1 Introduction
Uncertainty and random fluctuations are a very common feature of real dynamical systems. For example, most physical, financial, biochemical and engineering systems are subjected to timevarying external or internal random disturbances. These complex disturbances and their associated responses are most naturally described in terms of stochastic processes. A more realistic formulation of a dynamical system in terms of differential equations should involve such stochastic processes. This led to the fields of stochastic and random differential equations, where the latter deals with processes that are sufficiently regular. Random differential equations (RDEs) provide the most natural extension of ordinary differential equations to the stochastic setting and have been widely accepted as an important mathematical tool in modeling and analysis of numerous processes in physics and engineering systems (Bunke, 1972; Soong, 1973; Sobczyk, 1991; Rupp and Neckel, 2013).
These internal and external disturbances of RDEs are not only of stochastic nature, but they are also of causal nature. They are causal in the sense that the disturbance processes are affecting other processes of the system. This allows us to model interventions on RDEs by forcing certain processes to be of a certain form, e.g. moving an object to a fixed position. Perfect or surgical interventions break any other causal influences on the intervened processes, but other types of interventions also occur in practice.
Although at least in principle random differential equations could be used for modeling causal relationships between the processes, infering such causal models from data is often difficult. A significant practical drawback of this modeling class is that obtaining time series data with sufficiently high temporal resolution is often costly, impractical or even impossible. Another issue is that if one has only access to a subset of the system’s processes, for example due to practical limitations on the measurability of some of the processes, then in general there does not have to exist an RDE on this subset of processes that could be estimated. A similar issue arises when the RDE contains exogenous latent confounding processes.
Structural causal models (SCMs), also known as structural equation models, are another wellstudied causal modeling tool and have been widely applied in the genetics, economics, engineering and social sciences (Pearl, 2009; Spirtes et al., 2000; Bollen, 1989). One of the advantages of SCMs over other causal modeling tools is that they have the ability to deal with cyclic causal relationships (Spirtes, 1995; Hyttinen et al., 2012; Mooij et al., 2011; Forré and Mooij, 2017; Bongers et al., 2016). In particular, recent work has shown how one can apply Markov properties (Forré and Mooij, 2017), how one can deal with marginalization and how one can causally interpret these models in the cyclic setting (Bongers et al., 2016).
Over the years, several attempts have been made to interpret these structural causal models that include cyclic causal relationships. They can be derived from an underlying discretetime or continuoustime dynamical system (Fisher, 1970; Iwasaki and Simon, 1994; Dash, 2005; Lacerda et al., 2008; Mooij et al., 2013). All these methods assume that the dynamical system under consideration converges to a single static equilibrium, with the exception of the analysis by Fisher (1970), who assumes that observations are time averages of a dynamical system. These assumptions give rise to a more parsimonious description of the causal relationships of the equilibrium states and ignore the complicated but decaying transient dynamics of the dynamical system. The assumption that the system has to equilibrate to a single static equilibrium is rather strong and limits the applicability of the theory, as many dynamical systems have multiple equilibrium states.
In this paper, we relax this condition and capture, under certain convergence assumptions, every random equilibrium state of the RDE in an SCM. Conversely, we show that under suitable conditions, every solution of the SCM corresponds to a samplepath solution of the RDE. Intuitively, the idea is that in the limit when time tends to infinity the random differential equations converge exactly to the structural equations of the SCM. Moreover, we show that this construction is compatible with interventions under similar convergence assumptions. We like to stress that our construction automatically captures the stochastic behavior of the RDE in the associated SCM. It can deal with randomness in the initial conditions, the coefficients and via the random inhomogenous part (captured as additive noise in the SCM), thereby significantly extending the work by Mooij et al. (2013) who only considers the deterministic setting.
The advantage of SCMs over RDEs is that by not modeling the transient random dynamics of the RDE, one arrives at a more compact representation for learning and prediction purposes of random systems that have reached equilibrium. Another advantage is that one can marginalize over a subset of the systems variables and get a more parsimonious representation that preserves the causal semantics (Bongers et al., 2016). Yet another advantage is that it is easier to deal with confounders within the framework of SCMs, as we only need to model the equilibrium distribution of these confounders, and don’t need to model their dynamics.
