On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

Abstract

We consider a one-dimensional stochastic equation 𝑑𝑋_𝑡=𝑏(𝑡,𝑋_𝑡−)𝑑𝑍_𝑡+𝑎(𝑡,𝑋_𝑡)𝑑𝑡, 𝑡≥0, with respect to a symmetric stable process 𝑍 of index 0<𝛼≤2. It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation 𝑑𝐿_𝑡=𝐵(𝐿_𝑡−)𝑑𝑊𝑡 with respect to the semimartingale 𝑊=(𝑍,𝑡) and corresponding matrix 𝐵. In the case of 1≤𝛼<2 we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. The existence proofs are established using the method of Krylov's estimates for processes satisfying the 2-dimensional equation. On another hand, the Krylov's estimates are based on some analytical facts of independent interest that are also proved in the paper.

1. Introduction

Let 𝑍 be a one-dimensional symmetric stable process of index 0<𝛼≤2 with 𝑍₀=0. In this paper we will study the existence of solutions of the equation 𝑑𝑋_𝑡=𝑏𝑡,𝑋_𝑡−𝑑𝑍_𝑡+𝑎𝑡,𝑋_𝑡𝑑𝑡,𝑡≥0,𝑋₀=𝑥₀∈ℝ, (1.1) where 𝑎,𝑏∶[0,∞)×ℝ→ℝ are measurable functions. The existence of solutions is understood in weak sense. In the case of 1≤𝛼≤2, the coefficients 𝑎 and 𝑏 are assumed to be only measurable satisfying additionally some conditions of boundness.

Two important particular cases of (1.1) are the equations𝑑𝑋_𝑡=𝑏𝑡,𝑋_𝑡−𝑑𝑍_𝑡,𝑡≥0,𝑋₀=𝑥₀∈ℝ,𝑑𝑋_𝑡=𝑑𝑍_𝑡+𝑎𝑡,𝑋_𝑡𝑑𝑡,𝑡≥0,𝑋₀=𝑥₀∈ℝ. (1.2)(1.3)

If 𝛼=2, then 𝑍 is a Brownian motion, and this case has been extensively studied by many authors. The multidimensional analogue of (1.1) with only measurable (instead of continuous) coefficients was first studied by Krylov [1] who proved the existence of solutions assuming the boundness of 𝑎 and 𝑏 and nondegeneracty of 𝑏. The approach he used was based on 𝐿_𝑝-estimates for stochastic integrals of processes 𝑋 satisfying (1.1). Later, the results of Krylov were generalized to the case of nonbounded coefficients in various directions. We mention here only the results of Rozkosz and Slomiński [2, 3] who replaced, in particular, the assumption of boundness by the assumption of at most linear growth of the coefficients. The linear growth condition guaranteed the existence of nonexploding solutions. The case of exploding solutions was studied in [4] under assumptions of some local integrability of the coefficients 𝑎 and 𝑏.

In the one-dimensional case with 𝛼=2, the results are even stronger. For example, for the time-independent case of the coefficients Engelbert and Schmidt obtained very general existence and uniqueness results in [5]. For the case of the time-independent equation (1.2), one had found even sufficient and necessary conditions for the existence and uniqueness (in general, exploding) solutions [6]. The time-dependent equation (1.2) was studied by several authors; we mention here [2, 7] only.

There is less known in the case 𝛼<2. The time-independent equation (1.1) with 1<𝛼<2 was considered in [8] using the method of 𝐿₂-estimates for stable stochastic integrals with drift. To our knowledge, (1.1) in its general form and with measurable coefficients has not been studied except the particular cases (1.2) and (1.3). Thus, (1.2) in the case of 𝑏(𝑡,𝑥)=𝑏(𝑥) with arbitrary index 𝛼 was studied by Zanzotto in [9] where he, in particular, generalized the results of Engelbert and Schmidt to the case of 1<𝛼≤2. The time-dependent equation (1.2) with the index 1<𝛼<2 was treated in [10] using the method of Krylov’s estimates combined with the time change method. The time change method was also used in [11] where one obtained the sufficient conditions for the existence of solutions for the case of 0<𝛼<2 different from those in [10].

On another hand, the time-independent case of (1.3), that is when 𝑎(𝑡,𝑥)=𝑎(𝑥), was studied by Tanaka et al. in [12]. One obtained there the sufficient existence and uniqueness conditions assuming the drift coefficient 𝑎 to be bounded plus satisfying some additional conditions depending on the case whether 0<𝛼<1, 𝛼=1, or 1<𝛼<2. The method used by them was a purely analytical one relying on some properties of homogeneous Markov processes 𝑋 satisfying (1.3). More recently, Portenko [13] obtained a new existence result for the time-independent equation (1.3) for the case of 1<𝛼<2 assuming the function 𝑎 to be integrable on ℝ of the power 𝑝>1/(𝛼−1). The general case of (1.3) with 1<𝛼<2 was studied in [14] assuming 𝑎 being bounded.

The goal of this paper is to prove the existence of solutions of (1.1).

The paper is organized as follows. In Section 2 we recall the definitions and basic facts needed in the forthcoming sections. We also show that the existence of solutions of (1.1) is equivalent to the existence of solutions of a 2-dimensional stochastic equation driven by the semimartingale 𝑊_𝑡=(𝑍_𝑡,𝑡) with a corresponding matrix 𝐵. The approach is based on time change method. Section 3 is devoted to obtaining of various estimates. First, we will derive an analytic estimate for the value function associated with the control problem determined by solutions of the 2-dimensional equation. Using this estimate, we prove some variants of Krylov’s estimates for solutions of the 2-dimensional equation. The results of Section 3 apply to the case with 1≤𝛼≤2. Finally, in Section 4 we prove the existence of solutions of (1.1) combining the ideas of time change method with the results of Section 3.

2. Preliminaries and Time Change Method

We shall denote by 𝐃_[0,∞)(ℝ) the Skorokhod space, that is, the set of all real-valued functions 𝑧∶[0,∞)→ℝ with right-continuous trajectories and with finite left limits (also called cádlag functions). For simplicity, we shall write 𝐃 instead of 𝐃_[0,∞)(ℝ). We will equip 𝐃 with the 𝜎-algebra 𝒟 generated by the Skorokhod topology. Under 𝐃^𝑛 we will understand the 𝑛-dimensional Skorokhod space defined as 𝐃^𝑛=𝐃×⋯×𝐃 with the corresponding 𝜎-algebra 𝒟^𝑛 being the direct product of 𝑛 one-dimensional 𝜎-algebras 𝒟.

