A scheme for the analytical stochastization of ordinary differential equations (ODEs) is presented in this article. Using Itô calculus, an ODE is transformed into a stochastic differential equation (SDE) in such a way that the analytical solutions of the obtained equation can be constructed. Furthermore, the constructed stochastic trajectories remain bounded in the same interval as the deterministic solutions. The proposed approach is in a stark contrast to methods based on the randomization of solution trajectories and is not focused on the analysis of martingales. This article extends the theory of Itô calculus by directly implementing it into analytical schemes for the solution of differential equations based on the generalized operator of differentiation. The efficacy of the presented analytical stochastization techniques is demonstrated by deriving stochastic soliton solutions to the Riccati differential equation. The presented semi-analytical stochastization scheme is relevant for the investigation of the global dynamics of different biological and biomedical processes where the variation interval of the stochastic solution is predetermined by the rationale of the model.
1 Introduction and motivation
Ever since the seminal work by Einstein , studying the effects of noise on dynamical systems has been an important area of research in physics and other applied sciences. With the introduction of the Wiener process, the derivative of which is white noise,  and Itô calculus , formal mathematical formulation of stochastic differential equations (SDEs) became possible.
Since then, many areas of applications for SDEs have been the focus of research. One of the first and foremost historical areas is economics and finance. SDEs in this area range from the now-classical Black-Scholes equation for call and put option pricing  to more recent works. The price change of assets obeying more sophisticated factors than simple supply and demand is modeled via SDEs in . A stochastic differential game of two insurers investing into the same financial market is posed using SDEs in . A duopolistic competition model with sticky prices is considered in . A stochastic control model of investment based on SDEs is presented in . And agent-based model of a financial market is presented in .
In recent years, SDE models have been applied to a wide array of biological phenomena . An SDE model for the evolution of the MCF-7 breast cancer cell line treated by radiotherapy is developed in . Tumor-immune responses to chemotherapy are studied using SDEs in . Plant growth is modeled via a Gompertz-type SDE in . Biochemical reaction systems are studied using SDEs in .
With the recent outbreak of the novel coronavirus COVID-19, the study of stochastic models for the modeling of this phenomenon has emerged. Most are based on the well-known deterministic differential equation model: susceptible-exposed-infected-recovered model (SEIR) [15,16]. Adak et al. used Brownian motion to induce stochasticity into a SLIR (susceptible-latent-infected-recovered) model by adding stochastic differential directly to a system of ordinary differential equations (ODE) in . An adaptation of an SIR model to include stochastic transition is discussed in . A SIRS epidemic model including fractional white noise is presented in . A variant of stochastization is used in the generalized logistic equation to model COVID-19 evolution in .
It is clear that the analysis of SDEs is currently a particularly relevant topic. Furthermore, there arises a requirement to induce stochastization into previously deterministic models described via ODEs. While there are many ways to approach this problem, the aim of this article is to provide a technique for stochastization in such a way that if an analytical solution to the ODE can be constructed, an analytical solution to the SDE can also be constructed. Consider the following ODE of the form:
where is a continuously differentiable function. Let denote a Wiener process . The objective is to construct an SDE with respect to function (where is a scalar parameter) of the form:
Equation (2) and its solution possess the following properties:
and as tends to 0, the SDE (2) tends to (1) and conversely the solution to (2) tends to the solution of (1). Itô calculus is a classical yet powerful mathematical theory of SDEs. Introduced almost seven decades ago it is still widely used in a variety of technical, biomedical, and economical applications.
Note that other approaches to this problem also exist. The most straightforward approach would consist of adding noise to (1) in every step of integration of a numerical integrator. While this technique does enable the randomization of the solution trajectory, a solution of (1) bounded to an interval becomes no longer bounded, which presents a problem for many applications.
Another option for inputting randomness into an ODE is based on random differential equations, discussed in detail in . Here the time variable is replaced by a stochastic Wiener process , yielding the following random differential equation:
The main objective of this article is to construct a semi-analytical scheme for solving (2) and to apply this scheme to the paradigmatic Riccati equation . It is well known that the Riccati equation does admit the first-order solitary solution (the kink-type solution) . In its turn, soliton solutions (and the Riccati equation in general) do play an important role in defining the global dynamics of different models, including the prostate cancer model , the Hepatitis C treatment model , and COVID-19 within host model with immune response . Thus, the construction and analysis of solitary solutions to various differential equations have become a very active field of research in recent years. An overview of the latest studies conducted in this area is presented further. The dynamics of ion acoustic solitary wave solutions of two-dimensional nonlinear Kadomtsev-Petviashvili-Burgers equation in quantum plasma have been investigated in . Solitary wave solutions to the nonlinear Schrödinger equation were constructed in [29,30,31, 32,33,34]. In , the authors have proposed novel methods for the construction of exact traveling wave solutions of the modified Liouville equation and the symmetric regularized long wave equation. The stability analysis of solitary wave solutions for the fourth-order nonlinear Boussinesq water wave equation was performed in . Dispersive traveling wave solutions of the equal-width and modified equal-width equations were obtained in . The problem formulations of models for internal solitary waves in a stratified shear flow with a free surface were presented in . New traveling wave solutions for the fourth-order nonlinear Ablowitz-Kaup-Newell-Segur water wave dynamical equation were constructed and investigated in . In , the authors have developed a novel approach for the construction of solitary wave solutions to the nonlinear Nizhnik-Novikov-Vesselov equation. Exact traveling and solitary wave solutions of the Kudryashov-Sinelshchikov equation were obtained in . The propagation of soliton-like solutions to the coupled nonlinear -dimensional Maccari system applied in plasma physics was investigated in . The authors of  have developed a modification of the variational iteration algorithm for the study of the numerical solution to the dispersive water wave phenomena. The interaction properties of soliton molecules were investigated and Painleve analysis for nano bioelectronics transmission model was performed in .
