Random Lie-point symmetries of stochastic differential equations

TL;DR

This paper investigates the invariance properties of stochastic differential equations under random transformations, deriving conditions for symmetries in both Ito and Stratonovich frameworks.

Contribution

It introduces the determining equations for random Lie-point symmetries of stochastic differential equations, extending symmetry analysis to stochastic contexts.

Findings

Derived the determining equations for random Lie-point symmetries.

Established relations between symmetries in Ito and Stratonovich forms.

Connected results with previous literature on stochastic symmetries.

Abstract

We study the invariance of stochastic differential equations under random diffeomorphisms, and establish the determining equations for random Lie-point symmetries of stochastic differential equations, both in Ito and in Stratonovich form. We also discuss relations with previous results in the literature.

Equations

$${\partial }_t:=(\partial /\partial t),{\partial }_i:=(\partial /\partial x^i);{\hat{\partial }}_k:=(\partial /\partial w^k).$$

$$X=\tau (x,t;w){\partial }_t+{\phi }^i(x,t;w){\partial }_i.$$

$$dp(z,\theta )=\left[1/\sqrt{2\pi \theta }\right]e^{-z^2/\theta }dz.$$

$$t\to s=t+\epsilon \tau (t),$$

$$\hat{w}(s)=\sqrt{1+\epsilon {\tau }^{\prime }}w(s),$$

$$d\hat{p}(\zeta ,\theta )=\left[1/\sqrt{2\pi \theta }\right]e^{-{\zeta }^2/\theta }d\zeta .$$

$$d{w}^k\to d{w}^k+\epsilon \frac{1}{2}\left(\frac{d\tau }{dt}\right)d{w}^k:=d{w}^k+\epsilon \delta w^k.$$

$$d{x}^i=f^i(x,t)dt+{\sigma }_k^i(x,t)d{w}^k.$$

$$△u:=\sum _{k=1}^n\frac{{\partial }^{2}u}{\partial w^k\partial w^k}+\sum _{j,k=1}^n(\sigma {\sigma }^T{)}^{jk}\frac{{\partial }^{2}u}{\partial x^j\partial x^k}:={△}_{w}u+{△}_{x}u.$$

$$X={\phi }^i(x,t){\partial }_i,$$

$$Y={\phi }^i(x,t;w){\partial }_i.$$

$$\cases{\partial_{t}\varphi^{i}\ +\ f^{j}\,(\partial_{j}\varphi^{i})\ -\ \varphi^{j}\,(\partial_{j}f^{i})\ =\ -\,\frac{1}{2}\,\triangle(\varphi^{i})&,\cr\sigma^{j}_{\ k}\,(\partial_{j}\varphi^{i})\ -\ \varphi^{j}\,(\partial_{j}\sigma^{i}_{\ k})\ =\ 0\ .\cr}$$

$$d{x}^i$$

$$f^i(x,t)$$

$${\sigma }_k^i(x,t)$$

$$d{x}^i=[f^i(x,t)+\epsilon (\delta f{)}^i(x,t)]dt+{\sigma }_k^i(x,t)+\epsilon (\delta \sigma {)}_k^i(x,t)]d{w}^k,$$

$$(\delta f{)}^i(x,t)$$

$$(\delta {\sigma }^{i}k)(x,t)$$

$$\cases{(\partial_{t}\varphi^{i})\ +\ f^{j}(\partial_{j}\varphi^{i})\ -\ \varphi^{j}(\partial_{j}f^{i})\ =\ -\ \frac{1}{2}\,(\triangle\varphi^{i})&\cr(\widehat{\partial}_{k}\varphi^{i})\ +\ \sigma^{j}_{\ k}(\partial_{j}\varphi^{i})\ -\ \varphi^{j}(\partial_{j}\sigma^{i}_{\ k})\ =\ 0&.\cr}$$

$$X:={\partial }_t+f^j{\partial }_j,\hat{X}:={\hat{\partial }}_k+{\sigma }_k^j{\partial }_j;Y_{\phi }:={\phi }^j{\partial }_j,Z_{\phi }:=(△{\phi }^j){\partial }_j,$$

$$[X,Y_{\phi }]=-\frac{1}{2}{Z}_{\phi };[\hat{X},Y_{\phi }]=0.$$

$$Y=\tau (x,t;w){\partial }_t+{\phi }^i(x,t;w){\partial }_i+h^k(x,t;w){\hat{\partial }}_k.$$

$${\hat{\partial }}_k:=\partial /\partial w^k.$$

$$h^k=B_{\ell }^k(x,t;w)w^{\ell }$$

$$x^i$$

$$t$$

$$w^k$$

$$d{x}^i=\left[f^i(x,t)+\epsilon (\delta f{)}^i(x,t,w)\right]dt+\left[{\sigma }_k^i(x,t)+\epsilon (\delta \sigma {)}_k^i(x,t,w)\right]d{w}^k;$$

$$f^i[x+\epsilon \phi ,t+\epsilon \tau ]$$

$${\sigma }_k^i[x+\epsilon \phi ,t+\epsilon \tau ]$$

$$dF$$

$$Y_0:={\partial }_t+f^j{\partial }_j,Y_k:={\hat{\partial }}_k+{\sigma }_k^j{\partial }_j;L:=Y_0+\frac{1}{2}△.$$

$$d{\phi }^i=L[{\phi }^i]dt+Y_k({\phi }^i)d{w}^k,d\tau =L[\tau ]dt+Y_k(\tau )d{w}^k,d{h}^k=L[h^k]dt+Y_k(h^k)d{w}^k.$$

Full text

I Introduction

Symmetry analysis of differential equations is a powerful and by now rather standard tool in the study of deterministic nonlinear problems Olver1 ; symmref ; CGbook ; its use in the context of stochastic differential equations sderef ; Oksendal ; LArnold ; Stroock is comparatively much less developed (a partial exception being the versions of Noether theorem Olver1 ; noeref for stochastic variational problems varstoref ).

