Existence and Uniqueness of Solutions for a Class of Nonlinear Stochastic Differential Equations

Abstract

By using successive approximation, we prove existence and uniqueness result for a class of nonlinear stochastic differential equations. Moreover, it is shown that the solution of such equations is a diffusion process and its diffusion coefficients are found.

1. Introduction

Differential equations, which are not solved for the derivative, have found diverse applications in many fields. Examples of equations of this type are Lagrange equations of classical mechanics or Euler equations.

Consideration of real objects under the influence of random factors leads to nonlinear stochastic differential equations, which are not solved for stochastic differential. Such equations were introduced by Kolmanovskii and Nosov in [1] for construction of stochastic analogues of neutral functional differential equations. Works [1–5] were devoted to the problems of  existence, uniqueness, and properties of solutions of neutral stochastic differential (delay) equations in finite dimensional spaces. Existence of solutions for such equations in Hilbert spaces was studied in papers [6–9]. In the paper [9] the author considered a stochastic equation without delay. In the monograph of Kolmanovskiĭ and Shaĭkhet [10] conditions were obtained for optimality in control problems for these equations.

In this paper we will study the existence and uniqueness of solutions for a class of nonlinear stochastic differential equations, which are not solved for the stochastic differential.

Statement of the Problem. Let us consider a nonlinear stochastic differential equation: 𝑑(𝑥−𝐺(𝑡,𝑥))=𝑎(𝑡,𝑥)𝑑𝑡+𝜎(𝑡,𝑥)𝑑𝑊(𝑡), (1) which is not solved for the stochastic differential.

Here 𝑥∈𝑅^𝑑,𝑡≥𝑡₀,𝐺(𝑡,𝑥), and  𝑎(𝑡,𝑥) are 𝑑-dimensional vector functions; 𝜎(𝑡,𝑥) is a  𝑑×𝑚 matrix and 𝑊(𝑡) is an 𝑚-dimensional Wiener process with independent components. Let 𝑥 be a random vector in 𝑅^𝑑, such that 𝐸|𝑥|²<∞. Assume that 𝑥 does not depend on 𝑊(𝑡)−𝑊(𝑡₀), 𝐹_𝑡=𝜎{𝑥,𝑊(𝑠)−𝑊(𝑡₀),𝑡₀≤𝑠≤𝑡}.

Definition 1. An 𝐹_𝑡-dimensional process 𝑥(𝑡) is said to be the solution of (1) if 𝑥(𝑡₀)=𝑥  and 𝑥−𝐺(𝑡,𝑥(𝑡))=𝑥−𝐺𝑡₀+,𝑥_𝑡𝑡0𝑎(𝑠,𝑥(𝑠))𝑑𝑠+^𝑡_𝑡0𝜎(𝑠,𝑥(𝑠))𝑑𝑊(𝑠) (2) holds for every 𝑡≥𝑡₀ with probability 1.
A solution 𝑥(𝑡) is said to be unique if for any continuous solutions 𝑥(𝑡),𝑦(𝑡) such that 𝑥(𝑡₀)=𝑦(𝑡₀)=𝑥 one has 𝑃{sup_{𝑠∈[𝑡0,𝑡]}|𝑥(𝑠)−𝑦(𝑠)|>0}=0 for all  𝑡≥𝑡₀.
In this work we use the method of successive approximations to establish existence and uniqueness (pathwise) of the solution of (1). We study its probability properties. We prove that 𝑥(𝑡) is diffusion process and find coefficients of diffusion.

2. Main Results

Firstly we prove the theorem of existence and uniqueness of the solution.

Theorem 2. Assume that  𝐺(𝑡,𝑥) is a continuous function; 𝑎(𝑡,𝑥),𝜎(𝑡,𝑥) are measurable functions for 𝑥∈𝑅^𝑑,𝑡≥𝑡₀ and satisfy the following conditions:(1) there exists a constant  𝐶>0, such that |𝐺(𝑡,𝑥)|+|𝑎(𝑡,𝑥)|+‖𝜎(𝑡,𝑥)‖≤𝐶(1+|𝑥|) for 𝑥∈𝑅^𝑑,𝑡≥𝑡₀;(2) there exist constants 𝐿₁>0 and 𝐿>0 such that |𝐺(𝑡,𝑥)−𝐺(𝑡,𝑥^′)|≤𝐿₁|𝑥−𝑥^′| and𝑎(𝑡,𝑥)−𝑎𝑡,𝑥^′+𝜎(𝑡,𝑥)−𝜎𝑡,𝑥^′≤𝐿𝑥−𝑥^′ (3) for 𝑥, 𝑥^′∈𝑅^𝑑, 𝑡≥𝑡₀.