Stochastization of such models describing different biological and biomedical processes is an important research direction [17,45]. Stochastic models allow a better description of real-life processes and do help to represent the effect of local unpredictability caused by the noise and different uncertainties. However, computational analysis of SDEs containing a variable that represents random white noise (usually calculated as the derivative of Brownian motion or the Wiener process) results in the investigation of martingales . As a matter of fact is that a particular solution of an SDE can wander far away from its non-stochastic counterpart solution as time moves away from the initial conditions .
SDE analysis is transformed into analytical or computational investigation of martingales. In some situations, such wandering out of a predetermined variation interval is not tolerable. The ability to control the variation range while allowing the stochastic wandering around the deterministic counterpart solution is an open problem in mathematical sciences up to the best of our knowledge.
However, one needs to keep in mind that the range of values of the solution (especially if it is a soliton solution) is predetermined by the structure of the model (e.g., the concentration of the infected cells cannot be negative). Therefore, it is important to construct such stochastization schemes for ODEs which would guarantee that the stochastic soliton solution would remain in the predetermined range of values. The derivation of such soliton solutions to the stochastic Riccati equation is the main objective of this article. That will be achieved by extending the theory of Itô calculus by directly implementing it into analytical schemes for the solution of differential equations based on the generalized operator of differentiation.
This article is organized as follows. Preliminary results of Itô calculus and some operator methods for the solution of differential equations are presented in Section 2; the inverse balancing technique is adapted to SDEs in Section 3; the scheme for the stochastization of first-order ODEs is constructed in Section 4; the developed scheme is applied to obtain the stochastization of the Riccati equation in Section 5; concluding remarks are given in Section 6.
A short review of the concepts of Itô calculus required for the article’s main results is provided in this section.
The following notations are used throughout the text:
– Wiener process;
– random processes;
– Itô integral;
– functions describing an SDE;
– Itô function with respect to random process ;
– partial differentiation operator with respect to variable ;
– integration operator with respect to variable ;
– generalized differential operator;
– multiplicative operator.
2.1 Wiener process and Itô integral
Let us consider a non-differentiable Wiener process with the following properties :
Let be continuous and satisfy a global Lipschitz condition. The integral of with respect to the Wiener process is defined as the Itô integral :
where is a partition of the interval . The aforementioned limit is taken in the mean-square sense, ensuring that .
2.2 Itô’s lemma
Itô’s lemma is a fundamental result in the SDE theory. Suppose a process is given that has the following differential:
where and an additional function satisfy conditions detailed in the previous section. Then, the differential of process is given by:
The aforementioned equation is an analogy of the chain rule of differentiation for stochastic processes.
2.3 SDEs and their solutions
First, consider the following stochastic equation where the coefficients depend only on :
In this case, it is possible to compute the solution directly by applying the Itô integral (7):
Suppose that it is possible to determine such a function that taking and using (9) to compute its differential yields (10). Then, is given by (11). Furthermore, if some function satisfies the condition , then the solution to (8) is given by:
The substitutions described above are possible only if Itô’s condition holds true 
Thus, given (8), it is possible to apply Itô’s condition to compute . Then, Itô’s lemma yields:
from which the function can be determined as follows:
In the subsequent parts of this article the following notation is used:
and the only time-dependent function is referred to as the Itô function of stochastic process .
2.4 Operator solutions to ODEs
In order to obtain stochastization of ODEs, the analytical solutions of these ODEs must be considered. In this section, a short review of the generalized differential operator method for the construction of analytical solutions to ODEs is presented. The method was first presented in  and later developed in [50,51].
Consider the following ODE:
The generalized differential operator for (17) reads
where denotes the partial differentiation operator with respect to . Using the generalized differential operator (18), the following multiplicative operator is constructed:
Operator possesses the following property:
for any analytic bivariate function . Furthermore, the solution to (17) can be written as:
2.5 Operator solution to a particular partial differential equation
In subsequent sections, a partial differential equation arises that possesses the following form:
where are analytical functions. The following boundary condition is posed:
where is also analytic.
where is the solution to the following operator problem:
3 Inverse balancing technique for SDEs
The main idea of the inverse balancing technique is as follows: given a differential equation, and an analytical form of its solution to assume that the solution is known, and determine the parameters of the differential equation in terms of the solution parameters. This yields a robust approach to determining the necessary conditions of the existence of a particular solution to a class of differential equations. It has been applied in a variety of fields, ranging from astrophysics , and population dynamics  to medicine .
