Strong Markov Continuous Local Martingales and Solutions of One-Dimensional Stochastic Differential Equations (Part I)
✍ Scribed by H. J. Engelbert; W. Schmidt
- Publisher
- John Wiley and Sons
- Year
- 1989
- Tongue
- English
- Weight
- 944 KB
- Volume
- 143
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
The main purpose of the present paper is to investigate the connection between strong MARKOV continuous local martingales and solutions of one-dimensional stochastic differential equations driven by a WIENER process.
Thus this paper contributes to the general problem to clarify the structure of MABKOV processes and to describe them using well-understood, "elementary" objects.
Historically, the LEVY-KHINTCHTNE formula for processes with stationary and independent increments can be seen as a first analytical result. W. FELLER [16], [17], [ 181 characterized one-dimensional continuous strong MARKOV processes by the explicit structure of their infinitesimal generator. In the more-dimensional case there are also extensive investigations concerning the structure of the infinitesimal generator for certain classes of strong MARKOV processes: Cf., for example, A. V. SKOROKHOD [25], [26], F. B. KNIGHT [21], E. (hu, J. JACOD, P. PROTTEB, and M. J. SH~LRPE [2]. E. B. DYN~IN [5] and K. ITO and H. P. MCKE~LN [20] completed the analytical approach of W.
FEUER by probabilistic mpects: They gave a "constructive" description of the sample paths of onedimensional continuous strong MABKOV processes using the sample paths of "more elementary" stochastic processes. More precisely, starting from a Wmmat process they constructed continuous "regular" & ~K O V processes on the real line with given characteristics by random time change followed by spatial transformation and lulling.
Using modern methods of stochastic calculus S. MELEARD [23] gave an excellent probabilistic restatement for continuous "regular" strong lhhRKov processes.
The principal result of E. (knia and J. JACOD [l] is that every semimartingale HUNT process with values in the n-dimensional EucuDean space can be obtained by a random time change from a PL~KOV process that satisfies a stochastic differential equation driven by a WIENER process and a POISSON random measure.