The remainder of the paper is organized as follows: In Section 2 we review the necessary theory about stochastic processes in order to describe RDEs. In Section 3 we introduce random dynamical models, that define RDEs together with interventions, and we discuss convergence properties of those models. In Section 4 we introduce structural causal models. In Section 5 we present our main result, which builds the bridge between RDEs and SCMs. In Section 6 we give an example from chemical kinetics and Section 7 contains a discussion including some open problems. Proofs are provided in the Supplementary Material.
2 Preliminaries
Let be a set. A stochastic process is an valued function such that (which denotes ) is for each an measurable function^{1}^{1}1Assuming the Borel algebra on , that is the smallest algebra on which contains all open balls.
(i.e. a random variable) on a probability space
. We will always assume that there exists some background probability space on which the random variables and processes live. Furthermore, will always denote an interval in and has the meaning of time, if not stated otherwise. For each we have an valued function from to , which is called a sample path or realization of .Let two such processes and be equivalent, i.e., for all we have , then for all there are sets such that and holds for all . If, moreover, one can choose the sets independently of , that is such that holds for all , then we denote such an equivalence between the processes and by .
If the sample paths of a stochastic process are continuous on for almost all , then the process is called samplepath continuous on . For a samplepath continuous process and the function defined, with probability one, by
is a random variable, if it exists (Doob, 1953; Loéve, 1977). Moreover, a stochastic process is called samplepath differentiable on , if there exists a set such that and that for all the derivative
exists. The mapping is called the samplepath derivative of . Note that there is always a stochastic process such that . Similarly the samplepath integral is defined, that is, a stochastic process is called samplepath integrable on , if the integral^{2}^{2}2We use here Lebesgue integration, hence we assume the Lebesgue measure on the Lebesgue algebra of Lebesgue measurable sets of . exists for almost all .
A random vector
can itself be seen as a stochastic process, that is the process defined by . This stochastic process is by definition samplepath continuous. Moreover, if a process is samplepath continuous on and there exists a random variable such thatalmost surely, then we say that converges to and we will call the process convergent.
3 Random Differential Equations
Ordinary differential equations, which have the general form
(1) 
provide a simple deterministic description of the dynamics of dynamical systems. The solution of an initial value problem consisting of differential equation (1) together with an initial value
(2) 
represents the state of such a system at time , given that the state (2) was attained at time . The inclusion of random effects in the dynamical system leads to a number of modifications that can be made to the formulation of the initial value problem (1), (2) (Gard, 1988). The first, and simpler, case arises when the initial value is replaced by a random variable . The second case arises when the deterministic function has random coefficients, i.e. it is replaced by a random function , where is a stochastic process uncoupled with the solution process . As a special case, may be replaced by a random function with a random inhomogenous part (i.e., additive noise), that is, it is replaced by a random function . Of course, a combination of these cases could hold.
The inclusion of random effects in differential equations leads to two distinct classes of equations, for which the random processes have differentiable and nondifferentiable sample paths, respectively. If the random processes occuring in a differential equation (for example and ) are sufficiently regular, i.e. have differentiable sample paths, then the majority of problems can be analyzed by use of methods which are analogous to those in deterministic theory of differential equations; such equations are called random differential equations
. The second class occurs when the inhomogenous part is an irregular stochastic process such as Gaussian white noise. The equations are then written symbolically as stochastic differentials, but are interpreted as integral equations with Ito or Stratonovich stochastic integrals. These differential equations are called
stochastic differential equations. In this paper, we will focus on random differential equations.3.1 Observational random dynamical models
We will define a random differential equation in terms of an observational random dynamical model:
Definition 1.
An observational random dynamical model (oRDM) is a tuple
where

