## On solutions of one-dimensional stochastic differential equations without drift.(English)Zbl 0535.60049

We consider the stochastic differential equation $$dX_ t=b(X_ t)dW_ t$$, $$t\geq 0$$, where b is a real-valued (universally) measurable function and W is a Wiener process. In the previous paper, Stochastic differential systems, Proc. 3rd IFIP-WG 7/1, Lect. Notes Contr. Inf. Sci. 36, 47-55 (1981; Zbl 0468.60077), the authors have shown that a nontrivial weak solution of this equation exists for all initial distributions if and only if $$b^{-2}$$ is locally integrable. However, the uniqueness in law fails in general.
In the present paper we give a complete description of all solutions by construction from a so-called fundamental solution. The fundamental solution has no sojourn time in the zeros of b and the general solution can be obtained from it by time delay in the zeros of b. Furthermore, some properties of solutions are investigated. Thus we characterize the set of all strong Markov solutions and a certain class of Markov solutions. We construct examples of Markov solutions which are not strong Markov. Finally, we study the representation property of solutions.
In the appendix, a few results on the time change of arbitrary strong Markov continuous local martingales and perfect additive functionals of them are collected. The basic method of the paper consists in a systematic use of random time change.

### MSC:

 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J55 Local time and additive functionals

Zbl 0468.60077
Full Text:

### References:

 [1] Azéma, J., Yor, M.: Temps locaux. Astérisque 52-53 (1978) [2] Dellacherie, C., Capacités et processus stochastiques (1972), Springer: Berlin-Heidelberg-New York, Springer · Zbl 0246.60032 [3] Dellacherie, C.; Meyer, P. A., Probabilités et potentiel (1975), Paris: Hermann, Paris [4] Dynkin, E. B., Markov processes (1965), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York [5] Engelbert, H. J.; Hess, J., Stochastic integrals of continuous local martingales I, Math. Nachr., 97, 325-343 (1980) · Zbl 0453.60054 [6] Engelbert, H. J.; Hess, J., Stochastic integrals of continuous local martingales II, Math. Nachr., 100, 249-269 (1981) · Zbl 0466.60047 [7] Engelbert, H. J.; Hess, J., Integral representation with respect to stopped continuous local martingales, Stochastics, 4, 121-142 (1980) · Zbl 0443.60042 [8] Engelbert, H. J.; Schmidt, W., On the behaviour of certain functionals of the Wiener process and applications to stochastic differential equations, Lecture Notes in Control and Information Sciences 36, 47-55 (1981), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0468.60077 [9] Engelbert, H.J., Schmidt, W.: Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations. In preparation · Zbl 0699.60044 [10] Gihman, I. I.; Skorohod, A. V., Stochastic differential equations (1972), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0242.60003 [11] Gihman, I. I.; Skorohod, A. V., The theory of stochastic processes Vol. III (1979), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0404.60061 [12] Girsanov, I. V., An example of nonuniqueness of the solution of K. Itô’s stochastic integral equation (in Russian), Teor. Verojatnost i Primenen., 7, 336-342 (1962) [13] Ikeda, N.; Watanabe, S., Stochastic differential equations and diffusion processes (1981), Amsterdam-Oxford-New York: North Holland, Amsterdam-Oxford-New York [14] Itô, K.; McKean, H. P., Diffusion processes and their sample paths (1965), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0127.09503 [15] Kazamaki, N., Changes of time, stochastic integrals, and weak martingales, Z. Wahrscheinlich-keitstheorie verw. Gebiete, 22, 25-32 (1972) · Zbl 0213.19304 [16] McKean, H. P., Stochastic integrals (1969), New York-London: Academic Press, New York-London [17] Meyer, P. A., La perfection en probabilité, Lecture Notes in Mathematics 258, 243-252 (1972), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0238.60048 [18] Orey, S., Conditions for the absolute continuity of two diffusions, Trans. Amer. Math. Soc., 193, 130-140 (1974) [19] Stroock, D. W.; Yor, M., On extremal solutions of martingale problems, Ann. Sci. Ecole Norm. Sup. 4∘ serie, 13, 95-164 (1980) · Zbl 0447.60034 [20] Tanaka, H., Note on continuous additive functionals of the 1-dimensional Brownian path, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 1, 251-257 (1963) · Zbl 0129.30701 [21] Walsh, J. B., The perfection of multiplicative functionals, Lecture Notes in Mathematics 258, 233-242 (1972), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0241.60061 [22] Wang, A. T., Generalized Itô’s formula and additive functionals of a Brownian motion, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 41, 153-159 (1977) · Zbl 0349.60081 [23] Watanabe, S., Solution of stochastic differential equations by random time change, Appl. Math. Optim., 2, 90-96 (1975) · Zbl 0326.60066 [24] Yershov, M. P., On stochastic equations, Lecture Notes in Mathematics 330, 527-530 (1973), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.