An Analytic Study of the Ornstein–Uhlenbeck Process with Time-Varying Coefficients in the Modeling of Anomalous Diffusions

We consider the problem of modeling anomalous diffusions with the Ornstein–Uhlenbeck process with time-varying coefficients. An anomalous diffusion is defined as a process whose mean-squared displacement non-linearly grows in time which is nonlinearly growing in time. We classify diffusions into types (subdiffusion, normal diffusion, or superdiffusion) depending on the parameters of the underlying process. We solve the problem of finding the coefficients of dynamics equations for the Ornstein–Uhlenbeck process to reproduce a given mean-squared displacement function.


INTRODUCTION
The Ornstein-Uhlenbeck process is widely used in dynamic modeling problems under the influence of random factors (see, e.g., [1,2]) and also plays an important role in optimal control theory (in particular, it arises in the solution of the stochastic linear controller problem, see [3]). At first, the process with this name (see [4]) was used to describe the velocity of a particle moving in a random environment, which can be implemented in the form of a linear stochastic differential equation: where X t , t 0, is the velocity process; κ 1 , σ > 0 are constants reflecting respectively the rate of convergence of X t to its asymptotic average value EX ∞ = 0 and the contribution of the random component in the process dynamics; w t , t 0, is the standard one-dimensional Wiener process; x is the initial value of the velocity. One of the main characteristics in the study of such a model is the displacement process Y T = T 0 X t dt, where T 0 is the time of motion. In what follows we will need a modified definition of asymptotic equivalence of two functions. An important feature of (1) is that the mean-squared displacement D T equal to EY 2 T has order T (D T ∼ T ), i.e., it is close to the displacement Y (w) T defined by the Wiener process Y T ] 2 = T . Such processes are known as normal diffusions [2]. Advantages of the displace- = w T include the existence of derivative with respect to time: dY t = X t dt, which is known to be lacking for Y (w) t (trajectories w t with probability one are not differentiable [5]). At the same time, processes where D T ∼ T have been known for a long time; they should be called anomalous diffusions (see surveys in [6,7]). A fundamental property of an anomalous diffusion is a nonlinear change in time of the corresponding mean-squared displacement. As we have noted above, the linear case corresponds to the standard Ornstein-Uhlenbeck process (1) and to the Wiener process w t if as the model of the particle's position one takes the Brownian motion itself. For anomalous diffusions one often assumes a functional dependence of the mean-squared displacement D T on time T , which has a power law: D T = T α (α > 0). Here researchers distinguish subdiffusion (0 < α < 1) and superdiffusion when α > 1, also considering "superslow diffusions" with D T ∼ ln T and other slowly growing functions, see, e.g., [7]. In the present work, to define an anomalous diffusion we propose to consider a nonstationary Ornstein-Uhlenbeck process X t , t 0, with time-varying coefficients (in Eq. (1) the constants (−κ 1 ), σ are replaced with piecewise continuous functions of time a t , σ t ). Precise requirements on the coefficients will be shown below; so far we emphasize that these functions can be both unbounded for t → ∞ and can tend to zero. In special cases, modeling of anomalous diffusions with this kind of a process was done in [6,8] for power law type functions.
The purpose of this work is to classify diffusions into types (subdiffusion, normal diffusion, superdiffusion) based on known parameters of the underlying process X t and solutions to the inverse problem of finding coefficients X t for modeling a given mean-squared displacement D T . We note that by the definition of displacement Y T = T 0 X t dt, in the study of anomalous diffusions an important question is to examine Y T , the integrated Ornstein-Uhlenbeck process ("integrated" here is a term from probability and statistics used to denote integrals of Riemann type for random processes). Analysis of an integrated process also is of independent importance since it is widely used in practical applications (modeling cumulative volatility [9] and asset return [10], logarithm of population size [11], power consumption [12], geology [13] and others). Often it is Y t which is available for observation instead of X t [12,14], and estimation of the parameters of X t by known data on Y t is usually done under stationarity assumptions [12,14] for process X t , which obviously does not hold in many situations (see, e.g., [15][16][17]). One possible way to construct Y T with D T = EY 2 T ∼ T is dynamic scaling [13] of the displacement corresponding to the standard process (1), i.e., an external transformation that does not account for possible non-autonomous dynamics of the equations on X t . In this work we will show that in order to model a known meansquared displacement function D T it suffices that the process parameters obey a relation based on Riccati equations known from filtering theory. The paper is organized as follows. In Section 2 we formulate the basic assumptions on the coefficients of dynamics equations, give a formal definition of anomalous diffusion, show a statement that lets one determine the diffusion type based on the parameters of process X t , and give examples. Section 3 is devoted to solving the problem of finding parameters X t in order to construct a given mean-squared displacement function and examples of applying the resulting relations. autonomous linear stochastic differential equation where the initial state x is non-random; w t , t 0, is the standard Wiener process; a t , σ t , t 0, are piecewise continuous functions of time. Note that for σ t → 0, t → ∞, we have the case of the so-called attenuating disturbances (see [18,19] for applications in control theory); if, on the other hand, σ t → ∞, then disturbances grow with time (see, e.g., the cognitive model in [16] or a description of the population dynamics in [20]), which certainly also influences the dynamics and statistical characteristics of the process (2). A solution of (2) is a process of the form where Φ(t, s) = exp t s a v dv . Let us write down the main characteristics of the process (3); see, e.g., [21]: expectation covariance function We assume that function a t , t 0, guarantees stability with rate δ t for solutions of the corresponding deterministic equation. Namely, there exists a function δ t > 0, t 0, such that for some constant κ > 0; Note that function δ t can be naturally called the stability rate since according to (ii) this function determines the rate of decrease for the upper bound for Φ(t, s). Next we comment on the introduced conditions. Condition (i) means that the chosen stability rate δ t cannot be improved for the class of functions of the form δ (1) t = λδ t (λ > 0 is a constant). For example, for an unbounded (for t → ∞) a t it is impossible to have optimal stability rate which is equal to a constant. Relation (ii) together with (iii) yields a definition of asymptotic stability for the solutions of (2) when σ t ≡ 0, see [22].
Note that in case δ t ≡ κ 1 > 0 we get an exponential stability type; for δ t → 0, t → ∞, the resulting stability is weaker, subexponential, and for δ t → ∞, on the contrary, we see superexponential stability (the terminology arose in the analysis of nonlinear equations, see, e.g., [23]).
Assuming that the initial position is zero, we define the displacement process and find the mean-squared displacement We transform (6), using a formula for the variance of Riemann type integrals of random processes (see, e.g., [24]) substituting (5), we get It is interesting to study the behavior of D T for T → ∞, so if for some nondecreasing func-tionD T > 0 it holds thatD T ∼ D T (see Definition 1), this asymptotically equivalent function will also characterize the order of change in the mean-squared displacement D T . Note that due to (ii) and inequality Φ(t, s) κ 0 exp{− t sκ δ v dv} (κ 0 ,κ are some positive constants) mean-squared displacement D T can be estimated as where D (1) T and D (2) T are the mean-squared displacements defined based on (2) for the cases a t = −κδ t and a t = −δ t respectively. For these situations, we have also introduced separate notations for second moment functions (see (4)): Integrating by parts, we transform representation (7) to the form Next we formulate the definition of an anomalous diffusion. In the following statement we show conditions that let us classify diffusion (2) depending on the characteristics of processes that define mean-squared displacements D (1) T and D (2) T .
t and m (2) t have been defined in (9) and (10). Then the following diffusion types are possible: Proofs of Theorem 1 and the Theorem 2 shown below are given in the Appendix. Next we show sample applications of Theorem 1 for classification of diffusions under various coefficients a t and σ t .