Albeit some concrete results exists, in particular concerned with strongly conserved quantities related to symmetries Mis1 ; Mis2 ; AlbFei and to the linearization problem Meleshko , a great deal of activity has been so far devoted to discussing what the suitable definition of symmetry would be in the case of stochastic differential equations (SDEs in the following); here we refer e.g. to Mis1 ; Mis2 ; AlbFei ; Meleshko ; WSM ; symmstoref ; Kozlov ; GRQ1 ; GRQ2 . See also GGPR for a review.

As in the case of deterministic differential equations, these works considered smooth vector fields in the space of independent and dependent variables (also called the phase space or manifold); in fact, dealing with smooth vector fields is at the hearth of the Sophus Lie approach, in that it allows to deal with infinitesimal transformations and hence with linearized problems.

On the other hand, when dealing with SDEs one is from the beginning considering an object which is not just a smooth vector field; the evolution described by a SDE can be described in terms of random diffeomorphisms, i.e. a diffeomorphism which depends on a random process. Thus in this context it would be quite natural to consider invariance under the same class of transformations.

Actually this is exactly what has been done by L.Arnold and P.Imkeller in their seminal work on normal forms for SDEs ArnImk (see also the book by L.Arnold LArnold ); as well known, the theory of (Poincaré-Dulac in the case of general dynamical systems, or Birkhoff-Gustavsson for Hamiltonian ones) normal forms ArnGMDE ; refNF1 is intimately connected with symmetry properties CGbook ; ArnGMDE ; refNF2 , so that the success of their approach suggests that one can follow the same path in discussing general symmetry properties of SDEs outside the perturbation approach.

The goal of the present paper is indeed to apply the Arnold-Imkeller approach to the analysis of symmetries of SDEs. We will see this can be done without difficulties, and explicit determining equations for the symmetry of a given SDE can be obtained. The formulation of these is the main contribution of our paper. We will also consider some concrete examples and determine symmetries for them, choosing equations which have physical relevance.

We will assume the reader has some familiarity with the basic concepts in the theory of symmetry of (deterministic) differential equations (see e.g. Olver1 ; symmref ; CGbook ), and also with the basics of stochastic differential equations (see e.g. sderef ); as the former may be not so familiar to readers primarily interested in SDEs, we will very briefly go over basic concepts for standard (that is, deterministic) symmetries of SDEs.

We will first consider simple class of symmetries, in order to focus on the main point of our contribution, and only later on discuss the most general case. This will make the paper a little longer than it would be going directly to the most general case, but we trust it will help the reader – not to say that simple symmetries seem to be the most useful in applications.

The plan of the paper is as follows. After briefly introducing the class of maps to be considered, i.e. random diffeomorphisms (Sect.II), we will first discuss symmetries of SDEs in Ito form (Sects.III and IV) in increasing generality and examples of these (Sect.V), passing then to discuss the case of Stratonovich SDEs (Sect. VI) and examples of these (Sect.VII). We then discuss the relation between symmetries of an Ito equation and of its Stratonovich counterpart (Sect.VIII); we also discuss the (lack of) simple algebraic structure of symmetry generators for a given Ito equation and the (existence of) the same structure for a given Stratonovich equation (Sect.IX). Finally we draw our brief Conclusions in Sect.X.

All the functions and other mathematical objects (manifolds, vector fields) to be considered will be assumed – unless differently stated – to be smooth; by this we will always mean $\mathcal{C}^{\infty}$.

We will always use (unless differently stated) the Einstein summation convention; we will usually denote partial derivatives w.r.t. the $t$ and $x$ variables, and later on also w.r.t. the $w^{k}$ variables, by the shorthand notation

[TABLE]

II Allowed Maps

II.1 Random diffeomorphisms

Arnold and Imkeller ArnImk define a near-identity random map $h:\Omega\times M\to M$, with $M$ a smooth manifold and $\Omega$ a probability space, as a measurable map such that:

(i) $h(\omega,.)\in\mathcal{C}^{\infty}(M)$;

(ii) $h(\omega,0)=0$;

(iii) $(Dh)(\omega,0)=id$.

Property (i) means that we can consider this as a family of diffeomorphisms (i.e., passing to generators, of vector fields) on $M$, depending on elements $\omega$ of the probability space $\Omega$. The dependence is rather arbitrary, i.e. no request of smoothness is present.

We will also refer to the generator of such a map, with a slight abuse of notation, as a random diffeomorphism. Note that random diffeomorphisms (as well as random maps) only act in $M$, i.e. they do not act on the elements of $\Omega$.

In our case, $M={\bf R}\times M_{0}$, with ${\bf R}$ corresponding to the time coordinate, is the phase manifold for the system, while $\Omega$ will be the path space for the $n$-dimensional Wiener process $W(t)=\{w^{1}(t),...,w^{n}(t)\}$.

Moreover, as suggested by the notation above, we should consider $M$ as a fiber bundle over ${\bf R}$ (the fibers being $M_{0}$) and $h$ should not act on ${\bf R}$.

In the end, introducing local coordinates $x^{i}$ on $M_{0}$, we want to consider random diffeomorphisms generated by vector fields of the form

[TABLE]

A time-preserving random diffeomorphism will be characterized by having $\tau=0$, while the fibration-preserving ones (with reference to the fibration $M\to{\bf R}$) will be characterized by $\tau=\tau(t)$. One should also mention that special care is needed when considering time changes which depend on $x$ (which is itself a stochastic process) and/or $w$, which are random time changes RTC .

