On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

We consider a one-dimensional stochastic equation dXt b t, Xt− dZt a t, Xt 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 dLt B Lt− 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 themethod 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.


Introduction
Let Z be a one-dimensional symmetric stable process of index 0 < α ≤ 2 with Z 0 0. In this paper we will study the existence of solutions of the equation dX t b t, X t− dZ t a t, X t dt, t ≥ 0, X 0 x 0 ∈ 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.

International Journal of Stochastic Analysis
Two important particular cases of 1.1 are the equations dX t b t, X t− dZ t , t ≥ 0, X 0 x 0 ∈ R, 1.2 dX t dZ t a t, X t dt, t ≥ 0, X 0 x 0 ∈ R.

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 Slomi ński 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 L 2 -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.

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 ndimensional 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 Z 0 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 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 where 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 International Journal of Stochastic Analysis 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 It is known that for α < 2 for any g ∈ C 2 , where C 2 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.
be the Fourier transform of g.Clearly, the function g ξ 1 , ξ 2 can be seen as the result of taking the Fourier transform from the function g z 1 , z 2 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.
ii Let g be absolutely continuous on every compact subset of R and g ∈ L 1 R .Then 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 T 0 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 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 x 0 ∈ R if there exists a symmetric stable process Z with respect to the filtration F such that Z 0 0 and 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 x 0 ∈ R an arbitrary value.We introduce the matrix B defined as and set W t : Z t , t , L t A t , Y t , x 0 : 0, x 0 .Consider the 2-dimensional equation which, if written componentwise, is equivalent to the following two one-dimensional equations: 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 A 0 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 δ 1 > 0 and δ 2 > 0 such that δ 1 ≤ |b| ≤ δ 2 .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

2.16
It can be easily verified that the process A satisfies the relation 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 s, X s ds.

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 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 x 0 ∈ R, and there exists a constant δ 1 > 0 such that |b| > δ 1 .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 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 T t t ∧ A ∞ and T A t 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 x 0 Y T t for all t ∈ 0, A ∞ and X t Δ for all t ≥ A ∞ .Also let H t F A t 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 ≥ 0 |b| α s, X s ds P-a.s.

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Applying the same time change to processes in 2.15 yields

Some Estimates
Let δ 1 , δ 2 , 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 I 1 and I 2 we denote the classes of all F-predictable one-dimensional processes r t and γ t , respectively, such that δ 1 ≤ r t ≤ δ 2 and |γ t | ≤ K.For any t, x ∈ R 2 , λ > 0, and any nonnegative, measurable function Then, for all t, x ∈ R 2 , it holds where the constant N depends on δ 1 , K, and α only.
Proof.For any function h : R 2 → R such that h ∈ L 1 R 2 and any ε > 0 we define to be the ε-convolution of h with a smooth function q such that q ∈ C ∞ 0 R 2 and R 2 q t, x dt dx 1.

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It follows that

3.10
Obviously, f ε is square integrable, and 3.5 implies that f ε → f as ε ↓ 0 a.s. in R 2 .Now, applying Proposition 2.1, the Parseval identity and integration by parts to the inequality

3.12
Let δ : It follows from the assumptions that δ > 0. The inequality 3.12 can be rewritten then as

3.13
One sees easily that there exists a constant λ 0 > 0 such that Combining the inequalities 3.12 and 3.14 , we obtain for all λ ≥ λ 0

3.15
Let Clearly, the constant N 1 is finite and depends on δ 1 , δ 2 , K, and α only.
Using the estimate 3.15 and the inverse Fourier transform yields for all t, x ∈ R 2 and λ ≥ λ 0

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:

3.18
We are interested in L 2 -estimates of the form 13 , and the assumptions 3.18 hold.Then, for any t 0 , x 0 ∈ R 2 , λ ≥ λ 0 , and any measurable function f : R 2 → 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 R 2 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

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Then, for s ≥ 0, applying the It ô's formula to the expression

3.23
It remains to pass to the limit in the above inequality letting ε → 0, s → ∞ and using the Fatou's lemma.

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.
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 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 D 4 , D 4 such that H n k 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.

4.10
The following fact can be proven similar as 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 Y T 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.
is the Banach space of functions f : R → R with the norm f ∞ ess sup |f x |.
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 ≥ 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 18Applying 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.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 .
the class of all infinitely many times differentiable real-valued functions with compact support defined on R 2 defines the value function v t, x as which holds a.e. in R 2 .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 The inequality 3.19 can be extended in a standard way first to any function f ∈ L 2 R and then to any nonnegative, measurable function using the monotone class theorem arguments see, e.g., 19, Theorem 20 .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 |Y t | ≥ m} and N is a constant depending on δ, K, α, m, and t only.
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.
n 1, 2, . .., such that International Journal of Stochastic Analysis 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 D 4 , D 4 .Due to the Aldous' criterion 21 , we have only to show that lim 14, Lemma 4.2 .For any Borel measurable function f : R 2 → 0, ∞ and any t ≥ 0, there exists a sequence m k ∈ 0, ∞ , k 1, 2, . . .such that m k ↑ ∞ as k → ∞ and it holds 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 International Journal of Stochastic Analysis By Chebyshev's inequality and Lebesgue bounded convergence theorem, J 1 n,k → 0 as n → ∞.To show that J 2 n,k,m → 0 as n → ∞ and J 3 k,m → 0 as k → ∞, we use first the Chebyshev's inequality and then Corollary 3.3 and Lemma 4.2, respectively, to estimate N depends on K 1 , K 2 , 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., P τ m Y n < t −→ P τ m Y < t as n −→ ∞ 4.15 s , Y s ds ≤ N f 2,m k ,t , 4.11where the constant N depends on δ, K, α, t, and m k only.