If 𝐿₁<(1/6)^3/4, then there exists a unique continuous solution 𝑥(𝑡) of (1) with probability 1 for all 𝑡≥𝑡₀. Moreover it has a bounded second moment such that 𝐸|𝑥(𝑡)|²<∞  for all 𝑡≥𝑡₀.

Proof. To find a solution of the integral equation (2), we use the method of successive approximations. We start by choosing an initial approximation 𝑥₀(𝑡)=𝑥.
At the next step, 𝑥₁(𝑡)=𝑥+𝐺(𝑡,𝑥)−𝐺𝑡₀+,𝑥_𝑡𝑡0𝑎(𝑠,𝑥(𝑠))𝑑𝑠+^𝑡_𝑡0𝜎(𝑠,𝑥(𝑠))𝑑𝑊(𝑠). (4) Consequently, 𝑥_𝑛+1(𝑡)=𝑥+𝐺𝑡,𝑥_𝑛(𝑡)−𝐺𝑡₀+,𝑥_𝑡𝑡0𝑎𝑠,𝑥_𝑛(𝑠)𝑑𝑠+^𝑡_𝑡0𝜎𝑠,𝑥_𝑛(𝑠)𝑑𝑊(𝑠). (5) Consider an interval [𝑡₀,𝑇] such that 𝐵=3(𝐿²₁+5(𝑇−𝑡₀)𝐿²)<1. Note that it can be always achieved by choosing a sufficiently small 𝑇−𝑡₀ and by the conditions on 𝐿₁.
We prove that a solution exists on this interval.
From (4) it follows that 𝐸𝑥₁(𝑡)−𝑥²≤3𝐸𝐺(𝑡,𝑥)−𝐺𝑡₀,𝑥²+𝐸^𝑡_𝑡0𝑎(𝑠,𝑥)𝑑𝑠²+^𝑡_𝑡0𝐸‖𝜎(𝑠,𝑥)‖²(𝑠)𝐶𝑑𝑊≤3₁1+𝐸|𝑥|²+𝑇−𝑡₀²𝐶₁1+𝐸|𝑥|²+𝑇−𝑡₀𝐶₁1+𝐸|𝑥|², (6) where 𝐶₁ is a constant independent of 𝑡₀, 𝑇, and 𝑥.
Next, estimate 𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛+1(𝑡)−𝑥_𝑛(𝑡)²,𝑥₁(𝑡)−𝑥²=𝐺(𝑡,𝑥)−𝐺𝑡₀,𝑥+^𝑡_𝑡0𝑎(𝑠,𝑥)𝑑𝑠+^𝑡_𝑡0𝜎(𝑠,𝑥)𝑑𝑊(𝑠)²≤3𝐺(𝑡,𝑥)−𝐺𝑡₀,𝑥²+^𝑡_𝑡0𝑎(𝑠,𝑥)𝑑𝑠²+^𝑡_𝑡0𝜎(𝑠,𝑥)𝑑𝑊(𝑠)²≤3𝐶₁1+|𝑥|²+^𝑡_𝑡0𝑎(𝑠,𝑥)𝑑𝑠²+^𝑡_𝑡0𝜎(𝑠,𝑥)𝑑𝑊(𝑠)². (7) Therefore, 𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥₁(𝑡)−𝑥²≤3𝐶₁1+𝐸|𝑥|²+𝐸^𝑇_𝑡0𝑎(𝑠,𝑥)𝑑𝑠²+𝐸sup_{𝑡∈[𝑡0,𝑇]}^𝑡_𝑡0𝜎(𝑠,𝑥(𝑠))𝑑𝑊(𝑠)²≤3𝐶₁1+𝐸|𝑥|²+𝑇−𝑡₀^𝑇_𝑡0𝐶₁1+𝐸|𝑥|²𝑑𝑠+4^𝑇_𝑡0𝐶₁1+𝐸|𝑥|²𝑑𝑠≤31+𝑇−𝑡₀²+4𝑇−𝑡₀𝐶₁1+𝐸|𝑥|². (8) The estimation of the stochastic integral follows from its properties. Similar to [11, p.20], we have 𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛+1(𝑡)−𝑥_𝑛(𝑡)²≤3𝐿²₁sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛(𝑡)−𝑥_𝑛−1(𝑡)²+𝑇−𝑡₀^𝑇_𝑡0𝐿²𝑥_𝑛(𝑠)−𝑥_𝑛−1(𝑠)²𝑑𝑠+sup_{𝑡∈[𝑡0,𝑇]}^𝑡_𝑡0𝜎𝑠,𝑥_𝑛(𝑠)−𝜎𝑠,𝑥_𝑛−1(𝑠)𝑑𝑊(𝑠)². (9) Thus, we have 𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛+1(𝑡)−𝑥_𝑛(𝑡)²≤3𝐿²₁𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛(𝑡)−𝑥_𝑛−1(𝑡)²+𝑇−𝑡₀²𝐿²𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛(𝑡)−𝑥_𝑛−1(𝑡)²+4^𝑇_𝑡0𝐿²𝐸𝑥_𝑛(𝑠)−𝑥_𝑛−1(𝑠)²𝐿𝑑𝑠≤3²₁+𝑇−𝑡₀²𝐿²+4𝐿²𝑇−𝑡₀×𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛(𝑠)−𝑥_𝑛−1(𝑠)²=𝐵𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛(𝑡)−𝑥_𝑛−1(𝑡)²≤𝐵^𝑛𝐸sup_{𝑡∈[𝑡0,𝑇]}𝑥₁(𝑡)−𝑥²≤𝐵^𝑛31+5𝑇−𝑡₀𝐶₁1+𝐸|𝑥|². (10) Since 𝐵<1, it then follows that the series ^∞_𝑛=1𝑃sup_{𝑡∈[𝑡0,𝑇]}𝑥_𝑛+1(𝑡)−𝑥_𝑛(𝑡>1)𝑛²≤^∞_𝑛=1𝐵^𝑛𝑛⁴31+5𝑇−𝑡₀𝐶₁1+𝐸|𝑥|² (11) converges.
This implies the uniform convergence with probability 1 of the series 𝑥+^∞_𝑛=0𝑥_𝑛+1(𝑡)−𝑥_𝑛(𝑡), (12) on the interval [𝑡₀,𝑇]. Its sum is 𝑥(𝑡). So 𝑥_𝑛(𝑡) converges to some random process. Every 𝑥_𝑛(𝑡) is continuous with probability 1. Whence it follows that the limit 𝑥(𝑡) is continuous with probability 1, too.
Then prove uniqueness of this continuous solution. Assume that there exists a second continuous solution 𝑦(𝑡) of (1). Denote  by 𝜘_𝑁(𝑡) the random variable which equals 1 if |𝑥(𝑠)|≤𝑁, |𝑦(𝑠)|≤𝑁  and it equals 0 otherwise. Then 𝜘_𝑁(𝑡)𝑥(𝑡)−𝑦(𝑡)=𝜘_𝑁(𝑡)𝐺(𝑡,𝑥(𝑡))−𝐺𝑡,𝑦(𝑡)+𝜘_𝑁(𝑡)^𝑇_𝑡0𝜘_𝑁(𝑠𝑎((𝑠(𝑠)+)𝑠,𝑥))−𝑎𝑠,𝑦𝑑𝑠_𝑡𝑡0𝜘_𝑁(𝑠𝜎((𝑠(𝑠)(𝑠.)𝑠,𝑥))−𝜎𝑠,𝑦𝑑𝑊) (13) Therefore, 𝐸𝜘_𝑁(𝑡)𝑥(𝑡)−𝑦(𝑡)²≤3𝐸𝜘_𝑁(𝑡)𝐿²₁𝑥(𝑡)−𝑦(𝑡)²+𝑡−𝑡₀^𝑡_𝑡0𝐸𝜘_𝑁(𝑠)𝑥(𝑠)−𝑦(𝑠)²+𝑑𝑠_𝑡𝑡0𝐸𝜘_𝑁(𝑠)𝐿²𝑥(𝑠)−𝑦(𝑠)²𝑑𝑠. (14) The last inequality implies the estimation 1−3𝐿²₁𝐸𝜘_𝑁(𝑡)𝑥(𝑡)−𝑦(𝑡)²≤1+𝑇−𝑡₀𝐿^2𝑡_𝑡0𝐸𝜘_𝑁(𝑡)𝑥(𝑠)−𝑦(𝑠)²𝑑𝑠. (15) By the Gronwall-Bellman inequality and the above inequality together, we have 𝐸𝜘_𝑁(𝑡)𝑥(𝑡)−𝑦(𝑡)²=0. (16) So, 𝑃𝑥(𝑡)≠𝑦(𝑡)≤𝑃sup_{𝑠∈[𝑡0,𝑡]}|𝑥(𝑠)|>𝑁+𝑃sup_{𝑠∈[𝑡0,𝑡]}𝑦(𝑠)>𝑁. (17) Probabilities at the right-hand side of the above inequality approach to zero as 𝑁→∞, because 𝑥(𝑡)  and𝑦(𝑡) are continuous with probability 1. Therefore 𝑥(𝑡) and𝑦(𝑡) are stochastic equivalent.