is a time interval,

is a finite index set of endogenous processes,

is a finite index set of exogenous processes,

is the product of the codomains of the endogenous processes, where each codomain ,

is the product of the codomains of the exogenous processes, where each codomain ,

is a function that specifies the dynamics,

is an exogenous stochastic process.
The oRDM gives the observational random dynamics of the random dynamical system, without any intervention from outside. The random dynamics are described in terms of random differential equations (Bunke, 1972):
Definition 2.
A stochastic process is a samplepath solution of the random differential equations (RDE) associated to ,
(3) 
if the ordinary differential equations (ODE)
are satisfied for almost all .
Let and let be a dimensional random variable, where , such that a.s., then is called a samplepath solution of (3) with respect to the initial condition .
The samplepath solution of (3) with respect to the initial condition is called unique on if for an arbitrary pair , of samplepath solutions with respect to the initial conditions we have .
We associated an ordinary differential equation to any specific sample path . The solutions of these ordinary differential equations are the sample paths of a stochastic process , which is the samplepath solution of the random differential equation.
In particular, an oRDM is a deterministic dynamical model, if the background probability space is . In this setting, the associated RDE is just a single ODE.
Example 1 (Damped coupled harmonic oscillator).
Consider the wellknown damped coupled harmonic oscillator, consisting of a onedimensional system of point masses () with positions and momenta . They are coupled by springs, with spring constants and equilibrium lengths (), under influence of friction with friction coefficients and with fixed endpoints and (see Figure 1).
The equations of motion of this system are provided by the ODE:
Suppose that the lengths
are not constant, but are indepent normally distributed around
with a certain variance. Then this ODE with random coefficients is actually a random differential equation modeled by an oRDM.
If the oRDM is sufficiently regular, then the majority of problems for such models can be analyzed by use of methods which are analogous to those in the theory of ordinary differential equations (Bunke, 1972; Sobczyk, 1991; Rupp and Neckel, 2013).
Definition 3.
An oRDM is called regular if is continuous and is samplepath continous.
For ordinary differential equations a sufficient condition, for the existence and uniqueness of a solution with respect to an initial value, is the Lipschitz condition. Similarly, one can prove, by using results from the theory of ordinary differential equations, that there exists a similar sufficient condition for random differential equations of regular oRDMs.
Theorem 1.
Consider a regular oRDM . If for almost all there exists a continuous function such that for each and the condition
is satisfied, where means the Euclidean norm in , then for any initial condition there exists a unique samplepath solution of the RDE (3) w.r.t. .
3.2 Intervened random dynamical models
Interventions on an observational random dynamical model can be modeled in different ways. Here we will consider interventions on the endogenous processes. We model an intervention on a subset of the endogenous processes by forcing those processes to be . This can be seen as a “surgical” intervention, since they break the causal influences on the intervened processes (Eberhardt, 2014). The random dynamics of the other processes are still untouched and are described in terms of the RDE associated to those processes, that is^{3}^{3}3For , we adopt the notation for .
This yields the following random dynamical model including interventions.
Definition 4.
A random dynamical model (RDM) is a tuple
where

is a time interval,

is a finite index set of endogenous processes,

is a finite index set of exogenous processes,

is a subset of intervened processes,

is the product of the codomains of the endogenous processes, where each codomain ,

is the product of the codomains of the exogenous processes, where each codomain ,