We will start by considering “simple” (i.e. time-preserving) random symmetries in order to tackle the key problem in the simplest setting; later on (see Section IV) we will consider the general case.

Remark 1. In the literature one considers also transformations directly222As opposed to the “indirect” action due to the modification of the Wiener process induced by the action on the time variable; see below. acting on the Wiener processes as well; this is related to so called “W-symmetries” GRQ2 . We will consider also this class of transformations, in which case one considers diffeomorphisms (in the extended space $(x,t;w)$) generated by vector fields of the form $X=\tau(x,t;w)\partial_{t}+\varphi^{i}(x,t;w)\partial_{i}+h^{k}(x,t;w)\widehat{\partial}_{k}$. $\odot$

II.2 Maps acting on the time variable

If we consider vector fields which act on the time variable as well, we should take into account that the Wiener processes $w^{k}(t)$ are affected by a change in $t$. In the simplest case, this action on $t$ will be just a “global” reparametrization of time, i.e. will not depend on the $x^{i}(t)$ and $w^{k}(t)$ variables.333Actually, in order to end up with an equation possibly of the same type (not to say about it being exactly the same equation as the original one) the transformed processes $\widetilde{w}$ should be only a function of the (transformed) time $\widetilde{t}$, i.e. $\widetilde{w}=\widetilde{w}(\widetilde{t})$; this amounts to requiring the transformation of time does not depend on the space coordinates $x^{i}$. On the other hand, it could depend on the Wiener processes themselves.

This situation was discussed, in the context of symmetries for SDEs, in GRQ1 (see Appendix A there); we give a short account of this discussion here for the sake of completeness.

The probability that a Wiener process $w(t)$ undergoes a change $dw=z$ in the time interval $\theta=dt$ has a density

[TABLE]

Under a near-identity map (we will assume $\tau^{\prime}<1/\varepsilon$ for all $t$)

[TABLE]

we have $\theta=dt=[1/(1+\varepsilon\tau^{\prime})]ds$; thus the density $dp$ should now be expressed in terms of $\widehat{\theta}=ds=(1+\varepsilon\tau^{\prime})dt$.

Instead of going through computations, we note that if we consider $\zeta=\sqrt{1+\varepsilon\tau^{\prime}}z$ and the stochastic process

[TABLE]

the probability that $\widehat{w}(s)$ undergoes a change $\zeta=d\widehat{w}$ in the time interval $\theta=ds$ has a density

[TABLE]

Thus we conclude that the map (2) induces the map (3) on the standard Wiener process.

In the case of $\tau=\tau(x,t)$ extra care should be paid: in general this would produce a random non-smooth map, and only those expressed as integrals should be allowed RTC (the integration has a regularizing role); proceeding in a formal way as we will do in the following has indeed in general a formal value, and the actual well-posedness of the considered maps should be verified in each case.

When the considered non-autonomous map is acceptable, with $\tau=\tau(x,t)$ (see e.g. Theorem 8.20 in the book by Oksendal Oksendal ) or even $\tau=\tau(x,t;w)$, one proceeds in a similar way and obtains exactly the same result (see Section IV). This implies in particular that under (2),

[TABLE]

III Ito equations; simple symmetries

We will consider stochastic differential equations in Ito form, i.e.

[TABLE]

In the following it will be convenient to use the notation

[TABLE]

For a function depending only on the $(x,t)$ variables – as in the case of deterministic symmetries – the first term vanishes identically.

III.1 Deterministic symmetries

We will start by considering simple symmetries; in the deterministic case these are generated by vector fields

[TABLE]

while when we look for simple random symmetries we mean those generated by a vector field

[TABLE]

The determining equations for simple deterministic symmetries of Ito equations (that is, for $\tau=0$ and $\varphi=\varphi(x,t)$) were determined in GRQ1 (see also GRQ2 for extensions), and turned out to be, in the present notation,

[TABLE]

III.2 Random symmetries

We will now consider the case of simple random symmetries, i.e. for vector fields of the form (8); under this we have $x^{i}\to x^{i}+\varepsilon\varphi^{i}(x,w)$, and hence

[TABLE]

Plugging these into (5), the latter is mapped into a new Ito equation

[TABLE]

where the variations are given by

[TABLE]

Thus the equations remains invariant if and only if, for all $i$ and $k$,

[TABLE]

These are the determining equations for simple random symmetries – of the form (8) – for the Ito equation (5).

Note that introducing the vector fields

[TABLE]

the (11) are simply rewritten as

[TABLE]

Remark 2. The only difference w.r.t. the determining equations for deterministic symmetries (9) is the presence of the $\partial_{k}\varphi^{i}$ term in the second equation; but one should however recall that – despite the formal analogy – the term $\triangle\varphi^{i}$ does now also include derivatives w.r.t. the $w^{k}$ variables (i.e. the $\triangle_{w}\varphi^{i}$ term), which are of course absent in (9), where actually $\triangle\varphi^{i}=\triangle_{x}\varphi^{i}$. $\odot$

IV Ito equations; general random symmetries

So far we have considered (invariance of SDEs under) maps generated by vector fields of the special forms (7) or (8). We want now to remove this limitation, and consider general vector fields in the $(x,t;w)$ space, i.e.

[TABLE]

Here we started to use, as mentioned above, the shorthand notation

[TABLE]

We also write $X=\tau\partial_{t}+\varphi^{i}\partial_{i}$ for the restriction of $Y$ to the $(x,t)$ space.

