Existence and non-existence of solutions of one-dimensional stochastic equations

• Autorzy: Engelbert H. J.
• Języki publikacji: EN

Abstrakty

EN

We consider the one-dimensional stochastic equation [formula] for a continuous local martingale M with square variation [M] and measurable drift and diffusion coefficients b and σ. The main purpose of this paper is to derive a necessary condition for the existence of a solution X starting from x₀. As a result, we construct a diffusion coefficient σ such that the above stochastic equation has no solution X whatever the initial value x₀ and the non-zero, say, continuous drift coefficient b might be.

Słowa kluczowe

EN

stochastic equations; continuous local martingales; nonexistence; local time

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2000
• Tom: Vol. 20, Fasc. 2
• Strony: 343--358
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Engelbert H. J.

• Friedrich-Schiller-University of Jena, Institute for Stochastics, Ernst-Abbe-Platz 1-4, D-07743 Jena

Bibliografia

• [1] A. S. Cherny and H. J. Engelbert, Isolated singular points of stochastic differential equations, Preprint, February 2000.
• [2] C. Dellacherie, Capacites et processus stochastiques, Springer, 1972.
• [3] H. J. Engelbert and W. Schmidt, On solutions of one-dimensional stochastic differential equations without drift, Z, Wahrsch. verw. Gebiete 68 (1985), pp. 287-314.
• [4] H. J. Engelbert and W. Schmidt, On one-dimensional stochastic differential equations with generalized drift, Lecture Notes in Control and Inform. Sei. 69 (1985), pp. 143-155.
• [5] H. J. Engelbert and W. Schmidt, Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations, Part III, Math. Nachr. 151 (1991), pp. 149-197.
• [6] J. Jacod and J. Mémin, Existence of weak solutions for stochastic differential equations with driving semimartingales, Stochastics 4 (1981), pp. 317-337.
• [7] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, Berlin 1991.
• [8] M. Rutkowski, Strong solutions of stochastic differential equations involving local times, Stochastics 22 (1987), pp. 201-218.
• [9] M. Rutkowski, Fundamental solutions of stochastic differential equations with drift, Stochastics and Stochastics Reports 26 (1989), pp. 193-204.
• [10] M. Rutkowski, On solutions of stochastic differential equations with-drift, Probab. Theory Related Fields 85 (1990), pp. 387-402.
• [11] W. Schmidt, On stochastic differential equations with reflecting barriers, Math. Nachr. 142 (1989), pp. 135-148.
• [12] A. V. Skorohod, Studies in the Theory of Random Processes, Addison-Wesley, Reading, Massachusetts, 1965.