Construction of analytical solutions to systems of two stochastic di ﬀ erential equations

: A scheme for the stochastization of systems of ordinary di ﬀ erential equations (ODEs) based on Itô calculus is presented in this article. Using the presented techniques, a system of stochastic di ﬀ erential equations (SDEs) can be constructed in such a way that eliminating the stochastic component yields the original system of ODEs. One of the main bene ﬁ ts of this scheme is the ability to construct analytical solutions to SDEs with the use of special vector-valued functions, which signi ﬁ cantly di ﬀ ers from the randomization approach, which can only be applied via numerical integration. Moreover, using the presented techniques, a system of ODEs and SDEs can be constructed from a given di ﬀ usion function, which governs the uncertainty of a particular process


Introduction
Solutions to stochastic systems governed by multi-dimensional differential equations pose a clear interest due to important applications in science and engineering.In mathematics, the theory of stochastic processes remains to be an active topic of research for both theory and applications [1].Although the theory of onedimensional systems governed by stochastic differential equations (SDEs) is well developed, research into analytic solutions to nonlinear multi-dimensional SDEs continues.This is due not only to the problem's mathematical relevance but also to the plethora of applications of SDEs in various fields of research.A short review of recent works is given below.
A scheme based on SDEs is used to optimize design strategies of chemical experiments in the study by Huang et al. [2].SDE-based models are one of the main approaches in the modeling SARS-CoV-2 pandemic [3][4][5] and other infectious diseases [6,7] with various levels of detail.Due to the changing underwater landscape, SDEs are required to model underwater acoustic wave propagation [8].Since perturbations in electrical power systems can lead to reduced quality, safety, and economy of electrical supply, SDEs are required to model such perturbations and predict their effects on the entire system [9].Despite widespread applications, the construction of solutions to SDEs is not a trivial task.A well-known approach for approximating numerical solutions to stochastic engineering problems is the stochastic finite element method (SFEM).The straightforward approach to SFEM is based on the application of deterministic algorithms finite element method.For example, Monte-Carlo simulation methods were used in the study by Schuëller and Pradlwarter [10] to admit the stochastic characteristics in the form of a series of deterministic sample inputs (a clear advantage of such an approach is that no further solution methods have to be developed).
However, the Monte-Carlo simulation often requires a prohibitively large number of samples due to its slow convergence.The main attraction of the alternative approach to SFEM is the fact that it can solve the problem only once, providing the solution for the whole continuum of different realizations of the stochastic process [11].
Clearly, the SFEM has a few drawbacks too.It solves the problem in a high-dimensional physical-stochastic product space, whereby additional nodes and degrees of freedom must be created.Thus, the complexity of the problem grows exponentially as the number of random parameters increases [12,13].
Thus, the application of reduced-order techniques, allowing the reduction of the computational effort, is clearly the priority aim in stochastic computational methods.A good overview and comparison of different model reduction techniques is provided in the study by Besselink et al. [14].
The main objective of this article is to provide a technique for the stochastization of a reduced system of two coupled nonlinear differential equations and to construct analytical solutions to these stochastic equations.The apparent simplicity of the problem is deeply misleading.First, we are not interested in developing a numerical scheme for the solution of the stochastic system.Second, we are not interested in designing a randomization scheme that could yield estimates of martingales.On the contrary, the main objective of this study is to derive the necessary and sufficient conditions for the existence of analytic Ito integrals to the coupled system, and to construct the analytic solution itself.
Building on the research presented in the one-dimensional case [15], in this article, we consider the stochastization of the following systems of ordinary differential equations (ODEs): where = k 1, 2. Let ( ) ω t denote a Wiener process [16].As mentioned, the objective of this article is to construct a system of SDEs with respect to functions ( | ) ξ t α k (α is a scalar parameter) of the following form: (2) where = k 1, 2; ≥ t 0; < < α 0 1; and the following pointwise limits hold true: i.e., as parameter α tends to 0, the system of SDEs equation (2) tends to equation (1) and conversely the solution to equation (2) tends to the solution of equation (1).The trajectories ξ k are dependent on the realization of the Wiener process ω.Note that in this article, we focus on the special case of the Wiener process ( ) ω t being the same for both values of k.