We conclude that 𝑃sup_{𝑠∈[𝑡0,𝑇]}𝑥(𝑠)−𝑦(𝑠)>0=0. (18) Hence, the existence and uniqueness of the solution are proved on [𝑡₀,𝑇].
Next, we show boundedness of a second moment of the solution. Denote again by 𝜘_𝑁(𝑡) the indicator of the set 𝜔∶sup_{𝑠∈[𝑡0,𝑡]}|𝑥(𝑠)−𝑥|≤𝑁. (19) Then 𝜘_𝑁(𝑡)(𝑥(𝑡)−𝑥)=𝜘_𝑁(𝑡)𝐺(𝑡,𝑥(𝑡))−𝐺𝑡₀,𝑥+𝜘_𝑁(𝑡)^𝑡_𝑡0𝜘_𝑁(𝑠)𝑎(𝑠,𝑥(𝑠))𝑑𝑠+𝜘_𝑁(𝑡)^𝑡_𝑡0𝜘_𝑁(𝑠)𝜎(𝑠,𝑥(𝑠))𝑑𝑊(𝑠). (20) Similarly, we have 𝐸𝜘_𝑁(𝑡)(𝑥(𝑡)−𝑥)²≤3𝐸𝜘_𝑁(𝑡)𝐿₁|𝑥(𝑡)|+|𝐺(+𝐺𝑡−𝑥𝑡,𝑥)|₀,𝑥²+𝑇−𝑡₀^𝑡_𝑡0𝐸𝜘_𝑁(𝑠)|𝑎((𝑠(+|𝑎((𝑠𝑠,𝑥))−𝑎𝑠,𝑥)|𝑠,𝑥))|²+𝑑𝑠_𝑡𝑡0𝐸𝜘_𝑁(𝑠)𝜎((𝑠(+|𝜎((𝑠|𝑠,𝑥))−𝜎𝑠,𝑥)|𝑠,𝑥))|)²𝑑𝑠≤9𝐿²₁𝐸𝜘_𝑁(𝑡)|𝑥(𝑡)−𝑥|²+18𝐶₁1+𝐸|𝑥|²+6𝑇−𝑡₀^𝑡_𝑡0𝐸𝜘_𝑁(𝑠)𝐿²|𝑥(𝑠)−𝑥|²𝑑𝑠+6𝑇−𝑡₀²𝐶₁1+𝐸|𝑥|²+6^𝑡_𝑡0𝐸𝜘_𝑁(𝑠)𝐿²|𝑥(𝑠)−𝑥|²𝑑𝑠+6𝑇−𝑡₀²𝐶₁1+𝐸|𝑥|². (21) So, 1−9𝐿²₁𝐸𝜘_𝑁(𝑠)|𝑥(𝑠)−𝑥|²≤18𝐶₁1+𝐸|𝑥|²+12𝑇−𝑡₀𝐶₁1+𝐸|𝑥|²+6𝐿²𝑇−𝑡₀+6𝐿^2𝑡_𝑡0𝐸𝜘_𝑁(𝑠)|𝑥(𝑠)−𝑥|²𝑑𝑠. (22) Use the Gronwall-Bellman inequality to find  the estimation 𝐸𝜘_𝑁(𝑡)|𝑥(𝑡)−𝑥|²≤𝐷1+𝐸|𝑥|², (23) where 𝐷 is a constant independent of 𝑁. Applying the Fatou lemma to the last inequality and assuming that 𝑁→∞, we have 𝐸|𝑥(𝑡)−𝑥|²≤𝐷1+𝐸|𝑥|². (24) Since 𝐸|𝑥|²≤∞, then it follows from [8] that 𝐸|𝑥|²<∞  for every 𝑡∈[𝑡₀,𝑇].
Thus existence, uniqueness, and boundedness of the second moment of the solution 𝑥(𝑡) are proved on [𝑡₀,𝑇]. Since the constant 𝐵 is dependent only on 𝑇−𝑡₀ and 𝐸|𝑥|²<∞, and 𝑥(𝑇) is independent of  𝑊(𝑠)−𝑊(𝑇) for 𝑠⩾𝑇, then by similar manner we can prove the existence and uniqueness of the solution of the IVP with initial conditions (𝑇,𝑥(𝑇)) on the interval [𝑇,𝑇₁], where 𝑇₁ is chosen such that the inequality 3(𝐿²₁+5(𝑇₁−𝑇)𝐿²)<1.
This procedure can be repeated in order to extend the solution of (1) to the entire semiaxis 𝑡⩾𝑡₀. The theorem is proved.

