We consider a one-dimensional stochastic equation , , with respect to a symmetric stable process of index . It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation with respect to the semimartingale and corresponding matrix . In the case of 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 be a one-dimensional symmetric stable process of index with . In this paper we will study the existence of solutions of the equation where are measurable functions. The existence of solutions is understood in weak sense. In the case of , the coefficients and are assumed to be only measurable satisfying additionally some conditions of boundness.

Two important particular cases of (1.1) are the equations

If , then 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 and and nondegeneracty of . The approach he used was based on -estimates for stochastic integrals of processes 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 Slomiล„ski [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 and .

In the one-dimensional case with , 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 . The time-independent equation (1.1) with was considered in [8] using the method of -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 with arbitrary index was studied by Zanzotto in [9] where he, in particular, generalized the results of Engelbert and Schmidt to the case of . The time-dependent equation (1.2) with the index was treated in [10] using the method of Krylovโ€™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 different from those in [10].

On another hand, the time-independent case of (1.3), that is when , was studied by Tanaka et al. in [12]. One obtained there the sufficient existence and uniqueness conditions assuming the drift coefficient to be bounded plus satisfying some additional conditions depending on the case whether , , or . The method used by them was a purely analytical one relying on some properties of homogeneous Markov processes satisfying (1.3). More recently, Portenko [13] obtained a new existence result for the time-independent equation (1.3) for the case of assuming the function to be integrable on of the power . The general case of (1.3) with was studied in [14] assuming 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 with a corresponding matrix . 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 Krylovโ€™s estimates for solutions of the 2-dimensional equation. The results of Section 3 apply to the case with . 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 the Skorokhod space, that is, the set of all real-valued functions with right-continuous trajectories and with finite left limits (also called cรกdlag functions). For simplicity, we shall write instead of . We will equip with the -algebra generated by the Skorokhod topology. Under we will understand the -dimensional Skorokhod space defined as with the corresponding -algebra being the direct product of one-dimensional -algebras .

Let be a complete probability space carrying a process with , and let be a filtration on . The notation means that is adapted to the filtration . We call a symmetric stable process of index if trajectories of belong to and for all and . If , is a process of Brownian motion with the variance . For we have a Cauchy process with unbounded second moment. In general, for . The explicit form of the probability density function is known only for three values of .

For all , is a Markov process and can be characterized in terms of analytic characteristics of Markov processes. First, for any function and , we can define the operator where is the Banach space of functions with the norm . The family is called the family of convolution operators associated with . Formally, for a suitable class of functions , let called the infinitesimal generator of the process .

On another hand, in the case of , is a purely discontinuous Markov process that can be described by its Poisson jump measure (jump measure of on interval ) defined as the number of times before the time that has jumps whose size lies in the set . The compensating measure of , say , is given (see, e.g., [15, Propostionโ€‰โ€‰13.9],) by It is known that for for any , where is the set of all bounded and twice continuously differentiable functions and is a suitable constant. In contrary to the case of , the infinitesimal generator of a Brownian motion process 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 . Let and be the Fourier transform of . Clearly, the function can be seen as the result of taking the Fourier transform from the function first in one variable and then in another (in any order). The following facts will be used later (cf. [14, Propositionโ€‰โ€‰2.1]).

Proposition 2.1. Let be the infinitesimal generator of a symmetric stable process . We have he following.(i)Assume that and . Then(ii)Let be absolutely continuous on every compact subset of and . Then

Finally, let us discuss how one can construct a solution of (1.1) for any using the time change method. By the definition, a process is called a -time change if it is an increasing right-continuous process with such that is a -stopping time for any (cf. [15, chapter 6]). Define called the right-continuous inverse process to . It follows that is an increasing process starting at zero. Moreover, is a -adapted process if and only if is a -time change.

We shall here also recall the concept of exploding solutions for (1.1). Let be the one-point compactification of equipped with the -algebra of its Borel subsets. For any function we set called the explosion time of the trajectory and define (or simply ) to be the Skorohod space of exploding functions such that is right-continuous with finite left-hand limits on the interval and whenever .

We say that a stochastic process , defined on a probability space with filtration and with trajectories in , is a weak solution of (1.1) with initial state if there exists a symmetric stable process with respect to the filtration such that and for all , where is called the explosion time of . Since is a semimartingale for all , the stochastic integral in (2.11) can be defined for all appropriate integrands via semimartingale integration theory.

