A(t,B t ) is not a Semimartingale
A(t,B t ) is not a Semimartingale
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DOI：
10.1007/978-1-4684-0562-0_15
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发表时间：
1991
期刊：
影响因子：
--
通讯作者：
L. Rogers;J. Walsh
中科院分区：
文献类型：
--
作者：
L. Rogers;J. Walsh
Let (Bt)t≥0be Brownian motion on ℝ,B0= 0, and for each realxdefine $$A\left( {t,x} \right) \equiv \int_0^t {{I_{\left( { - \infty ,x} \right]}}} \left( {{B_s}} \right)ds = \int_{ - \infty }^x {L\left( {t,y} \right)} dy$$, where{L(t,y):t≤ 0,y∈ ℝ} is the local time process ofB. The processA(t,x) enters naturally into the study of the Brownian excursion filtration (see Rogers & Walsh [1],[2], and Walsh [4]). In [2], it was necessary to consider the occupation density of the processYt≡A(t,Bt), which would have been easy ifYwere a semimartingale; it is not, and the aim of this paper is to prove this.
Let (Bt)t≥0be Brownian motion on ℝ,B0= 0, and for each realxdefine $$A\left( {t,x} \right) \equiv \int_0^t {{I_{\left( { - \infty ,x} \right]}}} \left( {{B_s}} \right)ds = \int_{ - \infty }^x {L\left( {t,y} \right)} dy$$, where{L(t,y):t≤ 0,y∈ ℝ} is the local time process ofB. The processA(t,x) enters naturally into the study of the Brownian excursion filtration (see Rogers & Walsh [1],[2], and Walsh [4]). In [2], it was necessary to consider the occupation density of the processYt≡A(t,Bt), which would have been easy ifYwere a semimartingale; it is not, and the aim of this paper is to prove this.