Remark 3. Note that in (14) we are considering also the possibility of direct action on the $w^{k}$ variables (apart from the action induced by a change in time), as in the approach to W-symmetries GRQ2 . As already pointed out there, the requirement that the transformed processes $\widehat{w}^{k}(t)=w^{k}(t)+\varepsilon h^{k}(x,t,w)$ are still Wiener processes, implies that $\widehat{w}^{k}=M^{k}_{\ell}w^{\ell}$ with $M$ an orthogonal matrix, and hence that necessarily

[TABLE]

with $B$ a (real) antisymmetric matrix; see GRQ2 for details. This will be assumed from now on. (Note moreover that if $B$ does not depend on $w$ then $\triangle(h^{k})$ reduces to its “deterministic” part.) $\odot$

Remark 4. On physical grounds one would be specially interested in the case where the change of time does not depend on either the realization of the stochastic processes $w^{k}(t)$ or on the spatial coordinates $x^{i}$, i.e. on fiber-preserving maps. These will be obtained from the general case by simply setting $\tau=\tau(t)$. It should also be noted that, beside any physical considerations, a (non trivially) space dependent time change would provide a process which is not absolutely continuous w.r.t. the original one - thus definitely not of interest in the present context. See also the brief discussion in Sect.II.2. $\odot$

IV.1 The general case

The vector field (14) induces – taking into account also the discussion of the previous Section II.2 and in particular eq.(4) – the infinitesimal map

[TABLE]

With this, the Ito equation (5) will read

[TABLE]

we do of course aim at obtaining explicit expressions for $\delta f$ and for $\delta\sigma$.

Working, as always, at first order in $\varepsilon$, we have

[TABLE]

The differentials $d\varphi^{i}$, $d\tau$, $dh^{k}$ should be computed by the Ito formula; for a generic function $F(x,t;w)$ we have, making use of (5),

[TABLE]

where we have defined the Misawa vector fields $Y_{\mu}$ and the second order operator $L$ by

[TABLE]

The expressions for $d\varphi^{i}$, $d\tau$, $dh^{k}$ are immediately obtained specializing (19):

[TABLE]

Using (18) and (21) we can rewrite (17) in the form

[TABLE]

We like to write this in the form

[TABLE]

here, setting $\delta w^{k}=\psi dw^{k}$ (with $\psi=(1/2)(\partial_{t}\tau)$, see (4)), we have

[TABLE]

We thus conclude that the determining equation for (random) symmetries of the Ito equation (5) are

[TABLE]

These can also be finally rewritten, using the explicit form of $L$ and $\psi$, as

[TABLE]

Several special cases are considered in the following.

Remark 5. This is a system of $n+n^{2}$ linear equations for the $2n+1$ unknown functions $\{\tau,\varphi^{1},...,\varphi^{n};h^{1},...h^{n}\}$; these reduce to $n$ or $n+1$ functions if we consider simple symmetries or at least symmetries not acting directly on the $w$ variables. Thus the system is over-determined for all $n>1$, and in general we will have no symmetries; even in the case there are symmetries, the equations are not always easy to deal with, despite being linear, due to the dimension. For $n=1$ the counting of equations and unknown functions would suggest we always have symmetries, but the solutions could be only local in some of the variables. $\odot$

Remark 6. The solutions to the determining equations should then be evaluated on the flow of the evolution equation (the Ito SDE); this can lead some function to get less general, or even trivial; see Example 1 below. $\odot$

Remark 7. We focused on the definition of random symmetries of a SDE and on the determining equations for these; on the other hand, we have not considered how the symmetries can be used in the study of the SDE. The first use of symmetries for SDEs should be through the introduction of symmetry-adapted coordinates; (see Remark 8 in this respect). A more structured approach, relating simple symmetries to reduction pretty much as for deterministic equations, has been developed by Kozlov Kozlov in the case of deterministic symmetries of SDEs; we postpone investigation of the possibility to extend his results to the framework of random symmetries to future work. $\odot$

IV.2 Special cases

It is interesting to consider some special (simpler) cases.

(1) In the case of deterministic simple (time preserving) vector fields, i.e. $\varphi=\varphi(x,t)$, $\tau=h=0$, the equations (24) reduce to the (9) seen above.

(2) Similarly, in the case of simple random symmetries, i.e. $\varphi=\varphi(x,t;w)$, $\tau=h=0$, we get the equations (11) derived above.

(3) If we consider the case of deterministic fiber-preserving symmetries, i.e. $\varphi=\varphi(x,t)$, $\tau=\tau(t)$, $h=0$, the equations (9) reduce to

[TABLE]

These equations coincide with those derived in GRQ1 , see Theorem 2 there.

(4) When considering W-symmetries of SDEs GRQ2 one considered vector fields with, in the present notation, $\varphi=\varphi(x,t)$, $\tau=\tau(t)$, $h=h(t,w)$. In this case the equations (9) reduce to

[TABLE]

These equations were already obtained in GRQ2 , see the Corollary to Proposition 1 there.

(5) Let us consider the general case with $\varphi=\varphi(x,t;w)$, $\tau=\tau(t,w)$, $h=0$. The equations (9) are in this case

[TABLE]

(6) As mentioned above (see Remark 4) we are specially interested in the case where $\tau=\tau(t)$ while $\varphi$ and $h$ are in general form (up to the restriction on $h$ discussed in Remark 3). In this case the only simplifications in (9) are, of course, in the terms involving $\tau$, and amount to $Y_{0}(\tau)=(\partial_{t}\tau)$, $Y_{k}(\tau)=0$, and $\triangle(\tau)=0$. Thus in this case the (9) reduce to

[TABLE]

For $h=0$ (i.e. excluding W-symmetries) these further reduce to

[TABLE]

(7) Finally, in applications one is often faced with $n$-dimensional system, depending on $n$ Wiener processes,

[TABLE]

with $M$ and $\sigma$ constant matrices. It is also frequent that $\sigma$ is diagonal.