If , then is called a nonexploding solution, otherwiseโ€”exploding solution with the explosion time .

Let be a symmetric stable process of index defined on a probability space and an arbitrary value. We introduce the matrix defined as and set , , .

Consider the 2-dimensional equation which, if written componentwise, is equivalent to the following two one-dimensional equations: Notice that the process is a semimartingale; hence (2.13) can be seen as a stochastic differential equation with respect to a semimartingale.

Moreover, so that the matrix is nondegenerate since and by the definition of the coefficient . We also see that is a strictly increasing nonnegative process such that . Let . The properties of imply that the right inverse to process is a continuous process defined on the interval .

Proposition 2.2. Assume that there exist constants and such that . Then, (1.1) has a solution if and only if (2.13) has a solution.

Proof.. We notice that the assumptions on the coefficient imply that the solutions of both equations are nonexploding.
Suppose first that 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 -a.s. finite for all . Let It can be easily verified that the process satisfies the relation By its definition, the process is -adapted so that its right-inverse process is a -time change process defined for all . We notice that is a global time change (that is, for all ) because . Now define Applying the time change to the semimartingale 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 It remains to notice that the process is nothing but a symmetric stable process of the index (see [18], Theoremโ€‰โ€‰3.1). Hence is a solution of (2.13).
Now, let be a solution of (2.13) defined on a probability space with a filtration , where is a symmetric stable process adapted to . Let for all where is the right inverse to . It follows so that is a global time change. By applying the time change to the semimartingale in (2.15) we obtain
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 ) such that 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 required in Proposition 2.2 can be slightly relaxed.

Corollary 2.3. Let be a solution of (2.13), where , and there exists a constant such that . Then there exists a (possibly, exploding) solution of (1.1).

Proof. By assumptions, there exist a solution of (2.14) and a process satisfying (2.15), both adapted to the same filtration . For any , let be the right inverse of the process . By and we denote the limits of processes and as , respectively. Clearly, and are strictly increasing and continuous processes defined on intervals and , respectively. In particular, we have that and for all . We notice further that is a -time change, finite on and equal to infinity for . Define for all and for all . Also let for all . Our goal is to show that the process is a solution of (1.1).
By making a time change in the relation (2.14), we obtain for all Applying the same time change to processes in (2.15) yields Let us look at the process . By properties of stable integrals (see, e.g., [11, Propositionโ€‰โ€‰4.3] or [18, Theoremโ€‰โ€‰3.1]), there exists a symmetric stable process of the same index stopped at such that From the last relation and time change properties in stochastic integrals with respect to a semimartingale (see, e.g., [16, Theoremโ€‰โ€‰10.19]), it follows that Now, the relation (2.25) yields that, for all , the integral is finite hence there exists the stochastic integral (see, e.g., [11, Propositionโ€‰โ€‰4.3]). Using (2.28), we obtain then Enlarging the probability space, we can assume that is extended to a full symmetric stable process of index . The last relation combined with (2.26) verifies that is a solution of (2.11), possibly, exploding in .

Corollary 2.4. Let be a solution of (2.13) with , and assume that there exist constants and such that for all . Then there exists a nonexploding solution of (1.1).

3. Some Estimates

Let , , and be strictly positive constants and a symmetric stable process of index defined on a probability space with filtration . By and we denote the classes of all -predictable one-dimensional processes and , respectively, such that and .

For any , , and any nonnegative, measurable function ( denotes the class of all infinitely many times differentiable real-valued functions with compact support defined on ) defines the value function as where the processes and are given by Then, for the value function and the process , the Bellman principle of optimality can be formulated as follows (cf. [1]): for any -valued -stopping time it holds Using standard arguments, one can derive from the principle above the corresponding Bellman equation ( and are deterministic) which holds a.e. in . Here and denote the partial derivatives of the function in and , respectively.

Define . Then, the Bellman equation is equivalent to two equations

Lemma 3.1. Let and . Then, for all , it holds where the constant depends on , and only.

Proof. For any function such that and any we define to be the -convolution of with a smooth function such that and .
For , let It follows that so that Obviously, is square integrable, and (3.5) implies that as a.s. in .
Now, applying Proposition 2.1, the Parseval identity and integration by parts to the inequality yields Let . It follows from the assumptions that . The inequality (3.12) can be rewritten then as One sees easily that there exists a constant such that for all .
Combining the inequalities (3.12) and (3.14), we obtain for all Let Clearly, the constant is finite and depends on , , , and only.
Using the estimate (3.15) and the inverse Fourier transform yields for all and The result follows then by taking the limit in the above inequality and using the Lebesgue-dominated convergence theorem.

