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IX: 30, 496-514, LNM 465 (1975)

**SHARPE, Michael J.**

Homogeneous extensions of random measures (Markov processes)

Homogeneous random measures are the appropriate definition of additive functionals which may explode. The problem discussed here is the extension of such a measure given up to a terminal time into a measure defined up to the lifetime

Comment: The subject is taken over in a systematic way in Sharpe,*General Theory of Markov processes,* Academic Press 1988

Keywords: Homogeneous random measures, Terminal times, Subprocesses

Nature: Original

Retrieve article from Numdam

XIV: 08, 76-101, LNM 784 (1980)

**SHARPE, Michael J.**

Local times and singularities of continuous local martingales (Martingale theory)

This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh 1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$

Comment: See Calais-Genin 1717

Keywords: Local times, Local martingales, Semimartingales in an open interval

Nature: Original

Retrieve article from Numdam

Homogeneous extensions of random measures (Markov processes)

Homogeneous random measures are the appropriate definition of additive functionals which may explode. The problem discussed here is the extension of such a measure given up to a terminal time into a measure defined up to the lifetime

Comment: The subject is taken over in a systematic way in Sharpe,

Keywords: Homogeneous random measures, Terminal times, Subprocesses

Nature: Original

Retrieve article from Numdam

XIV: 08, 76-101, LNM 784 (1980)

Local times and singularities of continuous local martingales (Martingale theory)

This paper studies continuous local martingales $(M_t)$ in the open interval $]0,\infty[$. After recalling a few useful results on local martingales, the author proves that the sample paths a.s., either have a limit (possibly $\pm\infty$) at $t=0$, or oscillate over the whole interval $]-\infty,\infty[$ (this is due to Walsh 1133, but the proof here does not use conformal martingales). Then the quadratic variation and local time of $M$ are defined as random measures which may explode near $0$, and it is shown that non-explosion of the quadratic variation (of the local time) measure characterizes the sample paths which have a finite limit (a limit) at $0$. The results are extended in part to local martingale increment processes, which are shown to be stochastic integrals with respect to true local martingales, of previsible processes which are not integrable near $0$

Comment: See Calais-Genin 1717

Keywords: Local times, Local martingales, Semimartingales in an open interval

Nature: Original

Retrieve article from Numdam