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Teoriya Veroyatnostei i ee Primeneniya, 1974, Volume 19, Issue 4, Pages 665–687
(Mi tvp3972)
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This article is cited in 2 scientific papers (total in 2 papers)
The esistence of a martingale with given diffusion functional
M. P. Ershov Moscow
Abstract:
Let $\mathbf R_+=[0,\infty)$ and $C$ be the space of continuous functions on $\mathbf R_+$ “starting” from zero with the topology of uniform convergence on compacts.
Let $A\colon \mathbf R_+\times C\mapsto \mathbf R_+$ be a Borel functional such that
(i) for each $\mathbf x\in C$, $A(\,\cdot\,,\mathbf x)\in C$ and is non-decreasing,
(ii) the set
$$
\{\{A(t,\mathbf x)\}_{t\in \mathbf R_+}\mid\mathbf x\in C\}
$$
is relatively compact in $C$,
(iii) for each $t\in \mathbf R_+$, $A(t,\,\cdot\,)$ is continuous, and
(iv) for each $t\in \mathbf R_+$, $x_s=y_s$ $(0\le s\le t)$ implies
$$
A(t,\mathbf x)=A(t,y)\quad(\mathbf x=\{x_s\}_{s\in \mathbf R_+},y=\{y_s\}_{s\in \mathbf R_+}).
$$
Then we prove that (on some probability space) there exists a continuous martingale $\mathbf X$ such that its Meyer squared variation process
$$
\langle\mathbf X\rangle=A(\,\cdot\,,\mathbf X)\quad\text{a.s.}
$$
In particular, in case
$$
A(t,\mathbf x)=\int_0^ta^2(t,\mathbf x)\,ds
$$
where $a^2$ is a bounded non-anticipative function, it follows that in the conditions of D. W. Stroock and S. R. S. Varadhan [12] continuity in $(s,\mathbf x)$ may he replaced by that in $\mathbf x$ only.
Received: 18.12.1973
Citation:
M. P. Ershov, “The esistence of a martingale with given diffusion functional”, Teor. Veroyatnost. i Primenen., 19:4 (1974), 665–687; Theory Probab. Appl., 19:4 (1975), 633–655
Linking options:
https://www.mathnet.ru/eng/tvp3972 https://www.mathnet.ru/eng/tvp/v19/i4/p665
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