Now, let be a solution of (2.13), and there exist constants and such that the following assumptions are satisfied: We are interested in -estimates of the form where .

Theorem 3.2. Assume that , is a solution of (2.13), and the assumptions (3.18) hold. Then, for any , , and any measurable function , the estimate (3.19) is satisfied where the constant depends on , and only.

Proof. Assume first that so that there is a solution of (3.5) satisfying the inequality (3.6). By taking the -convolution on both sides of (3.5), we obtain for all and Then, for , applying the Itรดโ€™s formula to the expression yields By Lemma 3.1 It remains to pass to the limit in the above inequality letting , and using the Fatouโ€™s lemma.
The inequality (3.19) can be extended in a standard way first to any function and then to any nonnegative, measurable function using the monotone class theorem arguments (see, e.g., [19, Theoremโ€‰โ€‰20]).

Now, for arbitrary but fixed , , define to be the -norm of on . Applying (3.19) to the function , we obtain the following local version of Krylovโ€™s estimates.

Corollary 3.3. Let and be a solution of (2.13). Suppose that the conditions (3.18) are satisfied. Then, for any , , and any nonnegative measurable function , it follows that where and is a constant depending on , , , , and only.

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

Corollary 3.4. Let be a solution of (2.13) with . If the assumptions (3.18) are satisfied with arbitrary and , 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 is well studied, we restrict ourselves to the case .

Theorem 4.1. Let and assume that the assumptions (3.18) are satisfied where the constant is arbitrary for and for . Then, for any , 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 is bounded. Hence we can find a sequence of functions and , such that they are globally Lipshitz continuous and uniformly bounded by the constant . Moreover, and as pointwise and in norm for all , . For any , (2.13) has a unique strong solution (see, e.g., [20, Theoremโ€‰โ€‰9.1]). That is, for any fixed symmetric stable process defined on a probability space , there exists a sequence of processes , , such that where or, written componentwise, Let so that
Now we claim that the sequence of processes , , is tight in the sense of weak convergence in . Due to the Aldousโ€™ criterion ([21]), we have only to show that for all and for all , , every sequence of -stopping times , and every sequence of real numbers such that . We use to denote the Euclidean norm of a vector.
But both conditions are clearly satisfied because of the uniform boundness of the coefficients and for all .
Since the sequence is tight, there exists a subsequence , , a probability space and the process on it with values in such that converges weakly (in distribution) to the process as . For simplicity, let .
According to the embedding principle of Skorokhod (see, e.g., [20, Theoremโ€‰โ€‰2.7]), there exists a probability space and the processes , , , on it such that(i) as -a.s.(ii) in distribution for all Using standard measurability arguments ([1, chapter 2]), one can prove that the processes and are symmetric stable processes of the same index as the processes with respect to the augmented filtrations and generated by processes and , respectively.
Using the properties (i), (ii), and (4.1), one can show (cf. [1, chapter 2]) that On the other hand, the same properties and the quasileft continuity of the the processes yield Therefore, in order to show that the process is a solution of (2.13), it suffices to verify that, for all , The following fact can be proven similar as [14, Lemmaโ€‰โ€‰4.2].
Lemma 4.2. For any Borel measurable function and any , there exists a sequence , such that as and it holds where the constant depends on , , , , and only.
Without loss of generality, we can assume in Lemma 4.2 that . Now, to prove (31), it is enough to verify that for all and we have In order to prove (4.12) we estimate for a fixed By Chebyshevโ€™s inequality and Lebesgue bounded convergence theorem, as . To show that as and as , we use first the Chebyshevโ€™s inequality and then Corollary 3.3 and Lemma 4.2, respectively, to estimate where the constant depends on , , , , and only. Obviously, as implying that the right-hand sides in (4.14) converge to 0 by letting first and then .
Because of the property as -a.s., for all , . Therefore, the last two terms can be made arbitrarily small by choosing large enough for all due to the fact that the sequence of processes 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 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 will be a (nonexploding) solution of (1.1).

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


The author would like to thank Henrikas Pragarauskas for the valuable discussions on the subject of Krylovโ€™s estimates.