Notes. The existence and uniqueness of the solution can be obtained as corollary from work [1], where this result was proved for an SDE of neutral type by replacing a condition for Lipschitz constant  𝐿₁<(1/6)^3/4 with more weak condition 𝐿₁<1. But by using our method, paths of obtained solution are continuous with probability 1. Otherwise, in the pointed work only the measurability of the solution and boundedness of its second moment were stated.

Now, we state some probability properties of the solution obtained in Theorem 2. We prove that under assumptions of Theorem 2, the solution of (1) is a Markov random process. Moreover, if the coefficients are continuous then it is a diffusion process. We will find its diffusion coefficients.

Theorem 3. Under conditions (1)-(2) of Theorem 2 the solution 𝑥(𝑡) of (1) is a Markov process with a transition probability defined by 𝑃(𝑡,𝑥,𝑠,𝐴)=𝑃𝑥_𝑡,𝑥(𝑠)∈𝐴, (25) where 𝑥_𝑡,𝑥(𝑠) is a solution of that equation 𝑥_𝑡,𝑥(𝑠)=𝐺𝑠,𝑥_𝑡,𝑥(𝑠)()++𝑥−𝐺𝑡,𝑥_𝑠𝑡𝑎𝑢,𝑥_𝑡,𝑥(𝑢)𝑑𝑢+^𝑠_𝑡𝜎𝑢,𝑥_𝑡,𝑥(𝑢)𝑑𝑊(𝑢), (26) where 𝑠⩾𝑡⩾𝑡₀,𝑥∈𝑅^𝑑.

Proof. As in Theorem 2, we solve (1) by the method of successive approximations. It can be shown that  𝑥_𝑡,𝑥(𝑠)  is completely defined by the nonrandom initial value 𝑥 and the process 𝑊(𝑠)−𝑊(𝑡) for 𝑠>𝑡, which are independent of 𝐹_𝑡. Since 𝑥(𝑡) is a solution of (2), it is 𝐹_𝑡 measurable. Hence 𝑥_𝑡,𝑥(𝑠) is independent of 𝑥(𝑡) and events from 𝐹_𝑡. Note that, from the uniqueness of the solution 𝑥(𝑠) for 𝑠>𝑡, it follows that it is a unique solution of the equation 𝑥(𝑠)((𝑠(𝑡)((𝑡+=𝐺𝑠,𝑥))+𝑥−𝐺𝑡,𝑥))^𝑠_𝑡𝑎((𝑢+𝑢,𝑥))𝑑𝑢_𝑠𝑡𝜎(𝑢,𝑥(𝑢))𝑑𝑊(𝑢). (27) Since the process 𝑥_{𝑡,𝑥(𝑡)}(𝑠) is also a solution of this equation, then 𝑥(𝑠)=𝑥_{𝑡,𝑥(𝑡)}(𝑠) with probability 1.
As for the rest, the proof is the same as the proof of the theorem for ordinary stochastic equations [11]. The theorem is proved.