Let (Ω,ℱ,𝐏) be a complete probability space carrying a process 𝑍 with 𝑍₀=0, and let 𝔽=(ℱ_𝑡) be a filtration on (Ω,ℱ,𝐏). The notation (𝑍,𝔽) means that 𝑍 is adapted to the filtration 𝔽. We call (𝑍,𝔽) a symmetric stable process of index 𝛼∈(0,2] if trajectories of 𝑍 belong to 𝐃 and 𝐄𝑒^{𝑖𝜉(𝑍𝑡−𝑍𝑠)}∣ℱ_𝑠=𝑒^{−(𝑡−𝑠)|𝜉|𝛼} (2.1) for all 𝑡>𝑠≥0 and 𝜉∈ℝ. If 𝛼=2, 𝑍 is a process of Brownian motion with the variance 2𝑡. For 𝛼=1 we have a Cauchy process with unbounded second moment. In general, 𝐄|𝑍_𝑡|^𝛽<∞ for 𝛽<𝛼. The explicit form of the probability density function is known only for three values of 𝛼∈{1/2,1,2}.

For all 0<𝛼≤2, 𝑍 is a Markov process and can be characterized in terms of analytic characteristics of Markov processes. First, for any function 𝑓∈𝐿^∞(ℝ) and 𝑡≥0, we can define the operator 𝑃_𝑡𝑓(𝑥)∶=_Ω𝑓𝑥+𝑍_𝑡𝑑𝐏(𝜔), (2.2) where 𝐿^∞(ℝ) is the Banach space of functions 𝑓∶ℝ→ℝ with the norm ‖𝑓‖_∞=esssup|𝑓(𝑥)|. The family (𝑃_𝑡)_𝑡≥0 is called the family of convolution operators associated with 𝑍. Formally, for a suitable class of functions 𝑔(𝑥), let ℒ𝑔(𝑥)=lim_𝑡↓0𝑃_𝑡𝑔(𝑥)−𝑔(𝑥)𝑡, (2.3) called the infinitesimal generator of the process 𝑍.

On another hand, in the case of 𝛼∈(0,2), 𝑍 is a purely discontinuous Markov process that can be described by its Poisson jump measure (jump measure of 𝑍 on interval [0,𝑡]) defined as 𝜇(𝑈×[0,𝑡])=_𝑠≤𝑡1_𝑈𝑍_𝑠−𝑍_𝑠−, (2.4) the number of times before the time 𝑡 that 𝑍 has jumps whose size lies in the set 𝑈. The compensating measure of 𝜇, say 𝜈, is given (see, e.g., [15, Propostion  13.9],) by 𝜈(𝑈)=𝐄𝜇(𝑈×[0,1])=_𝑈1|𝑥|^1+𝛼𝑑𝑥. (2.5) It is known that for 𝛼<2ℒ𝑔(𝑥)=_ℝ⧵{0}𝑔(𝑥+𝑧)−𝑔(𝑥)−𝟏_{|𝑧|<1}𝑔^′(𝑥)𝑧𝑐_𝛼|𝑧|^1+𝛼𝑑𝑧 (2.6) for any 𝑔∈𝐶², where 𝐶² is the set of all bounded and twice continuously differentiable functions 𝑔∶ℝ→ℝ and 𝑐_𝛼 is a suitable constant. In contrary to the case of 𝛼∈(0,2), the infinitesimal generator of a Brownian motion process (𝛼=2) is the Laplacian, that is, the second derivative operator.

We notice also that the use of Fourier transform can simplify calculations when working with infinitesimal generator ℒ. Let 𝑔∈𝐿₁(ℝ²) and 𝑔𝜉₁,𝜉₂∶=_ℝ2𝑒^{𝑖𝑧1𝜉1+𝑖𝑧2𝜉2}𝑔𝑧₁,𝑧₂𝑑𝑧₁𝑑𝑧₂ (2.7) be the Fourier transform of 𝑔. Clearly, the function 𝑔(𝜉₁,𝜉₂) can be seen as the result of taking the Fourier transform from the function 𝑔(𝑧₁,𝑧₂) first in one variable and then in another (in any order). The following facts will be used later (cf. [14, Proposition  2.1]).

Proposition 2.1. Let ℒ be the infinitesimal generator of a symmetric stable process 𝑍. We have he following.(i)Assume that 𝑔∈𝐶²(ℝ) and ℒ𝑔∈𝐿₁(ℝ). Thenℒ𝑔(𝜉)=−𝜉^𝛼𝑔(𝜉). (2.8)(ii)Let 𝑔 be absolutely continuous on every compact subset of ℝ and 𝑔^′∈𝐿₁(ℝ). Then𝑔^′(𝜉)=−𝑖𝜉𝑔(𝜉). (2.9)

Finally, let us discuss how one can construct a solution of (1.1) for any 𝛼∈(0,2] using the time change method. By the definition, a process 𝑇 is called a 𝔽-time change if it is an increasing right-continuous process with 𝑇₀=0 such that 𝑇_𝑡 is a 𝔽-stopping time for any 𝑡≥0 (cf. [15, chapter 6]). Define 𝐴_𝑡=∶inf{𝑠≥0∶𝑇_𝑠>𝑡} called the right-continuous inverse process to 𝑇. It follows that 𝐴 is an increasing process starting at zero. Moreover, 𝐴 is a 𝔽-adapted process if and only if 𝑇 is a 𝔽-time change.

We shall here also recall the concept of exploding solutions for (1.1). Let (ℝ,ℬ(ℝ)) be the one-point compactification ℝ=ℝ∪{Δ} of ℝ equipped with the 𝜎-algebra ℬ(ℝ) of its Borel subsets. For any function 𝑧∶[0,+∞)→ℝ we set 𝜏_Δ(𝑧)=inf{𝑡≥0∶𝑧(𝑡)=Δ} (2.10) called the explosion time of the trajectory 𝑧 and define 𝐃([0,+∞)) (or simply 𝐃) to be the Skorohod space of exploding functions 𝑧∶[0,+∞)→ℝ such that 𝑧 is right-continuous with finite left-hand limits on the interval [0,𝜏_Δ(𝑧)) and 𝑧(𝑡)=Δ whenever 𝑡≥𝜏_Δ(𝑧).

