Abstract
Let Zj t , j = 1, . . . , d, be independent one-dimensional symmetric stable processes of index α ∈ (0,2). We consider the system of stochastic differential equations
where the matrix A(x)=(A ij (x))1≤ i, j ≤ d is continuous and bounded in x and nondegenerate for each x. We prove existence and uniqueness of a weak solution to this system. The approach of this paper uses the martingale problem method. For this, we establish some estimates for pseudodifferential operators with singular state-dependent symbols. Let λ2 > λ1 > 0. We show that for any two vectors a, b∈ ℝ d with |a|, |b| ∈ (λ1, λ2) and p sufficiently large, the Lp-norm of the operator whose Fourier multiplier is (|u · a|α - |u · b|α) / ∑ j=1 d |u i |α is bounded by a constant multiple of |a−b|θ for some θ > 0, where u=(u1 , . . . , u d ) ∈ ℝd. We deduce from this the Lp-boundedness of pseudodifferential operators with symbols of the form ψ(x,u)=|u · a(x)|α / ∑ j=1 d |u i |α, where u=(u1,...,u d ) and a is a continuous function on ℝd with |a(x)|∈ (λ1, λ2) for all x∈ ℝd.
References
Bass, R.F.: Local times for a class of purely discontinuous martingales. Z. Wahrsch. Verw. Gebiete 67, 433–459 (1984)
Bass, R.F.: Uniqueness in law for pure jump processes. Probab. Theory Relat. Fields 79, 271–287 (1988)
Bass, R.F.: Diffusions and Elliptic Operators. Springer-Verlag, New York, 1997
Bass, R.F.: Stochastic differential equations driven by symmetric stable processes. Séminaire de Probabilités XXXVI, 302–313. Springer, New York, 2003
Bass, R.F., Burdzy, K., Chen, Z.-Q.: Stochastic differential equations driven by stable processes for which pathwise uniqueness fails. Stoch. Process Appl. 111, 1–15 (2004)
Bertoin, J.: Lévy Processes. Cambridge Univ. Press, Cambridge, 1996
Billingsley, P.: Weak Convergence of Measures: Applications in probability. CBMS Regional Conference Series in Applied Mathematics, No. 5. SIAM, Philadelphia, 1971
Calderón, A.P., Zygmund, A.: On singular integrals. Am. J. Math. 78, 289–309 (1956)
Calderón, A.P., Zygmund, A.: Singular integral operators and differential equations. American J. Math. 79, 901–921 (1957)
Elliott, R.J.: Stochastic Calculus and Applications. Springer-Verlag, New York, 1982
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence. John Wiley & Sons, 1986
Hoh, W.: The martingale problem for a class of pseudo-differential operators. Math. Ann. 300, 121–147 (1994)
Hoh, W.: Pseudo differential operators with negative definite symbols and the martingale problem. Stochastics Stochastics Rep. 55, 225–252 (1995)
Hoh, W., Jacob, N.: Pseudo-differential operators, Feller semigroups and the martingale problem. In: Stochastic processes and optimal control (Friedrichroda, 1992), Gordon and Breach, Montreux, 1993, pp. 95–103
Jacob, N.: Pseudo Differential Operators and Markov Processes. Vol. I. Fourier analysis and semigroups. Imperial College Press, London, 2001
Jacob, N.: Pseudo Differential Operators and Markov Processes. Vol. II. Generators and their potential theory. Imperial College Press, London, 2002
Jacob, N., Schilling, R.: Lévy-type processes and pseudodifferential operators. In: Lévy processes, Birkhäuser Boston, Boston, MA, 2001, pp. 139–168
Kolokoltsov, V.: On Markov processes with decomposable pseudo-differential generators. Stoch. Rep. 76, 1-44(2004)
Komatsu, T.: Markov processes associated with certain integro-differential operators. Osaka J. Math. 10, 271–303 (1973)
Komatsu, T.: On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proc. Japan Acad. Ser. A Math. Sci. 58, 353–356 (1982)
Komatsu, T.: On the martingale problem for generators of stable processes with perturbations. Osaka J. Math. 21, 113–132 (1984)
Lepeltier, J.-P., Marchal, B.: Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel. Ann. Inst. H. Poincaré 12, 43–103 (1976)
Meyer, P.-A.: Un cours sur les intégrales stochastiques. In: Séminaire de Probabilités X, Springer, Berlin, 1976, pp. 245–400
Skorokhod, A.V.: Studies in the Theory of Random Processes. Reading, MA, Addison-Wesley, 1965
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, New Jersey, 1970
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, New Jersey, 1971
Stroock, D.: Diffusion processes associated with Lévy generators. Z. Wahrsch. Verw. Gebiete 32, 209–244 (1975)
Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer-Verlag, New York, 1979
Tsuchiya, M.: On some perturbations of stable processes. In: Proceedings of the Second Japan-USSR Symposium on Probability Theory (Kyoto, 1972), Springer, Berlin, 1973, pp. 490–497
Tanaka, H., Tsuchiya, M., Watanabe, S.: Perturbation of drift-type for Lévy processes. J. Math. Kyoto Univ. 14, 73–92 (1974)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by NSF grant DMS-0244737.
Research partially supported by NSF grant DMS-0303310.
Rights and permissions
About this article
Cite this article
Bass, R., Chen, ZQ. Systems of equations driven by stable processes. Probab. Theory Relat. Fields 134, 175–214 (2006). https://doi.org/10.1007/s00440-004-0426-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-004-0426-z
Mathematics Subject Classifications (2000)
- Primary: 60H10
- Secondary: 60G52
- 60J75
- 42B20
- 35S05
Key words or phrases
- Stable processes
- Stochastic differential equations
- Martingale problem
- Weak solution
- Weak uniqueness
- Pseudodifferential operators
- Method of rotations