In this section, the method is extended to include SDEs.
Suppose that the Itô function , and a constant are given.
Then, the Itô partial differential equation with respect to function
has the following general solution:
where is an arbitrary function.
Equation (28) can be rewritten as:
Then, the solution to (30) takes the following form:
where is an arbitrary function. Note that
For clarity in further derivations, (29) will be rewritten as:
The SDE with respect to process :
The Itô condition for these functions is given by (28), which results in the proof of the theorem.□
4 Stochastization of first-order ODEs
4.1 Construction of analytical solutions to (40)
In order to construct the analytical solutions to (40) the algorithm described in Section 2.3 is applied. Let (40) be given. A transformation of process must be determined in order to transform (40) into:
Functions are given by:
where are defined by (39).
which leads to the solution:
can be inverted with respect to . The following theorem addresses this problem.
The inverse function to (52) with respect to reads
where satisfies the condition and the following differential equation:
The function is inverse to with respect to if the following relation holds true:
Using (52) in the aforementioned equation yields
which can be rearranged as:
Let satisfy the following:
Differentiating (58) with respect to and rearranging yields
4.2 Scheme for stochastization of first-order ODEs
5 Stochastization of the Riccati equation
5.1 General case
Consider the Riccati differential equation :
where, by (27):
and the function satisfies . Note that when considering the stochastization of an ODE, can be selected arbitrarily.
Denoting and rearranging (65) lead to:
Note that the term can be written as a factor when the operands do not depend on . Thus, (66) is transformed to:
Thus, the final form of reads
5.2 Stochastization of equation (63): special case
In this section, a stochastization for (63) is obtained such that the function depends only on and the solution to the obtained stochastic equation is constructed.
Suppose that , then:
Since the right-hand side does not depend on , is only dependent on . Furthermore, inserting yields
Then, (70) can be rewritten as:
when . In that case, has the following form:
5.3 Analytical solution of (77)
Thus, (77) is transformed into:
with the solution
By Theorem 4.2, the function must be derived from the differential equation:
The solution to the aforementioned equation reads
By (53) we obtain
and the analytical solution to (77) is given by:
Note that comparing the above solution to (69) it can be seen that only the variable within the function has the Wiener process . This means that the solution of the stochastic Riccati equation belongs to the same set of values as the deterministic Riccati equation.
5.4 Numerical comparison: stochastization and randomization of the Riccati equation
In this section, two different approaches to induce randomness into the Riccati equation are compared.
The randomization procedure is described as follows. Let the Riccati equation (63), a scaling variable , and a sample of a Gaussian random variable with mean zero and unit variance be given. Consider any constant-step time-forward numerical integrator with step size . The randomized solution at points is denoted as .
We initialize the process by setting the first point equal to the initial condition: . In the th step, the value is computed by performing one-time forward integration step for following the differential equation:
The process of randomization described above yields a random solution trajectory with the mean close to the deterministic solution (Figure 1(a)). The non-stochastic solution represented by the thick black line in Figure 1 is validated in the article . However, note that the randomized solution leaves the interval to which the deterministic solution is bound. This is not the case for the stochastic solution, which remains bounded to the same interval as (Figure 1(b)).
Note that the statistical mean of function (86) is not equal to the solution (69) of the deterministic Riccati equation. As mentioned before, the deterministic Riccati equation solution is obtained when .
6 Concluding remarks
A scheme for the analytical stochastization of ODEs is presented in this article. Given an ODE, its SDE counterpart is constructed in such a way that it satisfies the Itô condition. This ensures that it is possible to construct an analytical solution to the obtained SDE via the application of Itô calculus.
The described technique of stochastization has two important properties: as the parameter that governs the influence of randomness on the SDE solution tends to zero, the solution tends to the deterministic ODE solution. Furthermore, if the ODE solution is bounded to an interval, the constructed stochastic trajectories can also only belong to that interval – which is not true for most other stochastization schemes.
Due to the fact that the presented semi-analytical stochastization scheme allows us to confine a stochastic solution to a particular variation interval, the scheme is especially relevant for the investigation of the global dynamics of different biological and biomedical processes where the variation interval of the stochastic solution is predetermined by the rationale of the model.
The scheme is applied to the paradigmatic Riccati equation which possesses kink soliton solutions. It is shown that the general analytical form of the deterministic solution is preserved in the stochastic solution after the transformation of the ODE to the SDE. Stochastic trajectories obtained in this manner are a generalization of kink soliton solutions in the stochastic sense.
The extension of the presented stochastization scheme to higher-order ODEs and systems of ODEs, as well as applications to real-world models, remains a definite objective of future research.
Funding information: This research was funded by a grant (No. S-COV-20-8) from the Research Council of Lithuania.
Conflict of interest: The authors declare no conflict of interest.
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