We say that a stochastic process (𝑋,𝔽), defined on a probability space (Ω,ℱ,𝐏) with filtration 𝔽=(ℱ_𝑡)_𝑡≥0 and with trajectories in 𝐃, is a weak solution of (1.1) with initial state 𝑥₀∈ℝ if there exists a symmetric stable process 𝑍 with respect to the filtration 𝔽 such that 𝑍₀=0 and𝑋_𝑡=𝑥₀+^𝑡₀𝑏𝑠,𝑋_𝑠−𝑑𝑍_𝑠+^𝑡₀𝑎𝑠,𝑋_𝑠𝑑𝑠on𝑡<𝜏_Δ(𝑋)𝐏-a.s. (2.11) for all 𝑡≥0, where 𝜏_Δ(𝑋) is called the explosion time of 𝑋. Since 𝑍 is a semimartingale for all 0<𝛼≤2, the stochastic integral in (2.11) can be defined for all appropriate integrands via semimartingale integration theory.

If 𝜏_Δ(𝑋)=∞, then 𝑋 is called a nonexploding solution, otherwise—exploding solution with the explosion time 𝜏_Δ(𝑋).

Let 𝑍 be a symmetric stable process 𝑍 of index 𝛼∈(0,2] defined on a probability space (Ω,ℱ,𝐏) and 𝑥₀∈ℝ an arbitrary value. We introduce the matrix 𝐵 defined as 0|𝑏|𝐵∶=^−𝛼1𝑎|𝑏|^−𝛼 (2.12) and set 𝑊_𝑡∶=(𝑍_𝑡,𝑡), 𝐿_𝑡=(𝐴_𝑡,𝑌_𝑡), 𝑥₀∶=(0,𝑥₀).

Consider the 2-dimensional equation 𝐿_𝑡=𝑥₀+^𝑡₀𝐵𝐿_𝑠−𝑑𝑊_𝑠, (2.13) which, if written componentwise, is equivalent to the following two one-dimensional equations: 𝐴_𝑡=^𝑡₀|𝑏|^−𝛼𝐴_𝑠,𝑌_𝑠𝑌𝑑𝑠,_𝑡=𝑥₀+𝑍_𝑡+^𝑡₀𝑎|𝑏|^−𝛼𝐴_𝑠,𝑌_𝑠𝑑𝑠. (2.14)(2.15) Notice that the process 𝑊 is a semimartingale; hence (2.13) can be seen as a stochastic differential equation with respect to a semimartingale.

Moreover, det(𝐵)=−|𝑏|^−𝛼 so that the matrix 𝐵 is nondegenerate since 𝑏⁻¹∶=1/𝑏 and 𝑏≠∞ by the definition of the coefficient 𝑏. We also see that 𝐴 is a strictly increasing nonnegative process such that 𝐴₀=0. Let 𝐴_∞∶=lim_𝑡→∞𝐴_𝑡. The properties of 𝐴 imply that the right inverse to 𝐴 process 𝑇 is a continuous process defined on the interval [0,𝐴_∞).

Proposition 2.2. Assume that there exist constants 𝛿₁>0 and 𝛿₂>0 such that 𝛿₁≤|𝑏|≤𝛿₂. Then, (1.1) has a solution if and only if (2.13) has a solution.

Proof.. We notice that the assumptions on the coefficient 𝑏 imply that the solutions of both equations are nonexploding.
Suppose first that 𝑋 is a solution of (1.1) which means that (2.11) is satisfied. The integrals on the right side of (2.11) are well defined and are 𝐏-a.s. finite for all 𝑡≥0. Let 𝑇_𝑡=^𝑡₀|𝑏|^𝛼𝑠,𝑋_𝑠𝐴𝑑𝑠,_𝑡=inf𝑠≥0∶𝑇_𝑠>𝑡. (2.16) It can be easily verified that the process 𝐴 satisfies the relation 𝐴_𝑡=^𝑡₀|𝑏|^−𝛼𝐴_𝑠,𝑋_𝐴𝑠𝑑𝑠. (2.17) By its definition, the process 𝑇 is 𝔽-adapted so that its right-inverse process 𝐴 is a 𝔽-time change process defined for all 𝑡≥0. We notice that (𝐴_𝑡) is a global time change (that is, 𝐴_𝑡∈[0,∞) for all 𝑡≥0) because 𝑇_∞=lim_𝑡↑∞𝑇_𝑡=∞. Now define 𝑌_𝑡=𝑋_𝐴𝑡,𝒢_𝑡=ℱ_𝐴𝑡. (2.18) Applying the time change 𝑡→𝐴_𝑡 to the semimartingale 𝑋 in (2.11) (see [16, Chapter 10]) and using the change of variables rule in Lebesgue-Stieltjes integral (see ch. 0, (4.9) in [17]) yield 𝑌_𝑡=𝑥₀+^𝐴𝑡₀𝑏𝑠,𝑋_𝑠−𝑑𝑍_𝑠+^𝑡₀𝑎𝐴_𝑠,𝑌_𝑠𝑑𝐴_𝑠. (2.19) It remains to notice that the process 𝑍_𝑡∶=^𝐴𝑡₀𝑏𝑠,𝑋_𝑠−𝑑𝑍_𝑠 (2.20) is nothing but a symmetric stable process of the index 𝛼 (see [18], Theorem  3.1). Hence 𝐿=(𝐴,𝑌) is a solution of (2.13).
Now, let 𝐿=(𝐴,𝑌) be a solution of (2.13) defined on a probability space (Ω,𝒢,𝐏) with a filtration 𝔾, where 𝑍 is a symmetric stable process adapted to 𝔾. Let 𝑋_𝑡=𝑌_𝑇𝑡,ℱ_𝑡=𝒢_𝑇𝑡 (2.21) for all 𝑡≥0 where 𝑇 is the right inverse to 𝐴. It follows 𝐴_∞=∞ so that 𝑇 is a global time change. By applying the time change 𝑡→𝑇_𝑡 to the semimartingale 𝑌 in (2.15) we obtain 𝑍_𝑇𝑡=𝑋_𝑡−𝑥₀−^𝑡₀𝑎𝑠,𝑋_𝑠𝑑𝑠. (2.22)
Using the standard arguments of time change in stochastic integrals with respect to symmetric stable processes (see, e.g., [11]), we conclude that there exists a symmetric stable process (defined on the same probability space as 𝑍) such that 𝑍_𝑇𝑡=^𝑡₀𝑏𝑠,𝑋_𝑠−𝑑𝑍_𝑠, (2.23) what finishes the proof.

