t2/2) .

Let us now look at Example 8. In this case f = 1, σ = y; hence (with the present notation) L=∂t+(1-x/2)∂x,M=∂w+x∂x . In this case we get that again Ker(L) and Ker(M) consist respectively of arbitrary functions P[z𝓁,w] and Q[zm, t], where now

z𝓁=(x-2)et/2, zm=xe-w.

With these, 𝒡 is trivial, i.e. reduces to constant functions.

Footnotes

a

By “smooth” we will always mean C∞, albeit in most steps it would be sufficient to consider C2 smoothness.

b

This and the following example are also given in [13]; see there for details.

c

An even more radical simplification is obtained by choosing c = − k and b(t) = −t3 / 6, i.e. Φ = −kw + ety−t3 / 6; with this choice the transformed equation is dx = tdw.

d

More generally, any function φ(y, t, w) = e−y η(ey + t − w) identifies a Lie-point symmetry for (8.7); see also the following Example 7 and Appendix B in this respect.

e

Note that in this case we obtain dx = dβ; indeed it follows directly from eq. (7.11) that choosing Φ to be just the integral part of (8.17) (i.e. setting b(t, w) = 0) we get f(x, t) = σ(x, t) = 0. Note also this is not the case in Example 6.

f

Equations with different β(x) could in principles admit more general symmetry generators, i.e. could have symmetries which are not simple ones. We will not discuss these here.

g

We stress this is written in terms of the x variable, and correspondingly the coefficient of the Wiener process in the equation is σ(x, t): in this framework – and hence in this Appendix – the Ito Laplacian is written exactly as in (2.10).

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TY  - JOUR
AU  - G. Gaeta
AU  - C. Lunini
PY  - 2021
DA  - 2021/01/06
TI  - Symmetry and integrability for stochastic differential equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 262
EP  - 289
VL  - 25
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2018.1452673
DO  - 10.1080/14029251.2018.1452673
ID  - Gaeta2021
ER  -