We have a corollary from this theorem.

Corollary 4. Suppose that conditions of Theorem 2 are satisfied. Then(1)if functions 𝐺,𝑎, and 𝜎 are independent of 𝑡, then the solution 𝑥(𝑡) is a homogeneous Markov process;(2)if functions 𝐺,𝑎, and 𝜎 are periodic functions with period 𝜃, then a transition probability is periodic function; that is, 𝑃(𝑡+𝜃,𝑥,𝑠+𝜃,𝐴)=𝑃(𝑡,𝑥,𝑠,𝐴).

Now, we investigate conditions for which the solution of (1) is a diffusion process. For this we must  find an additional estimate.

Lemma 5. Let 𝑥_𝑡,𝑥(𝑠) be a solution of (2) such that 𝑥_𝑡,𝑥(𝑡)=𝑥,𝑥∈𝑅^𝑑. If conditions of Theorem 2 are satisfied and 𝐺_𝑡(𝑡,𝑥) is continuous for 𝑡≥𝑡₀, 𝑥∈𝑅^𝑑, then an inequality 𝐸𝑥_𝑡,𝑥(𝑠)−𝑥⁴≤𝐻(𝑠−𝑡)²1+|𝑥|⁴ (28) holds. Here 𝐻>0 is a constant dependent only on 𝐶,𝐿₁,𝐿, and 𝐺_𝑡(𝑡,𝑥).

Proof. Denote by 𝜘_𝑁(𝑠) the indicator of the set 𝜔∶sup_{𝑢∈[𝑡,𝑠]}𝑥_𝑡,𝑥(𝑢)−𝑥≤𝑁. (29) Similarly to finding estimation (24), we have 𝐸𝜘_𝑁(𝑠)𝑥_𝑡,𝑥(𝑠)−𝑥⁴≤27𝐸𝜘_𝑁(𝑠)𝐺𝑠,𝑥_𝑡,𝑥(𝑠)()()()−𝐺𝑠,𝑥+𝐺𝑠,𝑥−𝐺𝑡,𝑥⁴+27𝐸^𝑠_𝑡𝜘_𝑁(𝑢)𝑎𝑢,𝑥_𝑡,𝑠(𝑢)𝑑𝑢⁴+27𝐸^𝑠_𝑡𝜘_𝑁(𝑢)𝜎𝑢,𝑥_𝑡,𝑥(𝑢)𝑑𝑊(𝑢)⁴≤6³𝐿⁴₁𝐸𝜘_𝑁(𝑠)𝑥_𝑡,𝑥(𝑠)−𝑥⁴+6³𝐺⁴_𝑡(𝑡,𝑥)(𝑠−𝑡)⁴+𝑜_𝑥(𝑡−𝑠)⁴+27(𝑠−𝑡)^3𝑠_𝑡𝐸𝜘_𝑁(𝑢)𝑎𝑢,𝑥_𝑡,𝑥(𝑢)⁴+()∫𝑑𝑢27)(36)(𝑠−𝑡^𝑠_𝑡𝐸𝜘_𝑁(𝑢)𝜎𝑢,𝑥_𝑡,𝑥(𝑢)⁴𝑑𝑢, (30) where 𝑜_𝑥 points dependence on 𝑥.
The last estimate follows from Hölder’s inequality and properties of stochastic integral. From inequality (30) and condition for 𝐿₁, we have 1−6³𝐿⁴₁𝐸𝜘_𝑁(𝑠)𝑥_𝑡,𝑥(𝑠)−𝑥⁴≤6³𝐺_𝑡(𝑡,𝑥)⁴(𝑠−𝑡)⁴+𝑜_𝑥(𝑡−𝑠)⁴+(27)(8)(𝑠−𝑡)^3𝑠_𝑡𝐿⁴𝐸𝜘_𝑁(𝑢)𝑥_𝑡,𝑥(𝑢)−𝑥⁴+(8)𝑑𝑢27)()(𝑠−𝑡^3𝑠_𝑡|𝑎(𝑢,𝑥)|⁴+(8)𝑑𝑢27)(36)()(𝑠−𝑡^𝑠_𝑡|𝜎(𝑢,𝑥)|⁴+(8)𝐿𝑑𝑢27)(36)()(𝑠−𝑡^4𝑠_𝑡𝐸𝜘_𝑁(𝑢)𝑥_𝑡,𝑥(𝑢)−𝑥⁴𝑑𝑢≤6³𝐺_𝑡(𝑡,𝑥)⁴(𝑠−𝑡)⁴+𝑜_𝑥(𝑠−𝑡)⁴+6³(𝑠−𝑡)⁴𝐶₂1+|𝑥|⁴+6⁵(𝑠−𝑡)²𝐶₂1+|𝑥|⁴+6⁵(𝑠−𝑡)𝐿^4𝑠_𝑡𝐸𝜘_𝑁(𝑢)𝑥_𝑡,𝑠(𝑢)−𝑥⁴𝑑𝑢, (31) where 𝐶₂ depends only on 𝐶.
Further, the right-hand side of (31) can be estimated by 1+|𝑥|⁴𝑅(𝑠−𝑡)²+𝑅₁^𝑠_𝑡𝐸𝜘_𝑁(𝑢)𝑥_𝑡,𝑥(𝑢)−𝑥⁴𝑑𝑢, (32) where 𝑅=6³|𝐺_𝑡(𝑡,𝑥)|⁴+6³𝐶₂+6⁵𝐶₂+1 and  𝑅₁=6⁵𝐿⁴,𝑡≤𝑠≤𝑡+1.
Using the lemma from [11, p.38], we obtain 𝐸𝜘_𝑁(𝑢)𝑥_𝑡,𝑥(𝑢)−𝑥⁴≤𝐻(𝑠−𝑡)²1+|𝑥|⁴, (33) where 𝐻>0 depends only on 𝐶₂, 𝐿, and 𝐺_𝑡(𝑡,𝑥).
Now, assume that 𝑁→∞  and obtain estimate (28). This completes the proof of the Lemma.