is a function that specifies the dynamics of the processes,

is an intervened stochastic process,

is an exogenous stochastic process.
If , then we call also a nonintervened random dynamical model, otherwise we will call it an intervened random dynamical model.
The (intervened) random dynamical model gives the (intervened) random dynamics of the random dynamical system, where the random dynamics are described by the following set of equations:
Definition 5.
A stochastic process is a samplepath solution of the (intervened) random differential equations associated to the (intervened) RDM ,
(4)  
if and the ordinary differential equations
are satisfied for almost all .
Let and let be a dimensional random variable, where , such that a.s., then is called a samplepath solution of the (intervened) RDE (4) with respect to the initial condition .
The samplepath solution of (4) with respect to the initial condition is called unique on if for an arbitrary pair , of samplepath solutions with respect to the initial conditions we have .
In particular, the nonintervened model , where is the terminal process , yields the same samplepath solutions as the observational random dynamical model . They describe the same random dynamics and in this sense the class of observational random dynamical models can be seen as a subclass of the class of random dynamical models.
Definition 6.
We call an RDM linear, if the function is given by
where and are matrices.
The function that defines the dynamics of the RDM encodes a functional structure that can be represented by a directed mixed graph.
Definition 7.
We define the functional graph of an RDM as the directed mixed graph with nodes , directed edges if and only if is a functional parent^{4}^{4}4Let and and consider a function . We call an a functional parent of w.r.t. , if there does not exist a function such that . of w.r.t. and bidirected edges if and only if there exists a such that is a functional parent of both and w.r.t. .
For a linear RDM one would draw an edge if is nonzero and if both and are nonzero for some .
The causal semantics of a random dynamical model can be modeled using interventions:
Definition 8.
Given an RDM , a subset and a stochastic process , the intervention maps to the intervened RDM .
Note that interventions on disjoint subsets of the endogenous processes commute.
Example 2 (Damped coupled harmonic oscillator).
Consider the damped coupled harmonic oscillator of Example 1. Its functional graph is depicted in Figure 2. We can perform an intervention on by moving the position of the mass to a fixed position . This is modeled by replacing the equation of motion of the position by the process , that defines the motion of moving the mass to the fixed position . Performing a similar intervention on the momentum usually does not lead to an RDM with samplepath solutions that converge to a certain random variable.
We can define a regularity condition for the RDM similar to the one for oRDMs.
Definition 9.
An RDM is called regular if is continuous and both and are samplepath continous.
The existence and uniqueness Theorem 1 generalizes to the RDM .
Corollary 1.
Consider a regular RDM . If for almost all there exists a continuous function such that for each and the condition
is satisfied, then for any initial condition such that a.s. there exists a unique samplepath solution of the RDE (4) w.r.t. .
3.3 Steady random dynamical models
Here we consider an important subclass of regular RDMs that satisfies certain convergence properties.
Definition 10.
We call an RDM steady, if is regular, , the process converges to a random variable and the process converges to a random variable .
The class of steady RDMs is not stable under arbitrary interventions, that is a steady RDM does not have to stay steady under an intervention, however it is stable under the following class of interventions:
Definition 11.
We call an intervention a perfect intervention if the process converges to a random variable .
Note that for any perfect intervention , the process is samplepath continuous by definition.
Although steadiness of an RDM guarantees that the exogenous and intervened processes converge, it does, in general, not guarantee that any of its samplepath solutions converges. However:
Definition 12.
Given a steady RDM . If a samplepath solution converges to a random variable , then we say that the samplepath solution equilibrates and we call an equilibrium variable of the samplepath solution .
If a samplepath solution , that describes the behaviour of the system, equilibrates, then in particular we have
almost surely.
4 Structural Causal Models
Structural causal models (SCMs), also known as structural equation models, provide a probabilistic description of the causal semantics of a system. They are widely used for causal modeling purposes (Pearl, 2009; Spirtes et al., 2000; Bollen, 1989). In this paper, we will follow the terminology of Bongers et al. (2016).
Definition 13.
A structural causal model (SCM) is a tuple
where

is a finite index set of endogenous variables,

is a finite index set of exogenous variables,

is the product of the codomains of the endogenous variables, where each codomain ,

is the product of the codomains of the exogenous variables, where each codomain ,

