The esistence of a~martingale with given diffusion functional

M. P. Ershov

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 4, pp. 665-687

Résumé

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.