Theorem 6. If conditions (1)-(2) of Theorem 2 are satisfied and functions 𝑎(𝑡,𝑥), 𝜎(𝑡,𝑥), 𝐺(𝑡,𝑥), 𝐺_𝑡(𝑡,𝑥), 𝐺_𝑥𝑖(𝑡,𝑥), and 𝐺_{𝑥𝑖𝑥𝑗}(𝑡,𝑥), 𝑖,𝑗=1,𝑑, are continuous for 𝑡⩾𝑡₀, 𝑥∈𝑅^𝑑. Functions 𝐺_𝑡(𝑡,𝑥), 𝐺_𝑥𝑖(𝑡,𝑥), and 𝐺_{𝑥𝑖𝑥𝑗}(𝑡,𝑥) satisfy Lipschitz condition with respect to 𝑥 in neighborhood of every point (𝑡,𝑥). Then the solution of (1) is diffusion process with 𝑎(𝑡,𝑥)=𝐼−𝐺_𝑥(𝑡,𝑥)⁻¹×𝐺_𝑡()()+1𝑡,𝑥+𝑎𝑡,𝑥2𝐺_𝑥𝑥(𝑡,𝑥)𝐼−𝐺_𝑥(𝑡,𝑥)⁻¹×𝜎(𝑡,𝑥)𝜎^𝑇(𝑡,𝑥)𝐼−𝐺_𝑥(𝑡,𝑥)^𝑇−1 (34) and diffusion matrix with 𝜎(𝑡,𝑥)=𝐼−𝐺_𝑥(𝑡,𝑥)⁻¹𝜎(𝑡,𝑥)𝜎^𝑇(𝑡,𝑥)𝐼−𝐺_𝑥(𝑡,𝑥)^𝑇−1. (35)