Example 1. Let us consider Eq. (2) for an important special case [8] of power functions
where δ t = (1 + t) α−1 with power α > 0, and classify the resulting diffusions. We use Theorem 1 for this purpose. First, (2) = 0 in the case α = 0. Then by Theorem 1 we have that: the value 0 α < 1 corresponds to a subdiffusion; α = 1, to normal diffusion; α > 1, to superdiffusion. Indeed, it is easy to see (see also [8]) that the mean-squared displacement In many cases, the statement of Theorem 1 readily enables us to classify the diffusion based on analyzing the relations between parameters δ t and σ t .
Example 2. Let σ 2 t ∼ δ t and suppose that σ 2 t = k 2 δ t , where k > 0 is a constant. In this case an increment of process X t defined by Eq. (2) is proportional (with a time-varying coefficient δ t ) to the differential of the deviation of the displacement process Y t from the scaled Brownian motion Y (W ) t , see [6], which is also used as the displacement model in case of an anomalous diffusion in [25].
Note that the faster δ t increases, i.e., the larger is the stability rate in Eq. (2), the more stable (in terms of the distribution) will the process Y (W ) t be and the smaller will be its contribution to the dynamics of Z t . Indeed, the quadratic variation is Y with probability one for t → ∞, see [26], where Y (W ) ∞ is a Gaussian random variable. Under the assumptions of the corollary, we have constants s 1 = s 2 > 0. Then it becomes obvious that for subexponential stability we have the resulting superdiffusion, exponential stability type leads to normal diffusion; superexponential, to subdiffusion.