Wiener and Itô processes
Consider a stochastic Wiener process ( ) ω t that satisfies the following well-known identities [16]: where δ jk is the Kronecker delta.Note that applied to differentials, the above equalities yield that [16]: An Itô process ( ) ξ t is a stochastic process that can be expressed via the following integral: where ( ) ξ 0 is a scalar initial condition.Note that ( ) = ω 0 0, which means that the value ( ) ξ 0 is not a random variable, but a scalar.Functions ( ( )) a t ξ t , and ( ( )) σ t ξ t , represent drift and diffusion, respectively.An equivalent definition of an Itô process is obtained by the differentiation of equation ( 8), which results in the following SDE: One of the most well-known results in stochastic calculus is Itô's lemma, which states that a differential of ( ( )) f t ξ t , , where ( ) ξ t is defined via equation ( 9), is given as follows [16]: x ξ x ξ An approach to stochastization by applying equation (10) to a single ODE has already been discussed in the study by Navickas et al. [15].In the remainder of this article, systems of stochastic equations will be considered.
While the stochastization of a system of ODEs follows a similar idea to that of a single ODE, such generalization is far from trivial, as Itô's lemma (10) is no longer true for a pair of stochastic processes.

Two-dimensional Itô's lemma
Let = k 1, 2. Suppose that two Itô processes (11) and functions × → F : k are given.Note that the parameter α present in equation (2) is not yet intro- duced.Then, ( ) are Itô processes given by [17]: Since F k are arbitrary functions, the initial conditions for η k are equal to: ( ) .

Solutions to the systems of SDEs of type (11)
Let us first consider the following system of SDEs where drift and diffusion for both Itô processes are constant: k k k k k (14) In this case, solution ( ) η t k to equation ( 14) can be obtained directly by applying the Itô integral (8): Now let us consider a generalized version of equation ( 14) that is defined in equation (11).Suppose that it is possible to determine a vector function ( and its inverse . The functions F and G satisfy the canonical relations for inverse mappings: If the vector functions F and G are considered coordinate-wise, then it can be denoted that: Then, combining equations ( 15) and ( 17) yields the solution to equation ( 11): (18) Note that the first relation in equation (17) implies equation (13), which together with equation ( 14) yields , , ; where  γ ξ ξ , Note that mixed partial derivatives of equation ( 22) must satisfy the relation: which yields the following PDE: From equation ( 21), it is obtained that Inserting these relations into equation ( 24) yields two PDEs: Solving the above equations via the method of characteristics yields the following result [18]: , and k is an arbitrary differentiable function.

Solution of equation (19)
Computing second-order derivatives of equation ( 22) and inserting the obtained expressions into equation ( 19) yields: , , , where the term . Applying equation ( 27) transforms equation (29) into: , , , Stochastization of ODE systems  5 which can be simplified as follows: Derivations presented above could be summarized as the following Lemma.
Lemma 2.1.Consider two systems of SDEs:  (34) reads as follows: where , and the function ( where C 1 and C 2 are arbitrary univariate functions.Noting that equation (27) holds true, the above equations become: This yields that for arbitrary constants  C 1 and  C 2 , the following relations hold true: Inserting equations (39) and (40) into equation (38), we obtain The above equation yields that which finalizes the proof.□

Main results
The goal of this section is to prove the following theorem: Theorem 3.1.Let the following system of ODEs be given as follows: (44) Then, the system of SDEs 1, is a stochastization of equation (44) (i.e., it satisfies equations (3)-( 5)) if relations (65), (67), and (69) hold true.
Moreover, the solution to the system of SDEs (45) reads as follows: , and = k 1, 2.
The notations required for Theorem 3.1 are introduced in the remaining parts of this section, which are dedicated entirely to the theorem's proof.

Derivation of the functions ( ( ) ) G x y ,
Suppose that the conditions of Lemma 2.1 hold and (47) In addition, let functions ( ) g ξ 1 1 and ( ) g ξ 2 2 satisfy the following equations: where (50 where I is an identity matrix.
if the following system of equations holds true: where = k 1, 2.