Actually, as Proposition 2.2 indicates, to prove the existence of solutions of (1.1), we need only to assume that (2.13) has a solution. In this sense the assumptions on the coefficient 𝑏 required in Proposition 2.2 can be slightly relaxed.

Corollary 2.3. Let 𝐿 be a solution of (2.13), where 𝑥₀∈ℝ, and there exists a constant 𝛿₁>0 such that |𝑏|>𝛿₁. Then there exists a (possibly, exploding) solution 𝑋 of (1.1).

Proof. By assumptions, there exist a solution 𝐴 of (2.14) and a process 𝑌 satisfying (2.15), both adapted to the same filtration 𝔽. For any 𝑡≥0, let 𝑇_𝑡=inf𝑠≥0∶𝐴_𝑠>𝑡 (2.24) be the right inverse of the process 𝐴. By 𝐴_∞ and 𝑇_∞ we denote the limits of processes 𝐴 and 𝑇 as 𝑡→∞, respectively. Clearly, 𝐴 and 𝑇 are strictly increasing and continuous processes defined on intervals [0,𝑇_∞) and [0,𝐴_∞), respectively. In particular, we have that 𝐴_𝑇𝑡=𝑡∧𝐴_∞ and 𝑇_𝐴𝑡=𝑡∧𝑇_∞ for all 𝑡≥0. We notice further that 𝑇 is a 𝔽-time change, finite on [0,𝐴_∞) and equal to infinity for 𝑡≥𝐴_∞. Define 𝑋_𝑡=𝑥₀+𝑌_𝑇𝑡 for all 𝑡∈[0,𝐴_∞) and 𝑋_𝑡=Δ for all 𝑡≥𝐴_∞. Also let ℋ_𝑡=ℱ_𝐴𝑡 for all 𝑡≥0. Our goal is to show that the process (𝑋,ℍ) is a solution of (1.1).
By making a time change in the relation (2.14), we obtain for all 𝑡≥0𝑇_𝑡=^𝑇𝑡₀|𝑏|^𝛼𝐴_𝑠,𝑥₀+𝑌_𝑠|𝑏|^−𝛼𝐴_𝑠,𝑥₀+𝑌_𝑠𝑑𝑠=^𝑇𝑡₀|𝑏|^𝛼𝐴_𝑠,𝑥₀+𝑌_𝑠𝑑𝑇_𝑠=^𝐴𝑇𝑡₀|𝑏|^𝛼𝑠,𝑥₀+𝑌_𝑇𝑠𝑑𝑠=^{𝑡∧𝐴∞}₀|𝑏|^𝛼𝑠,𝑋_𝑠𝑑𝑠𝐏-a.s. (2.25) Applying the same time change to processes in (2.15) yields 𝑋_{𝑡∧𝐴∞}=𝑥₀+𝑍_𝑇𝑡+^{𝑡∧𝐴∞}₀𝑎𝑠,𝑋_𝑠𝑑𝑠. (2.26) Let us look at the process 𝑍_𝑇𝑡. By properties of stable integrals (see, e.g., [11, Proposition  4.3] or [18, Theorem  3.1]), there exists a symmetric stable process (𝑍,ℍ) of the same index 𝛼 stopped at 𝐴_∞ such that 𝑍_𝑡=^𝑇𝑡₀𝑏⁻¹𝐴_𝑠,𝑥₀+𝑌_𝑠−𝑑𝑍_𝑠,𝑡≥0. (2.27) From the last relation and time change properties in stochastic integrals with respect to a semimartingale (see, e.g., [16, Theorem  10.19]), it follows that 𝑍_𝑡=^𝑇𝑡₀𝑏⁻¹𝐴_𝑠,𝑥₀+𝑌_𝑠−𝑑𝑍_𝑠=^𝑡₀𝑏⁻¹𝑠,𝑋_𝑠−𝑑𝑍_𝑇𝑠,𝑡<𝐴_∞,𝐏-a.s. (2.28) Now, the relation (2.25) yields that, for all 𝑡<𝐴_∞, the integral ∫^𝑡₀|𝑏|^𝛼(𝑠,𝑋_𝑠)𝑑𝑠 is finite hence there exists the stochastic integral ∫^𝑡₀𝑏(𝑠,𝑋_𝑠−)𝑑𝑍_𝑠 (see, e.g., [11, Proposition  4.3]). Using (2.28), we obtain then ^𝑡₀𝑏𝑠,𝑋_𝑠−𝑑𝑍_𝑠=^𝑡₀𝑏𝑠,𝑋_𝑠−𝑏⁻¹𝑠,𝑋_𝑠−𝑑𝑍_𝑇𝑠=𝑍_𝑇𝑡,𝑡<𝐴_∞,𝐏-a.s. (2.29) Enlarging the probability space, we can assume that 𝑍 is extended to a full symmetric stable process of index 𝛼. The last relation combined with (2.26) verifies that 𝑋 is a solution of (2.11), possibly, exploding in 𝐴_∞.

Corollary 2.4. Let 𝐿 be a solution of (2.13) with 𝑥₀∈ℝ, and assume that there exist constants 𝛿₁>0 and 𝛿₂>0 such that 𝛿₁≤|𝑏(𝑡,𝑥)|≤𝛿₂ for all (𝑡,𝑥). Then there exists a nonexploding solution 𝑋 of (1.1).

3. Some Estimates

Let 𝛿₁, 𝛿₂, and 𝐾 be strictly positive constants and 𝑍 a symmetric stable process of index 0<𝛼≤2 defined on a probability space (Ω,ℱ,𝐏) with filtration 𝔽. By ℐ₁ and ℐ₂ we denote the classes of all 𝔽-predictable one-dimensional processes 𝑟_𝑡 and 𝛾_𝑡, respectively, such that 𝛿₁≤𝑟_𝑡≤𝛿₂ and |𝛾_𝑡|≤𝐾.