In this case (for general, i.e. non necessarily diagonal, $\sigma$) the determining equations for simple random symmetries read

[TABLE]

We start by considering the second set of equations; assuming moreover that $\sigma$ is diagonal, $\sigma=\mathrm{diag}(\lambda_{1},...,\lambda_{n})$, these yield

[TABLE]

where we have defined $z^{k}:=x^{k}-\lambda_{k}w^{k}$ (no sum on $k$). For functions of this form we get immediately (using again the ansatz on $\sigma$) that $\triangle\varphi=0$, and hence the first set of determining equations read simply

[TABLE]

This is equivalent to

[TABLE]

The first set of equations implies that

[TABLE]

while the second one states that $\nabla\varphi^{i}$ is in the kernel of $M^{T}$.

V Examples I: symmetries of Ito equations

V.1 Simple random symmetries

We start by considering simple random symmetries of Ito equations; in this case the relevant determining equations are (11). We will consider examples which were already studied – for what concerns deterministic symmetries – in GRQ1 , so that comparison with results in the deterministic case is immediate.

Example 1. We start by considering a rather trivial example, i.e. $n=1$ and

[TABLE]

with $\sigma_{0}\not=0$. In this case we just have a system of two equations for the single function $\varphi=\varphi(x,t;w)$, and (11) read

[TABLE]

The solution to the second of these is $\varphi=F(z,t)$, where $F$ is an arbitrary (smooth) function of $z:=x-\sigma_{0}w$ and $t$. Plugging this into the first equation, we get

[TABLE]

This is an autonomous linear equation, and it is readily solved (e.g. by considering the Fourier transform of $F$), showing that there are nontrivial simple random symmetries. Note however these will grow exponentially fast in time.

It should also be noted that $dz=0$ on solutions to our equation (34), see Remark 6.

Example 2. We consider another one-dimensional example, i.e.

[TABLE]

this was considered in GRQ1 , where it was shown it admits no deterministic symmetries. The (11) read now

[TABLE]

the second equation yields

[TABLE]

inserting this in the first equation – and recalling that now the coefficients of different powers of $x$ must vanish separately, as $\psi=\psi(t,z)$, we get two equations,

[TABLE]

Solving these, we have $\psi(z,t)=(c_{1}/z)\exp[-t/2]$, with $c_{1}$ an arbitrary constant; and hence we conclude that the equation (35) admits a simple random symmetry:

[TABLE]

Example 3. We pass to consider examples in dimension two; we will write the vector indices (in $x$, $w$, $\varphi$) as lower ones in order to avoid any misunderstanding. The first case we consider is a system related to work by Finkel Finkel , i.e.

[TABLE]

here $a_{1},a_{2}$ are two non-zero real constants.

The first set of (11) reads in this case

[TABLE]

while the second set of determining equations (11) reads

[TABLE]

These of course imply that, setting $z_{k}:=x_{k}-w_{k}$,

[TABLE]

Plugging these into the equations (37), and again recalling that – as $\eta_{i}=\eta_{i}(t,z_{1},z_{2})$ – the coefficient of different powers of $x_{1}$ must vanish separately, the first of those equations enforces

[TABLE]

while in the second we get $\partial\eta_{2}/\partial z_{1}=0$ and the equation reads

[TABLE]

Again this autonomous linear equation is readily solved, showing that there are simple random symmetries.

Example 4. Finally we will consider another two-dimensional example, which is an Ornstein-Uhlenbeck type process related to the Kramers equation:

[TABLE]

note we have here a single Wiener process $w(t)$, and correspondingly we will look for solutions $\varphi^{i}=\varphi^{i}(x_{1},x_{2},t;w)$.

It was shown in GRQ1 that this system admits some deterministic symmetries; in particular there are the symmetries

[TABLE]

As in the previous example, we will start from the second set of equations in (11); for our system these read

[TABLE]

These of course rule out any possible dependence on $w$, i.e. show that there is no simple random symmetry.

V.2 General random symmetries

Example 5. We will consider again the equations of Example 2, i.e.

[TABLE]

we have seen this does not admit any deterministic symmetry but it admits one simple random symmetry. We will now check this admits some more general random symmetry; in order to keep computations simple, we will restrict to the time-independent case $\tau=0$ and $\varphi_{t}=h_{t}=0$.

In this case the equations (24) read

[TABLE]

The second equation requires

[TABLE]

plugging this into the first one we get

[TABLE]

Solutions to these are provided by

[TABLE]

with $k$ an arbitrary constant and $\beta$ an arbitrary smooth function.

The random symmetries we obtained in this way are

[TABLE]

Example 6. We consider the system

[TABLE]

this is manifestly covariant under simultaneous rotations in the $(x_{1},x_{2})$ and the $(w_{1},w_{2})$ planes GRQ2 .

In order to simplify (slightly) the computations, we will look for symmetries which are time-preserving and time-independent; that is, we assume $\tau=0$, $(\partial_{t}\varphi^{i})=0=(\partial_{t}h^{k})$. The first set of (24) provides now

[TABLE]

Setting $z_{k}:=x_{k}-w_{k}$, these give

[TABLE]

where the $\rho_{i}$ are arbitrary smooth functions of $(z_{1},z_{2})$.