is a measurable function that specifies the causal mechanism,

is a random variable.^{5}^{5}5We slightly deviate from Bongers et al. (2016), where instead they take an exogenous probability measure on .
The solutions are described in terms of structural equations.
Definition 14.
A random variable is a solution of the SCM if the structural equations
are satisfied almost surely.
The causal mechanism encodes a functional structure that can be represented by a directed mixed graph.
Definition 15.
We define the functional graph of an SCM as the directed mixed graph with nodes , directed edges if and only if is a functional parent of w.r.t. and bidirected edges if and only if there exists a such that is a functional parent of both and w.r.t. .
4.1 Intervened structural causal models
The causal semantics of a structural causal model can be modeled using perfect interventions (Pearl, 2009).
Definition 16.
Given an SCM , a subset and an endogenous variable , the intervention maps to the intervened model where the intervened causal mechanism is defined by:
We call an intervention a perfect intervention if .
Note that interventions on disjoint subsets of endogenous variables commute.
5 From Steady RDMs to SCMs
We now have set the stage for constructing an SCM from an RDM under some convergence properties. Here, we will consider steady RDMs, as discussed in Section 3.3, for which the exogenous and intervened processes are wellbehaved as time tends to infinity. For this class of RDMs we will see that the random differential equations, that determine the samplepath solutions of the RDM, play an analogous role to the structural equations, that determine the solutions of the SCM.
Definition 17.
Given a steady RDM . Define the SCM associated to to be where the associated causal mechanism is defined by
with
and
Note that the steadiness of implies the measurability of . This leads to our first main result:
Theorem 2.
Given a steady RDM . If there exists a samplepath solution of that equilibrates to , then is a solution of the associated SCM .
The converse does not hold in general, however we have the following sufficient condition:
Proposition 1.
Consider a steady RDM such that (i.e., is constant in time). If is a solution for the associated SCM , then there exists a samplepath solution of that equilibrates to .
We can weaken the condition that has to be constant over time by imposing the following additional assumption on the model.
Proposition 2.
Consider a steady RDM for which (i) there exists an and a such that for all and (ii) for almost all there exists a continuous function such that for each and the condition
is satisfied. If is a solution for the associated SCM , then there exists a unique samplepath solution of that equilibrates to .
Consider the diagram in Figure 3. So far, we have defined each mapping in this diagram separately (see Definition 8, 16 and 17). The next result shows that this diagram commutes:
Theorem 3.
Given a steady RDM , a subset and a process such that equilibrates to and equilibrates to . Then:
In other words, perfect intervention commutes with the mapping from steady RDM to SCM.
Example 3.
Consider a linear RDM where is of the form as in Definition 6 and is a random variable, that is a stochastic process that is constant in time. Then the associated SCM is where the causal mechanism is defined by
where .
Example 4 (Damped coupled harmonic oscillator).
Consider again the damped coupled harmonic oscillator of Example 1. The structural equations of the associated SCM are given by
These equations describe the equilibria of the positions and momenta. Figure 2 reflects the intuition that at equilibrium the position of each mass has a direct causal effect on the position of its neighbors. This can be seen more clearly by marginalizing over the momentum variables. Observing that the momentum variables always vanish at equilibrium, we can focus on the position variables as the variables of interest. We can marginalize over the momentum variables by solving each equation of w.r.t. itself and then substituting these in the equations of (Bongers et al., 2016). This yields the marginal model with the following structural equations
Resolving the selfloops of this marginal model by solving each equation w.r.t. itself gives the structural equations
and this model yields the same causal semantics for the position variables as the original model (Bongers et al., 2016). The functional graph associated to this model is depicted in Figure 3(a). If we now perform a perfect intervention on by moving the mass to a fixed position , then we get the graph as depicted in Figure 4(b). Because these models are uniquely solvable and linear we can perform separation w.r.t. both graphs and conclude that holds in the intervened model but not in the observational model (Forré and Mooij, 2017).
This example demonstrates that the equilibrium variables of the RDM can be studied by statistical tools applicable to SCMs. This sheds some new light on the concept of causality as expressed within the framework of structural causal models.
6 Application: Chemical Kinetics