For any (𝑡,𝑥)∈ℝ², 𝜆>0, and any nonnegative, measurable function 𝑓∈𝐶^∞₀(ℝ²) (𝐶^∞₀(ℝ²) denotes the class of all infinitely many times differentiable real-valued functions with compact support defined on ℝ²) defines the value function 𝑣(𝑡,𝑥) as 𝑣(𝑡,𝑥)=sup_{𝑟∈ℐ1,𝛾∈ℐ2}𝐄^∞₀𝑒^−𝜆𝑠𝑓𝑠+𝜓^𝑟_𝑠,𝑥+𝑋^𝛾_𝑠𝑑𝑠, (3.1) where the processes 𝜓^𝑟 and 𝑋^𝛾 are given by 𝜓^𝑟_𝑡=^𝑡₀𝑟_𝑠𝑑𝑠,𝑋^𝛾_𝑡=𝑍_𝑡+^𝑡₀𝛾_𝑠𝑑𝑠. (3.2) Then, for the value function 𝑣 and the process (𝜓^𝑟,𝑋^𝛾), the Bellman principle of optimality can be formulated as follows (cf. [1]): for any [0,∞)-valued 𝔽-stopping time 𝜏 it holds 𝑣(𝑡,𝑥)=sup_{𝑟∈ℐ1,𝛾∈ℐ2}𝐄^𝜏₀𝑒^−𝜆𝑠𝑓𝑡+𝜓^𝑟_𝑠,𝑥+𝑋^𝛾_𝑠𝑑𝑠+𝑒^−𝜆𝜏𝑣𝑡+𝜓^𝑟_𝜏,𝑥+𝑋^𝛾_𝜏. (3.3) Using standard arguments, one can derive from the principle above the corresponding Bellman equation (𝑟 and 𝛾 are deterministic) sup_{𝛿1≤𝑟≤𝛿2}sup_{|𝛾|≤𝐾}𝑟𝑣_𝑡(𝑡,𝑥)+ℒ𝑣(𝑡,𝑥)−𝜆𝑣(𝑡,𝑥)+𝛾𝑣_𝑥(𝑡,𝑥)+𝑓(𝑡,𝑥)=0, (3.4) which holds a.e. in ℝ². Here 𝑣_𝑡 and 𝑣_𝑥 denote the partial derivatives of the function 𝑣(𝑡,𝑥) in 𝑡 and 𝑥, respectively.

Define 𝑄={(𝑡,𝑥)∶𝑣_𝑡(𝑡,𝑥)>0}. Then, the Bellman equation is equivalent to two equations 𝛿₂𝑣_𝑡+ℒ𝑣−𝜆𝑣+𝐾𝑣_𝑥𝛿+𝑓=0on𝑄,₁𝑣_𝑡+ℒ𝑣−𝜆𝑣+𝐾𝑣_𝑥+𝑓=0on𝑄^𝑐. (3.5)

Lemma 3.1. Let 1<𝛼≤2 and 𝛿₂/𝛿₁∈[1,2+√2). Then, for all (𝑡,𝑥)∈ℝ², it holds 𝑣(𝑡,𝑥)≤𝑁𝑓₂∶=𝑁_ℝ2𝑓²𝑠,𝑦𝑑𝑠𝑑𝑦^1/2, (3.6) where the constant 𝑁 depends on 𝛿₁,𝐾, and 𝛼 only.

Proof. For any function ℎ∶ℝ²→ℝ such that ℎ∈𝐿₁(ℝ²) and any 𝜀>0 we define ℎ^(𝜀)()=1𝑡,𝑥𝜀²_ℝ²ℎ(𝑡,𝑥)𝑞𝑡−𝑠𝜀,𝑥−𝑦𝜀𝑑𝑠𝑑𝑦 (3.7) to be the 𝜀-convolution of ℎ with a smooth function 𝑞 such that 𝑞∈𝐶^∞₀(ℝ²) and ∫_ℝ2𝑞(𝑡,𝑥)𝑑𝑡𝑑𝑥=1.
For 𝜀>0, let 𝑓^(𝜀)∶=𝜆𝑣^(𝜀)−𝛿₂𝑣_𝑡^(𝜀)−ℒ𝑣^(𝜀)−𝐾𝑣_𝑥^(𝜀)on𝑄,𝜆𝑣^(𝜀)−𝛿₁𝑣_𝑡^(𝜀)−ℒ𝑣^(𝜀)−𝐾𝑣_𝑥^(𝜀)on𝑄^𝑐. (3.8) It follows that 𝛿₂𝑣_𝑡^(𝜀)+𝐿𝑣^(𝜀)−𝜆𝑣^(𝜀)=𝛿₂𝑣_𝑡^(𝜀)+𝐿𝑣^(𝜀)−𝜆𝑣^(𝜀)𝟏_𝑄+𝛿₂𝑣_𝑡^(𝜀)+𝐿𝑣^(𝜀)−𝜆𝑣^(𝜀)𝟏_𝑄𝑐=−𝑓^(𝜀)+𝐾𝑣_𝑥^(𝜀)+𝛿₂−𝛿₁𝑣_𝑡^(𝜀)𝟏_𝑄𝑐 (3.9) so that 𝛿₂𝑣_𝑡^(𝜀)+ℒ𝑣^(𝜀)−𝜆𝑣^(𝜀)2≤4𝑓^(𝜀)2+𝐾²𝑣_𝑥^(𝜀)2+2𝛿₂−𝛿₁²𝑣_𝑡^(𝜀)2. (3.10) Obviously, 𝑓^(𝜀) is square integrable, and (3.5) implies that 𝑓^(𝜀)→𝑓 as 𝜀↓0 a.s. in ℝ².
Now, applying Proposition 2.1, the Parseval identity and integration by parts to the inequality _ℝ2𝛿₂𝑣_𝑡^(𝜀)+ℒ𝑣^(𝜀)(𝑡,𝑥)−𝜆𝑣^(𝜀)(𝑡,𝑥)²𝑑𝑡𝑑𝑥≤4_ℝ2𝑓^(𝜀)2(𝑡,𝑥)+𝐾²𝑣_𝑥^(𝜀)2(𝑡,𝑥)𝑑𝑡𝑑𝑥+2𝛿₂−𝛿₁²_ℝ2𝑣_𝑡^(𝜀)2(𝑡,𝑥)𝑑𝑡𝑑𝑥 (3.11) yields _ℝ2𝑣^(𝜀)(𝜁,𝜉)²|𝜉|^𝛼+𝜆²+𝛿²₂𝜁²𝑑𝜁𝑑𝜉≤4_ℝ2𝑓^(𝜀)(𝜁,𝜉)²+𝐾²𝜉²𝑣^(𝜀)(𝜁,𝜉)²𝑑𝜁𝑑𝜉+2𝛿₂−𝛿₁²_ℝ2𝜁²𝑣^(𝜀)(𝜁,𝜉)²𝑑𝜁𝑑𝜉. (3.12) Let 𝛿∶=𝛿²₂−2(𝛿₂−𝛿₁)². It follows from the assumptions that 𝛿>0. The inequality (3.12) can be rewritten then as _ℝ2𝑣^(𝜀)(𝜁,𝜉)²𝜉^𝛼+𝜆²+𝛿𝜁²𝑑𝜁𝑑𝜉≤4_ℝ2𝑓^(𝜀)(𝜁,𝜉)²𝑑𝜉𝑑𝜁+4𝐾²_ℝ2𝜉²𝑣^(𝜀)(𝜁,𝜉)²𝑑𝜁𝑑𝜉. (3.13) One sees easily that there exists a constant 𝜆₀>0 such that 𝜉^𝛼+𝜆₀²+𝛿𝜁²≥𝜉^𝛼+𝜆₀²≥8𝐾²𝜉² (3.14) for all (𝜁,𝜉)∈ℝ².
Combining the inequalities (3.12) and (3.14), we obtain for all 𝜆≥𝜆₀12_ℝ²𝑣^(𝜀)(𝜁,𝜉)²𝜉^𝛼+𝜆+𝛿𝜁²𝑑𝜁𝑑𝜉≤4_ℝ2𝑓^(𝜀)(𝜁,𝜉)²𝑑𝜁𝑑𝜉. (3.15) Let 𝑁₁∶=_ℝ2𝑑𝜁𝑑𝜉𝜉^𝛼+𝜆²+𝛿𝜁². (3.16) Clearly, the constant 𝑁₁ is finite and depends on 𝛿₁, 𝛿₂, 𝐾, and 𝛼 only.
Using the estimate (3.15) and the inverse Fourier transform yields for all (𝑡,𝑥)∈ℝ² and𝜆≥𝜆₀𝑣^(𝜀)(𝑡,𝑥)²≤𝑁₁4𝜋²_ℝ²𝑣^(𝜀)(𝜁,𝜉)²𝜉^𝛼+𝜆²+𝛿𝜁²≤𝑁𝑑𝜁𝑑𝜉₁𝜋²_ℝ²𝑓^(𝜀)(𝑠,𝑧)²𝑑𝑠𝑑𝑧. (3.17) The result follows then by taking the limit 𝜀→0 in the above inequality and using the Lebesgue-dominated convergence theorem.