DIFFUSION PARAMETER DETERMINATION FOR A GIVEN FUNCTION OF MEAN-SQUARED DISPLACEMENT
Suppose that we know the function D T , T 0, and the problem is to find a t = −δ t and σ t for Eqs. (2) for which the corresponding displacement process would be asymptotically equivalent to D T in mean squares. It turns out that this problem always has a solution under the natural condition that the mean-squared displacement function increases monotonely. D t be a three times differentiable function, and D t > 0, t 0. Then there exists  a pair of functions (δ t , σ 2 t ), where δ t > 0 defines the stability rate and σ 2 t > 0, related bẏ

Theorem 2. Let
(δ > 0 is an arbitrary initial condition), such that for the displacement process it holds thatD T ∼ D T . Here the function m (13) is defined by (10) with x = 0.

Remark 1.
A known comparison theorem for solutions of Riccati equations (see [27, Theorem 4.1]) implies that δ t monotonically increases with respect to σ 2 t . Let δ (1) t and δ 2 t be the solutions of Eqs. (12) with the same initial conditionδ for different σ (1) t and σ (2) whose solution can be written down in a closed form: By the comparison theorem [27,Theorem 4.1] it follows that δ t δ t , where δ t is the solution of (12) for the case when 2σ 2 t − D t 0. For example, for D T = T we have the bound δ t 1/(δ −1 + t). The resulting boundary (14) will correspond to the case of diffusion with coefficientσ 2 Remark 2. When finding δ t and σ 2 t in case of a normal diffusion (D T = T ) Eq. (12) takes the form 2σ 2 t = 2δ 2 t +δ t . Suppose that coefficients δ t and σ 2 t are known, and we want to find out which diffusion type they generate. Let 2σ 2 t = 2δ 2 t +δ t > 0. Then the condition from Theorem 1 can be rewritten as lim i.e., we compare the actual diffusion coefficient σ t and theoretical oneσ t corresponding to regular diffusion for a fixed stability rate δ t .
Example 3. As an example, consider determination of the stability rate δ t for a constant coefficient σ t ≡ σ > 0 for three types of diffusions with the following mean-squared displacement functions: (a) D T = T ; (b) D T = (T + 1) 2 ; (c) D T = ln(T + 1). We use Theorem 2 and in all cases will assume that the initial value of stability rate δ 0 = σ is the same. Then the following situations hold: (a) for D T = T Eq. (12) becomes a Riccati equation with constant coefficientṡ δ t + 2δ 2 t = 2σ 2 and for initial condition δ 0 = σ it has a solution δ t ≡ σ, t 0, i.e., it defines the exponential stability type; (b) for D T = (T + 1) 2 Eq. (12) takes the forṁ and has a solution δ t > 0, t 0, for σ > 0; based on our analysis of the dual problem for a linear controller (see the proof of Theorem 2) we find i.e., stability will be subexponential, and the mean-squared displacement corresponding to δ (c) if D T = ln(T + 1) then (12) becomes the following equation: with a positive solution δ t > 0, t 0, for σ > 1. Considering the dual problem for a linear controller, we conclude that δ t ∼ √ t + 1 = δ (0) t is the superexponential stability type. It is easy to see that in this case D (0) T ∼ ln(T + 1). In conclusion we note that δ t found in Example 3 and the corresponding constant diffusion coefficients σ in the modeling of D T for cases (a)-(c) are in accordance with the conclusions made in Corollary: (a) σ 2 /δ t = σ 2 /σ is constant, and δ t = σ does indeed generate normal diffusion (D T = T ); (b) δ t ∼ 1/ √ t + 1 and σ 2 /δ t → ∞, t → ∞, and we get superdiffusion (D T = (T + 1) 2 ); (c) δ t ∼ √ t + 1, σ 2 /δ t → 0, t → ∞, then subdiffusion is obvious, and it does hold with D T = ln(T + 1).

CONCLUSION
In this work we have considered the problem of modeling anomalous diffusions with the Ornstein-Uhlenbeck process with time-varying coefficients. The statement of Theorem 1 on the classification of diffusions lets us conclude that anomalous diffusions may result from a disproportional change in the time variance of the process compared to the stability rate. The result formulated as a Riccati equation (12), where two parameters δ t and σ t turn out to be connected by a single relation, matches the remark made in [12] that it is impossible to independently determine the stability rate and diffusion coefficient when observing the displacement process (the work [12] analyzed a standard Ornstein-Uhlenbeck process (1)). At the same time, the relation shown in Theorem 2 between stability rate and the diffusion coefficient corresponding to mean-squared displacement D T can also be used to solve the problem when one of the parameters (2) is already known. More precisely, if we fix δ t or σ 2 t then the second parameter is defined by (12) provided that the resulting function has given properties: σ 2 t > 0 (for known δ t ) or δ t > 0, t 0 δ v dv → ∞, t → ∞ (for known σ 2 t ).

ACKNOWLEDGMENTS
The study has been funded by the Russian Academic Excellence Project "5-100." We construct a deterministic control problem (linear controller, see [28]) by passing to reverse time. Letp t = p −t . Then functionp t for t 0 is a solution of a Riccati equation with boundary condition at time moment T = 0:ṗ