Plugging these into the first set of (24) we obtain two equations involving $\varphi^{i}$ and derivatives of the $\rho^{i}$,

[TABLE]

These equations can then be solved for the $\varphi^{i}$ in terms of the $\rho^{i}$, yielding some complicate expression we do not report. This shows we have random symmetries in correspondence with arbitrary functions $\rho_{i}(z_{1},z_{2})$.

When these are linear,

[TABLE]

and writing $\chi:=[-1+3(x_{1}^{2}+x_{2}^{2})]$, the resulting random symmetries are identified by

[TABLE]

With the choice

[TABLE]

we get just simultaneous rotations in the $(x_{1},x_{2})$ and $(w_{1},w_{2})$ planes GRQ2 .

Remark 8. It may be interesting, also in view of Remark 7, to change coordinates as suggested by the symmetry; we will set $x_{1}=\rho\cos(\vartheta)$, $x_{2}=\rho\sin(\vartheta)$; and similarly $w_{1}=\chi\cos(\lambda)$, $w_{2}=\chi\sin(\lambda)$. With these coordinates, the equations (41) read simply

[TABLE]

The invariance under simultaneous rotations in the $(x_{1},x_{2})$ and $(w_{1},w_{2})$ planes (i.e. simultaneous shifts in $\vartheta$ and $\lambda$) is now completely explicit. $\odot$

VI Symmetries of Stratonovich equations

So far we have considered SDE in Ito form; as well known, in some framework it is convenient to consider instead SDE in Stratonovich form,

[TABLE]

In particular, these behave “normally” under change of coordinates (on the other hand, the Stratonovich integral is not a martingale and its rigorous meaning is not immediate).

Stratonovich equations were also considered by pioneers in the symmetry analysis of differential equations Mis1 ; Mis2 ; AlbFei ; we are only aware of works dealing with deterministic symmetries of Stratonovich equations, so we believe a short discussion of their random symmetries is also of interest; this is given in the present Section.

VI.1 Simple symmetries

VI.1.1 Simple deterministic symmetries

We will first consider (also in order to familiarize with the notation) the action of a deterministic vector field (7) on Stratonovich equations.

Under the action of $X$, the equation (42) is mapped into

[TABLE]

taking into account (42) and expanding the term $d\varphi$, we have that terms of first order in $\varepsilon$ cancel out if and only if

[TABLE]

if now we substitute for $dx$ according to (42), this yields

[TABLE]

which is finally rewritten as

[TABLE]

The vanishing of this (for all realizations of the Wiener processes $w^{k}$) is possible if and only if the $(n+n^{2})$ equations

[TABLE]

are satisfied. These are the determining equations for the simple deterministic symmetry generators – of the form (7) – of the Stratonovich SDE (42).

Remark 9. We can introduce, as suggested by Misawa Mis1 , the $(n+1)$ vector fields

[TABLE]

With this notation, the determining equations (45) read simply

[TABLE]

VI.1.2 Simple random symmetries

We will now consider again the equation (42), but now discuss its variation under a vector field of the form (8); we will go through the same computation as in the previous subsection.

Under the action of $Y$, the equation (42) is mapped into

[TABLE]

taking into account (42) and expanding the term $d\varphi$, we have that terms of first order in $\varepsilon$ cancel out if and only if

[TABLE]

the last term in the l.h.s. is the only difference with respect to the computation in the deterministic case. Considering $x$ on the solutions to (42), we get

[TABLE]

which is also rewritten as

[TABLE]

and the determining equations for the random simple symmetry generators (of the form (8)) of the Stratonovich SDE (42) are therefore

[TABLE]

Remark 10. In order to express this in compact terms, it is convenient to modify slightly the definition of the (Misawa) vector fields associated with the SDE; we will now write

[TABLE]

Then the determining equations (50) read simply

[TABLE]

VI.2 Symmetries acting on the time variable

The computations presented in Section VI.1 above can be extended to cover the case where the considered transformations act on time as well; in this case the discussion of Section II.2 should be taken into account.

VI.2.1 Deterministic symmetries

In the simpler case, i.e. a smooth transformation not depending on the random variables (deterministic symmetries), the role of $X$ in (7) will be taken by

[TABLE]

Note that under this we get $t\to s=t+\varepsilon\tau(t)$.

As discussed above (see Section II.2), $w^{k}(t)$ is mapped into $\widetilde{w}^{k}(t)=\sqrt{1+\varepsilon\tau_{t}}w^{k}(t)$, and hence $d\widetilde{w}^{k}=[1+\varepsilon(\tau_{t}/2)]dw^{k}$. Making use of this fact, and proceeding in the same way as above, we get at first order in $\varepsilon$

[TABLE]

Substituting now for $dx^{j}$ according to (42), we get

[TABLE]

the determining equations are therefore

[TABLE]

VI.2.2 Random symmetries

When we consider random symmetries the computations are slightly more complex. Proceeding in the same way as earlier on, we obtain the determining equations in the form

[TABLE]

Remark 11. If we want to express this in terms of commutation properties, we introduce the vector fields

[TABLE]

then the determining equations are rewritten as

[TABLE]

VII Examples II: Stratonovich equations

Example 7. Let us consider the equation

[TABLE]

in this case the Misawa vector fields are

[TABLE]

The requirement that $X:=\varphi(x,t,w)\partial_{x}$ commutes with both $Y_{0}$ and $Y_{1}$ yields

[TABLE]

Example 8. Let us consider the system

[TABLE]

The Misawa vector fields are now

[TABLE]

Requiring the vector field

[TABLE]

to commute with $Y_{1}$ and $Y_{2}$ enforces

[TABLE]

where we have defined $z_{k}:=[(aw_{k}-\log|x_{k}|)/a]$. Requiring now that $X$ also commutes with $Y_{0}$, we get that actually it must be $\eta^{1}=\eta^{2}=c$; thus in conclusion the only simple random symmetry of the system under consideration is

[TABLE]

this is actually, obviously, a simple deterministic symmetry.