Now, let 𝐿=(𝐴,𝑌) be a solution of (2.13), and there exist constants 𝐾>0 and 𝛿>0 such that the following assumptions are satisfied: |𝑎(𝑡,𝑥)|≤𝐾|𝑏(𝑡,𝑥)|^𝛼∀()∈[)|𝑏|𝑡,𝑥0,∞×ℝ,𝛿≤^−𝛼(𝑡,𝑥)≤𝛽𝛿where𝛽∈1,2+√2,(𝑡,𝑥)∈[0,∞)×ℝ. (3.18) We are interested in 𝐿₂-estimates of the form𝐄^∞₀𝑒^−𝜆𝑢𝑓𝑡₀+𝐴_𝑢,𝑥₀+𝑌_𝑢𝑑𝑢≤𝑁𝑓₂, (3.19) where (𝑡₀,𝑥₀)∈ℝ²,𝜆>0.

Theorem 3.2. Assume that 1<𝛼≤2, 𝐿 is a solution of (2.13), and the assumptions (3.18) hold. Then, for any (𝑡₀,𝑥₀)∈ℝ², 𝜆≥𝜆₀, and any measurable function 𝑓∶ℝ²→[0,∞), the estimate (3.19) is satisfied where the constant 𝑁 depends on 𝛿,𝐾, and 𝛼 only.

Proof. Assume first that 𝑓∈𝐶^∞₀(ℝ²) so that there is a solution 𝑣 of (3.5) satisfying the inequality (3.6). By taking the 𝜀-convolution on both sides of (3.5), we obtain for all 𝛿≤𝑟≤𝛽𝛿 and |𝛾|≤𝐾𝑟𝑣_𝑡^(𝜀)+ℒ𝑣^(𝜀)−𝜆𝑣^(𝜀)+𝛾𝑣_𝑥^(𝜀)+𝑓^(𝜀)≤0. (3.20) Then, for 𝑠≥0, applying the Itô’s formula to the expression 𝑣^(𝜀)𝑡₀+𝐴_𝑠,𝑥₀+𝑌_𝑠𝑒^−𝜆𝑠 (3.21) yields 𝐄𝑣^(𝜀)𝑡₀+𝐴_𝑠,𝑥₀+𝑌_𝑠𝑒^−𝜆𝑠−𝑣^(𝜀)𝑡₀,𝑥₀=𝐄^𝑠₀𝑒^−𝜆𝑢𝑣_𝑡^(𝜀)|𝑏|^−𝛼𝐴_𝑢,𝑌_𝑢+ℒ𝑣^(𝜀)−𝜆𝑣^(𝜀)+𝑎|𝑏|^−𝛼𝐴_𝑢,𝑌_𝑢𝑣_𝑥^(𝜀)𝑡₀+𝐴_𝑢,𝑥₀+𝑌_𝑢𝑑𝑢≤−𝐄^𝑠₀𝑒^−𝜆𝑢𝑓^(𝜀)𝑡₀+𝐴_𝑢,𝑥₀+𝑌_𝑢𝑑𝑢. (3.22) By Lemma 3.1𝐄^𝑠₀𝑒^−𝜆𝑢𝑓^(𝜀)𝑡₀+𝐴_𝑢,𝑥₀+𝑌_𝑢𝑑𝑢≤sup_𝑡0,𝑥0𝑣^(𝜀)𝑡₀,𝑥₀≤𝑁𝑓^(𝜀)₂. (3.23) It remains to pass to the limit in the above inequality letting 𝜀→0, 𝑠→∞ and using the Fatou’s lemma.
The inequality (3.19) can be extended in a standard way first to any function 𝑓∈𝐿₂(ℝ) and then to any nonnegative, measurable function using the monotone class theorem arguments (see, e.g., [19, Theorem  20]).