Chemical kinetics is the study of rates of chemical processes. The chemical processes are described by the chemical reactions which are often modeled through ordinary differential equations. A wellknown chemical reaction is the basic enzyme reaction which is schematically represented in Figure 6 (Murray, 2002).
It describes an enzyme , binding to a substrate , to form a complex , which in turn releases a product while regenerating the original enzyme. The ’s, called the rate constants, quantify the rate of a chemical reaction. These chemical reactions satisfy the law of mass action, which states that the rate of a reaction is proportional to the product of the concentrations of the reactants. Applying this to the concentration processes , , and of the basic enzyme reaction, gives the RDE:
Although this RDE has no random coefficients (or random inhomogenous part), randomness can enter the RDE via the initial conditions. In Figure 4(a) we simulated the RDE with rate constants and random initial conditions. The randomness of the initial conditions evolves over time and is captured in the associated SCM at equilibrium. That is, they are described by the associated SCM:
where we removed the selfloops for convenience. This is an example of an SCM that is not uniquely solvable, which is illustrated in Figure 4(a) by the dispersion of the concentration and at large , hence this example cannot be treated with the theory of Mooij et al. (2013) which assumes no dependence on initial conditions.
Let us for the moment fix the concentration of the complex by performing a perfect intervention on
as illustrated in Figure 4(b). From the functional graph of the associated intervened SCM in Figure 6(a) we can read off that performing another perfect intervention on the substrate should have no effect on the product , as it would lead to the functional graph in Figure 6(b) where there is no directed path from to . This prediction, based on the functional graph of the SCM associated to the RDM, is indeed verified by the simulations in Figures 4(b)–4(c). Intuitively, this is also what one would expect, since the complex is the only element in the system that is capable of releasing the product.This illustrates that random differential equations are capable of modeling randomness through the initial conditions, while the causal semantics at equilibrium of the dynamical system are parsimoniusly described by the associated SCM.
7 Discussion
In this paper we built a bridge between the world of random differential equations and the world of structural causal models. This allows us to study a plethora of physical and engineering systems subject to timevarying random disturbances within the framework of structural causal models. We naturally extend the work of Mooij et al. (2013) to the stochastic setting, which allows us to address both cycles and confounders. In particular, we relaxed the condition that the dynamical system has to equilibriate to a single static equilibrium, and show that if an RDE is sufficiently regular all equilibrium samplepath solutions of the RDE are described by the solutions of the associated SCM, while pertaining the causal semantics.
There are two possible interesting directions for future research. The first is relaxing the regularity assumption. Earlier work has shown that SCMs can be derived from stochastic differential equations (Hansen and Sokol, 2014), however they restrict to the acyclic case. The second is relaxing the convergence assumption. Although the convergence assumption is a convenient and simplifying assumption, convergence of the stochastic processes is not always satisfied in practice. Recent work has shown that dynamic asymptotic behaviour of ordinary differential equations can be captured by dynamic structural causal models (Rubenstein et al., 2016). Other related work on discretetime dynamical system and causality which does not require a single static equilibrium assumption is (Voortman et al., 2010).
Acknowledgements
Stephan Bongers and Joris Mooij were supported by NWO, the Netherlands Organization for Scientific Research (VIDI grant 639.072.410).
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Supplementary Material
Proofs
Proof of Theorem 1
Proof.
Continuity of and samplepath continuity of implies that for almost all the function is continuous on . Moreover, continuity of implies separate continuity of . That is, for each the function is continuous in and in particular measurable in . Hence for all the function is measurable. Applying theorem 1.2 in Bunke (1972) proves the result. ∎
Proof of Corollary 1
Proof.
Apply Theorem 1 to the regular oRDM . ∎
Proof of Theorem 2
Proof.
Let be a samplepath solution that equilibrates to . Then
which gives
where we used continuity of in the second equality, and steadiness in the last equality. This gives
and hence
∎
Proof of Proposition 1
Proof.
Let be a solution of . Then the stochastic process defined by is a samplepath solution of that equilibrates to . ∎
Proof of Proposition 2
Proof.
Let be a solution of . Then by Corollary 1 there exists a unique samplepath solution w.r.t. the initial condition . Hence is the unique samplepath solution that equilibrates to . ∎
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