Example 9. We consider again the equation

[TABLE]

as in Example 2 above. The corresponding Stratonovich equation is

[TABLE]

the determining equations (50) for simple random symmetries of this Stratonovich equation read

[TABLE]

It is immediate to check these, or more precisely the first of these, do not correspond to the equations obtained in Example 2. But, this set of equations does admit as solution

[TABLE]

which is just the same solution we found in Example 2.

Example 10. When dealing with symmetries of Stratonovich equations, it is customary to consider the system, first introduced by Misawa Mis1 ,

[TABLE]

it is well known – and immediately apparent – that this admits the simple symmetry generated by

[TABLE]

(and many others, as discussed by Albeverio and Fei AlbFei ). Note that this involves only one Wiener process, which will induce a non-symmetric expression for the equivalent Ito system.

Using (58), the equivalent system of Ito equations turns out to be

[TABLE]

It is immediate to check that the determining equations (11) are not satisfied by $X$; more precisely, the second set of (11) are (of course) satisfied, while the first set is not: in fact, we get (for all $i=1,2,3$)

[TABLE]

where we have written

[TABLE]

VIII Symmetries of Stratonovich vs. Ito equations

As well known, there is a correspondence between stochastic differential equations in Stratonovich and in Ito form. In particular, the Stratonovich equation (42) and the Ito equation (5) are equivalent if and only if the coefficients $b$ and $f$ satisfy the relation

[TABLE]

Note this involve implicitly the metric (to raise the index in $\sigma$); as we work in ${\bf R}^{n}$ we do not need to worry about this. Moreover, for $\sigma$ (and hence also $\sigma^{T}$) a constant matrix, we get $\rho=0$ i.e. $b^{i}=f^{i}$.

Note also that $\sigma$ is the same in (42) and in (5); thus (58) can be used in both directions. In particular, we can immediately use it to rewrite the determining equations for symmetries (of different types) of the Stratonovich equation (42) in terms of the coefficients in the equivalent Ito equation.

One would be tempted to study symmetries of an Ito equation by studying the symmetries of the corresponding Stratonovich one. This would be particularly attractive in view of the fact that the determining equations (55) for random symmetries of Stratonovich equations are substantially simpler than the determining equations (24) for symmetries of Ito equations; the same holds at the level of determining equations for simple random symmetries, as seen by comparing (50) and (11).

Unfortunately, this way of proceeding would give incorrect results (as also shown by Example 10 above); this is already clear in the case of simple random – and actually even deterministic – symmetries, so that we will just discuss this case

In fact, the determining equations (50) for random symmetries of (42) are immediately rewritten in terms of the coefficients $f^{i}$ of the equivalent Ito equation as ($i,k=1,...,n$)

[TABLE]

where $\rho^{i}(x,t)$ is defined in (58).

Note that the equations (59) can be expressed in the compact form (52) of commutation with the vector fields $Y_{\mu}$ defined in (51), except that now the same vector field $Y_{0}$ should now better (but equivalently) be defined as

[TABLE]

The determining equations for deterministic symmetries of (5) are also obtained in the same way from (45), or directly from the above (59) by setting to zero the derivatives with respect to the $w^{k}$ variables.

However, it is immediate to check that the equations (59) do not coincide with the correct equations (11). The difference is due to

[TABLE]

Note that this inequality generally holds (for $\sigma_{x}\not=0$) even in one dimension, and even for deterministic vector fields (i.e. for $\widehat{\partial}_{k}\varphi^{i}\equiv 0$).

In fact, in the one-dimensional deterministic case we get

[TABLE]

The non correspondence between the symmetries of an Ito equation and of the corresponding Stratonovich equation might seem rather surprising at first; however, first of all the notion of correspondence between an Ito and the associated Stratonovich equation is not so trivial, as discussed e.g. in the last chapter of the book by Stroock Stroock (see in particular Sect.8.1.2 there), and second one should in any case not expect identity of symmetries, but rather a correspondence between the two; thus the difference between the symmetries of the two is not so strange.

On the other hand, an Ito equation and the associated Stratonovich equation do carry the same statistical information. In view of the discussion and results in GRQ1 , we would expect there is a correspondence between symmetries of the Fokker-Planck equation (symmetries of scalar Fokker-Planck equations were classified in CicVit , see also ShtSto ) which are also symmetries of the Ito equation and symmetries of the equivalent Stratonovich equation. This is indeed the case.

Proposition. Given an Ito equation and the associated Fokker-Planck equation, the symmetries of the latter which are also symmetries of the Ito equation, are also symmetries of the associated Stratonovich equation.

Proof. In GRQ1 it was shown that symmetries of the Fokker-Planck equation

[TABLE]

with $A=-[(1/2)\sigma\sigma^{T}]$, $B^{i}=f^{i}+2\partial_{j}A^{ij}$, $C=(\partial_{i}\cdot f^{i})+\partial_{ij}^{2}A^{ij}$ have to satisfy the system

[TABLE]

where

[TABLE]

The symmetry of the Fokker-Planck is also a symmetry of the Ito equation if and only if $\Gamma_{j}^{k}=0$ for all $j,k$, since the condition for a symmetry of the Ito equation are given by

[TABLE]

On the other hand, the symmetries of the Stratonovich equation are given by (59), which can now be written as

[TABLE]

where $\rho^{i}=(1/2)(\partial/\partial x^{k})[(\sigma^{T})^{i}_{\ j}\sigma^{kj}]$.