Proof. Take any point (𝑡,𝑥) from a region 𝑡⩾𝑡₀, 𝑥∈𝑅^𝑑. Consider a stochastic ITO equation in a closed neighborhood of this point 𝑑𝑦=𝐼−𝐺_𝑥𝑠,𝑦⁻¹×𝐺_𝑡+1𝑠,𝑦+𝑎𝑠,𝑦2𝐺_𝑥𝑥𝑠,𝑦𝐼−𝐺_𝑥𝑠,𝑦⁻¹×𝜎𝑠,𝑦𝜎^𝑇𝑠,𝑦𝐼−𝐺_𝑥𝑠,𝑦^𝑇−1+𝑑𝑠𝐼−𝐺_𝑥𝑠,𝑦⁻¹𝜎𝑠,𝑦𝑑𝑊(𝑠). (36)
From the conditions of the theorem, it follows that there exists a closed neighborhood of the point  (𝑡,𝑥), such that coefficients of the equation are Lipschitz with respect to 𝑥. Fix this neighborhood. Extend these coefficients on the all region 𝑡≥𝑡₀, 𝑥∈𝑅^𝑑 such that they remain continuous by both variables, Lipschitz and linear with respect to 𝑥.
Then the equation 𝑑𝑦=𝑎₁𝑠,𝑦𝑑𝑠+𝜎₁𝑠,𝑦𝑑𝑊(𝑠) (37) with these extended coefficients has a unique solution of the IVP 𝑦(𝑡)=𝑥   for  𝑠>𝑡.
From ITO formula and coincidence of coefficients of (36) and (37) in this neighborhood of the point (𝑡,𝑥), one can show that processes 𝑥_𝑡,𝑥 and 𝑦(𝑠) coincide in this neighborhood.
It is known that under these assumptions the process 𝑦(𝑠) is diffusion. Its diffusion coefficients in the point (𝑡,𝑥) are defined by coefficients of (36).
Next, show that 𝑥(𝑠) has the same diffusion coefficients. To do this it is sufficient to estimate the following limits and use estimation (28) lim_𝑠→𝑡1𝑠−𝑡_{|𝑦−𝑥|≤𝜀0}𝑦−𝑥𝑃𝑡,𝑥,𝑠,𝑑𝑦,lim_𝑠→𝑡1𝑠−𝑡_{|𝑦−𝑥|≤𝜀0}𝑧,𝑦−𝑥²𝑃𝑡,𝑥,𝑠,𝑑𝑦, (38) where 𝜀₀ is chosen such that the point (𝑠,𝑦) is laying in this neighborhood.
We get lim_𝑠→𝑡1𝑠−𝑡_{|𝑦−𝑥|≤𝜀0}𝑦−𝑥𝑃𝑡,𝑥,𝑠,𝑑𝑦=lim_𝑠→𝑡1𝑠−𝑡_{{𝑤∶|𝑥𝑡,𝑥(𝑠)−𝑥|≤𝜀0}}𝑥_𝑡,𝑥(𝑠)−𝑥𝑃(𝑑𝑤)=lim_𝑠→𝑡1𝑠−𝑡_{{𝑤∶|𝑦(𝑠)−𝑥|≤𝜀0}}𝑦(𝑠)𝑃()=−𝑥𝑑𝑤𝐼−𝐺_𝑥(𝑡,𝑥)⁻¹𝐺_𝑡()()+1𝑡,𝑥+𝑎𝑡,𝑥2𝐺_𝑥𝑥()×𝑡,𝑥𝐼−𝐺_𝑥(𝑡,𝑥)⁻¹𝜎(𝑡,𝑥)×𝜎^𝑇(𝑡,𝑥)𝐼−𝐺_𝑥(𝑡,𝑥)^𝑇−1. (39) Similarly, obtain the second limit in (38): lim_𝑠→𝑡1𝑠−𝑡_{|𝑦−𝑥|≤𝜀0}𝑧,𝑦−𝑥²𝑃=𝑡,𝑥,𝑠,𝑑𝑦𝐼−𝐺_𝑥(𝑡,𝑥)⁻¹𝜎(𝑡,𝑥)𝜎^𝑇()×𝑡,𝑥𝐼−𝐺_𝑥(𝑡,𝑥)^𝑇−1𝑧,𝑧. (40) So, the existence of limits in (38) is proved for every point (𝑡,𝑥) and fixed 𝜀₀, chosen for this point. It is clear that from (28), these limits exist for every 𝜀>0 and they are independent of 𝜀. From the above and the definition of diffusion process [12, p.67] we can finish the proof. The theorem is proved.