Now, for arbitrary but fixed 𝑡>0, 𝑚∈ℕ, define 𝑓_2,𝑚,𝑡=^𝑡₀^𝑚_−𝑚𝑓(𝑠,𝑥)²𝑑𝑥𝑑𝑠^1/2 (3.24) to be the 𝐿₂-norm of 𝑓 on [0,𝑡]×[−𝑚,𝑚]. Applying (3.19) to the function 𝑓(𝑠,𝑥)=𝑓(𝑠,𝑥)𝟏_{[0,𝑡]×[−𝑚,𝑚]}(𝑠,𝑥), we obtain the following local version of Krylov’s estimates.

Corollary 3.3. Let 1<𝛼≤2 and 𝐿=(𝐴,𝑌) be a solution of (2.13). Suppose that the conditions (3.18) are satisfied. Then, for any 𝑡≥0, 𝑚∈ℕ, and any nonnegative measurable function 𝑓, it follows that 𝐄^{𝑡∧𝜏𝑚}₀^(𝑌)𝑓𝐴_𝑢,𝑌_𝑢𝑑𝑢≤𝑁𝑓_2,𝑚,𝑡, (3.25) where 𝜏_𝑚(𝑌)∶=inf{𝑡≥0∶|𝑌_𝑡|≥𝑚} and 𝑁 is a constant depending on 𝛿, 𝐾, 𝛼, 𝑚, and 𝑡 only.

From Theorem 3.2 and Corollary 3.3 we also obtain the following.

Corollary 3.4. Let (𝐴,𝑌) be a solution of (2.13) with 𝛼=1. If the assumptions (3.18) are satisfied with arbitrary 𝛿 and 𝐾<1/2√2, then the estimate (3.25) holds.

4. Existence of Solutions

Now we turn our attention to the existence of solutions of (1.1) and (2.13). Since the case of 𝛼=2 is well studied, we restrict ourselves to the case 𝛼<2.

Theorem 4.1. Let 1≤𝛼<2 and assume that the assumptions (3.18) are satisfied where the constant 𝐾 is arbitrary for 1<𝛼<2 and 𝐾<1/2√2 for 𝛼=1. Then, for any 𝑥₀∈ℝ, there exists a (nonexploding) solution of (1.1).