Thus it will suffice to show that

[TABLE]

An explicit computation shows that

[TABLE]

this completes the proof. $\triangle$

IX Algebraic structure of symmetries

It is well known that the Lie-point symmetries (or more precisely the Lie-point symmetry generators) for a given deterministic differential equation form a Lie algebra Olver1 ; symmref ; CGbook . One may wonder if the same holds in the case of stochastic differential equation. The question was answered in the negative by Wafo Soh and Mahomed WSM for (first order) Ito equations.

Here we want to discuss this point for Ito equations – confirming of course the result of WSM – and the analogous problem for Stratonovich ones; we will be satisfied with a discussion in the framework of simple symmetries (the negative result in the Ito case will a fortiori hold for general symmetries).

IX.1 Ito equations

We will consider vector fields $Y_{\xi}=\xi^{j}\partial_{j}$ and $Y_{\eta}=\eta^{j}\partial_{j}$ which are symmetries for a given SDE; to these we associate the vector fields $Z_{\xi}$ and $Z_{\eta}$ as in (12). By assumption we have

[TABLE]

We now want to consider

[TABLE]

and wonder if this is also a symmetry for the same SDE.

It is immediate to check that $[\widehat{X},Y_{\varphi}]=0$, just by Jacobi identity. As for the first of (13), here Jacobi identity implies that $[X\ ,\ Y_{\varphi}]=[X,[Y_{\xi},Y_{\eta}]]=[Y_{\xi},[X,Y_{\eta}]]-[Y_{\eta},[X,Y_{\xi}]]$; using now (69), this reads

[TABLE]

In order to check if the first of (13) is satisfied, we must express $Z_{\varphi}$ in terms of the $\{Y_{\xi},Y_{\eta},Z_{\varphi},Z_{\eta}\}$. Using (70) and some simple algebra, we get

[TABLE]

where we have defined

[TABLE]

This computation shows that

[TABLE]

Combining this with (71), the first of (13) reads simply

[TABLE]

But we have already used the condition that $Y_{\xi},Y_{\eta}$ are symmetries of the SDE identified by $X$, hence (73) has no reason to be true in general.

Example 11. In order to check and substantiate this claim, we can consider Example 5 above, i.e. the Ito equation (39); in that case we have seen that symmetry generators are written in the form (40). Let us consider two different symmetries $Y_{i}$ ($i=1,2$) given by

[TABLE]

By explicit computation, we have

[TABLE]

This is (in general) not in the form (40), and hence it is (in general) not a symmetry for the equation (39).

IX.2 Stratonovich equations

The situation is quite different for equations in Stratonovich form. This is rather evident comparing (11) and (50) (with our previous computation in hindsight).

Lemma. The Lie-point simple symmetry generators of a given Stratonovich SDE form a Lie algebra.

Proof. We can proceed as above, and define now the vector fields

[TABLE]

(Note that here we change slightly our notation w.r.t. Sect.VI, in order to keep uniformity with the previous subsection and to have a notation better suited to the present task.)

The determining equations (50) are now written as

[TABLE]

It is then immediate to check that if $Y_{\varphi}$ is given by (70), and $Y_{\xi},Y_{\eta}$ satisfy the determining equations

[TABLE]

then – just by Jacobi identity – (75) is also satisfied. $\triangle$

X Discussion and conclusions

Symmetry methods are widely recognized as one of our most effective tools in studying nonlinear deterministic equations Olver1 ; symmref ; the literature devoted to symmetry methods for stochastic differential equations is comparatively smaller, and moreover only considers invariance of SDEs (in Ito or Stratonovich form) only under deterministic transformations.

In this note we have considered – following the approach by Arnold and Imkeller in their analysis LArnold ; ArnImk of normal forms for SDEs transformations – the transformations of SDEs under random diffeomorphsims, i.e. diffeomorphisms depending on a random (multi-dimensional Wiener) process, and obtained the determining equations for random Lie-point symmetries of Ito stochastic differential equations.

The case of Stratonovich equations is also treated, in Section VI, and the determining equations are also obtained in this case.

We have also discussed the relation between symmetries of an Ito equation and those of the corresponding Stratonovich one; we have shown that in general – in particular, at the exception of the case where the matrix $\sigma(x,t)$ is actually independent of the space variables $x^{i}$ – these do not admit the same symmetries. The reason for this lies in the actual meaning of the “correspondence” between Ito and Stratonovich equations Stroock . On the other hand, an Ito equation and the corresponding Stratonovich one do carry the same statistical information, so that one would expect correspondence between symmetries to hold when considering symmetries of the associated Fokker-Planck equation. This is indeed the case, in a sense made precise by our Proposition in Sect.VIII.

We have considered a number of concrete examples (both in the Ito and the Stratonovich case), choosing equations with a physical significance, and explicitly shown that the determining equations we have written down can be analyzed and explicitly solved, i.e. that our theory is concretely applicable.

As stressed above (see Remark 7), here we only focused on the proper definition of random symmetries of a SDE and on the equations which have to be solved to constructively determine them; that is, we have not considered how the symmetries can be used in the study of the SDE (this appears to be a common feature of a large part of literature devoted to symmetry of SDEs).

On the other hand it seems that the use of symmetries in the framework of SDEs should go through the same general ideas as in the case of deterministic equations; that is, beyond any specific technique, the presence of symmetries suggests first of all that the analysis will be simpler if using symmetry-adapted coordinates. A glimpse of this is provided in Example 6 (and Remark 8) above.

More structured results do exist in the case of deterministic symmetries of SDEs Kozlov ; we will investigate in future work how these result can be extended to the framework of the random symmetries introduced here.