Relation between systems of SDEs with variable and constant coefficients
In this section, a technique for the construction of two systems of SDEs: as well as functions ( ) is presented.Schematic diagrams of this technique are depicted in Figures 1 and 2.
Consider the case where the following functions and parameters are given: that satisfy the relation (47); -Parameters  ∈ a k , where = k 1, 2. Then, the results of the previous sections yield functions and parameters ( ) as follows: -Functions ( ) a ξ ξ , k 1 2 are computed by solving the system of linear equations (31), yielding: where -Functions ( ) are derived from equation (50); -Functions ( ) ͠ ξ ξ Ψ , k 1 2 are obtained by solving the system of algebraic equations (51).

Stochastization of the systems of ODEs
Consider the following system of ODEs: where = k 1, 2. The objective of this section is to construct a system of SDEs of the form: where = k 1, 2; ≥ t 0; < < α 0 1, and the relations (3)-( 5) hold true.Naturally, more than one set of ( | ) a ξ ξ α , exist that satisfy equations (3)- (5).In this article, the most straightforward assumption is considered: the parameter α linearly increases and decreases the stochastization intensity, resulting in an increase of both additive and multiplicative noise.Let us denote: where = k 1, 2; < < α 0 1; and ( ) h x k is a univariate function.Higher values of α lead to a higher level of stochastization compared to lower values of the same parameter.
Moreover, the solution to the system of SDEs (71) reads as follows: Proof.The first result of the theorem follows directly from the derivations presented above.The solution equation ( 72) is obtained by inserting equation (66) as well as (18).□ A schematic diagram of Theorem 3.1 is depicted in Figure 3.In this section, the stochastization of a system of ODEs is derived using the techniques presented in previous sections and compared to another approach of introducing randomness into ODE systems.

Stochastization of an ODE system
First, the following functions and parameters are selected: -The simplest nonlinear case of diffusion functions (65)quadratic polynomialsis considered where < < α 0 1.Note that equation ( 3) is satisfied. - , are selected, which when inserted into equation (66) yields -The next step is selecting functions ( ) U ξ ξ , 1 1 2 and ( ) U ξ ξ , 2 1 2 do satisfy the relation (36) for some constants ξ 10 and ξ 20 .Note that this selection is not unique, and the following is an example: , 0.
The above functions satisfy the relation (36 . -As a final step, four constants  γ jk must be selected.As in the previous case, this selection is not unique: 4; 2; 1.  . (note that these are distinct from ξ 10 and ξ 20 ) and applying equation (35) yields functions ( ) 2 .with respect to y for a given x) [19].
The functions P 1 and P 2 that define the ODE system can be obtained by using condition (4): Then, according to Theorem 3.1, the system of SDEs is a stochastization of the system of ODEs: are values satisfying the following relation: yields the results depicted in Figure 4.Note that not only the deterministic, but also the stochastic solutions have fixed limits as → ±∞ t .As can be seen from the solution expressions (83), the influence of the Wiener process is decreasing as → +∞ t , since the rate of change of the deterministic part of the exponent outpaces the Wiener process.

Comparison to the randomization approach
The presented technique can be compared to the most straightforward approach to introducing stochasticity into a system of ODE: randomization.In this approach, the ODE system (82) is integrated numerically, but a random number from a Gaussian distribution is added to the system function at each step.
Consider the system (82), a scaling variable > ε 0, and two samples ( ) ( ) , of Gaussian random variables with a mean of 0 and a variance of 1.The randomization process can be described as follows: a numerical integration algorithm is used to integrate (82), but at each integration step, the ODE system is modified to have the following expression: (85 where n is the number of integration step.This process yields randomized solutions  ξ 1 and  ξ 2 to system (82), as shown in Figure 5.

C 1
and  C 2 are arbitrary, they can be chosen in such a way to rewrite the first two integrals on both equations of equation (41) to have different lower bounds equal to  ξ 10 and  ξ 20 : applying equation (49), where:1: schematic diagram, illustrating functions and considered in the technique presented in Section 3.2.Given functions and parameters are shown in green, whereas the derived ones are depicted in red.

Figure 2 :
Figure 2: A schematic diagram of the technique presented in Section 3.2.Given functions and parameters are shown in green, whereas the derived ones depicted in red.
and parameters shown in red in Figure1are derived as described in Sections 3.2 and 3.3: -Drift functions (67) read as follows:|