Proof. We first prove the existence of solutions of the equation (2.13).
It follows from the assumptions that the coefficient 𝐵 is bounded. Hence we can find a sequence of functions 𝑞_𝑛 and 𝑝_𝑛,𝑛≥1, such that they are globally Lipshitz continuous and uniformly bounded by the constant min{2𝛿,2𝐾𝛿}. Moreover, 𝑞_𝑛→𝑎|𝑏|^−𝛼 and 𝑝_𝑛→|b|^−𝛼 as 𝑛→∞ pointwise and in ‖⋅‖_{2,𝑡,𝑚−} norm for all 𝑡>0, 𝑚∈ℕ. For any 𝑛=1,2,…, (2.13) has a unique strong solution (see, e.g., [20, Theorem  9.1]). That is, for any fixed symmetric stable process 𝑍 defined on a probability space (Ω,ℱ,𝐏), there exists a sequence of processes 𝐿^𝑛=(𝐴^𝑛,𝑌^𝑛), 𝑛=1,2,…, such that 𝐿^𝑛_𝑡=𝑥₀+^𝑡₀𝐵_𝑛𝑠,𝐿^𝑛_𝑠𝑑𝑊_𝑠,𝑡≥0,𝐏-a.s., (4.1) where 𝐵_𝑛∶=0𝑝_𝑛1𝑞_𝑛 (4.2) or, written componentwise, 𝐴^𝑛_𝑡=^𝑡₀𝑝_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝑌𝑑𝑠,_𝑛𝑡=𝑥₀+𝑍_𝑡+^𝑡₀𝑞_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝑑𝑠. (4.3) Let 𝑆^𝑛_𝑡∶=^𝑡₀𝑞_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝑑𝑠, (4.4) so that 𝑌^𝑛=𝑥₀+𝑍+𝑆^𝑛,𝑛≥1. (4.5)
Now we claim that the sequence of processes 𝐻^𝑛∶=(𝑌^𝑛,𝑆^𝑛,𝐴^𝑛,𝑍), 𝑛≥1, is tight in the sense of weak convergence in (𝐃⁴,𝒟⁴). Due to the Aldous’ criterion ([21]), we have only to show that lim_𝑙→∞limsup_𝑛→∞𝐏sup_{0≤𝑠≤𝑡}𝐻^𝑛_𝑠>𝑙=0 (4.6) for all 𝑡≥0 and limsup_𝑛→∞𝐏𝐻^𝑛_{𝜏𝑡∧𝑛+𝑟𝑛}−𝐻^𝑛_{𝑡∧𝜏𝑛}>𝜀=0 (4.7) for all 𝑡≥0, 𝜀>0, every sequence of 𝔽-stopping times 𝜏^𝑛, and every sequence of real numbers 𝑟_𝑛 such that 𝑟_𝑛↓0. We use ‖⋅‖ to denote the Euclidean norm of a vector.
But both conditions are clearly satisfied because of the uniform boundness of the coefficients 𝑞_𝑛 and 𝑝_𝑛 for all 𝑛≥1.
Since the sequence {𝐻^𝑛} is tight, there exists a subsequence {𝑛_𝑘}, 𝑘=1,2,…, a probability space (Ω,ℱ,𝐏) and the process 𝐻 on it with values in (𝐃⁴,𝒟⁴) such that 𝐻^𝑛𝑘 converges weakly (in distribution) to the process 𝐻 as 𝑘→∞. For simplicity, let {𝑛_𝑘}={𝑛}.
According to the embedding principle of Skorokhod (see, e.g., [20, Theorem  2.7]), there exists a probability space (Ω,ℱ,𝐏) and the processes 𝐻=(𝑌,𝑆,𝐴,𝑍), 𝐻^𝑛=(𝑌^𝑛,𝑆^𝑛,𝐴^𝑛,𝑍^𝑛), 𝑛=1,2,…, on it such that(i)𝐻^𝑛→𝐻 as 𝑛→∞𝐏-a.s.(ii)𝐻^𝑛=𝐻^𝑛 in distribution for all 𝑛=1,2,… Using standard measurability arguments ([1, chapter 2]), one can prove that the processes 𝑍^𝑛 and 𝑍 are symmetric stable processes of the same index as the processes 𝑍^𝑛 with respect to the augmented filtrations 𝔽^𝑛 and 𝔽 generated by processes 𝐻^𝑛 and 𝐻, respectively.
Using the properties (i), (ii), and (4.1), one can show (cf. [1, chapter 2]) that 𝑌^𝑛_𝑡=𝑥₀+𝑍^𝑛_𝑡+^𝑡₀𝑞_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝐴𝑑𝑠,𝑡≥0,𝐏-a.s.,_𝑛𝑡=^𝑡₀𝑝_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝑑𝑠,𝑡≥0,𝐏-a.s. (4.8) On the other hand, the same properties and the quasileft continuity of the the processes 𝐻^𝑛 yield lim_𝑛→∞𝐴^𝑛_𝑡=𝐴_𝑡,lim_𝑛→∞𝑌^𝑛_𝑡=𝑌_𝑡𝑡≥0,𝐏-a.s. (4.9) Therefore, in order to show that the process 𝐿=(𝐴,𝑌) is a solution of (2.13), it suffices to verify that, for all 𝑡≥0, lim_𝑛→∞^𝑡₀𝑝_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝑑𝑠=^𝑡₀|𝑏|^−𝛼𝐴_𝑠,𝑌_𝑠𝑑𝑠𝐏-a.s.,lim_𝑛→∞^𝑡₀𝑞_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝑑𝑠=^𝑡₀𝑎|𝑏|^−𝛼𝐴_𝑠,𝑌_𝑠𝑑𝑠𝐏-a.s. (4.10) The following fact can be proven similar as [14, Lemma  4.2].
Lemma 4.2. For any Borel measurable function 𝑓∶ℝ²→[0,∞) and any 𝑡≥0, there exists a sequence 𝑚_𝑘∈(0,∞), 𝑘=1,2,… such that 𝑚_𝑘↑∞ as 𝑘→∞ and it holds 𝐄^{𝑡∧𝜏𝑚𝑘(}₎𝑌₀𝑓𝐴_𝑠,𝑌_𝑠𝑑𝑠≤𝑁𝑓_{2,𝑚𝑘,𝑡}, (4.11) where the constant 𝑁 depends on 𝛿, 𝐾, 𝛼, 𝑡, and 𝑚_𝑘 only.
Without loss of generality, we can assume in Lemma 4.2 that {𝑚_𝑘}={𝑚}. Now, to prove (31), it is enough to verify that for all 𝑡≥0 and 𝜀>0 we have lim_𝑛→∞𝐏^𝑡₀𝑝_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝑑𝑠−^𝑡₀|𝑏|^−𝛼𝐴_𝑠,𝑌_𝑠𝑑𝑠>𝜀=0. (4.12) In order to prove (4.12) we estimate for a fixed 𝑘∈ℕ𝐏^𝑡₀𝑝_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠𝑑𝑠−^𝑡₀|𝑏|^−𝛼𝐴_𝑠,𝑌_𝑠≤𝐏𝑑𝑠>𝜀_𝑡0𝑝_𝑘𝐴^𝑛_𝑠,𝑌^𝑛_s𝑑𝑠−^𝑡₀𝑝_𝑘𝐴_𝑠,𝑌_𝑠>𝜀𝑑𝑠3+𝐏^{𝑡∧𝜏𝑚(𝑛}𝑌₎₀𝑝_𝑘−𝑝_𝑛𝐴^𝑛_𝑠,𝑌^𝑛_𝑠>𝜀𝑑𝑠3+𝐏^{𝑡∧𝜏𝑚(}₎𝑌₀𝑝_𝑘−|𝑏|^−𝛼𝐴_𝑠,𝑌_𝑠>𝜀𝑑𝑠3+𝐏𝜏_𝑚𝑌^𝑛<𝑡+𝐏𝜏_𝑚𝑌=𝐽<𝑡_1𝑛,𝑘+𝐽²_{𝑛,𝑘,𝑚}+𝐽³_𝑘,𝑚+𝐏𝜏_𝑚𝑌^𝑛<𝑡+𝐏𝜏_𝑚𝑌<𝑡. (4.13) By Chebyshev’s inequality and Lebesgue bounded convergence theorem, 𝐽¹_𝑛,𝑘→0 as 𝑛→∞. To show that 𝐽²_{𝑛,𝑘,𝑚}→0 as 𝑛→∞ and 𝐽³_𝑘,𝑚→0 as 𝑘→∞, we use first the Chebyshev’s inequality and then Corollary 3.3 and Lemma 4.2, respectively, to estimate 𝐽²_{𝑛,𝑘,𝑚}≤3𝜀𝑁𝑝_𝑘−𝑝_{𝑛2,𝑚,𝑡},𝐽³_𝑘,𝑚≤3𝜀𝑁𝑝_𝑘−|𝑏|^−𝛼_2,𝑚,𝑡, (4.14) where the constant 𝑁 depends on 𝐾₁, 𝐾₂, 𝛼, 𝑚, and 𝑡 only. Obviously, ‖𝑝_𝑛−|𝑏|^−𝛼‖_2,𝑚,𝑡→0 as 𝑛→∞ implying that the right-hand sides in (4.14) converge to 0 by letting first 𝑛→∞ and then 𝑘→∞.
Because of the property 𝜏_𝑚(𝑌^𝑛)→𝜏_𝑚(𝑌) as 𝑛→∞𝐏-a.s., 𝐏𝜏_𝑚𝑌^𝑛<𝑡⟶𝐏𝜏_𝑚𝑌<𝑡as𝑛⟶∞ (4.15) for all 𝑚∈ℕ, 𝑡>0. Therefore, the last two terms can be made arbitrarily small by choosing large enough 𝑚 for all 𝑛 due to the fact that the sequence of processes 𝑌^𝑛 satisfies the property (4.6). This proves (4.12). The proof of (4.10) is similar, and we omit the details.
We have shown that 𝐿=(𝐴,𝑌) is a solution of (2.13). To finish the proof of the theorem, it is enough to use Corollary 2.4 that implies that the process 𝑋_𝑡=𝑌𝑇_𝑡 will be a (nonexploding) solution of (1.1).

Remark 4.3. If 𝛽=1, then the existence conditions of Theorem 4.1 coincide with those found in [14] where (1.1) with 𝑏=1 was considered.

Acknowledgment

The author would like to thank Henrikas Pragarauskas for the valuable discussions on the subject of Krylov’s estimates.