Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $W_t$ the coordinate functionnal defined by $W_t(w)=w_t$, and define the stochastic intégral as usual.It is a classical result that the linear span of the set $e^{\int u_v dW_t -\int \frac{u^2}{2} dt}$, u in $L^2[0,1]$ is dense in $L^2(\mu)$. My question is :

Is that true that the linear span f the set $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ , u in $L^2[0,1]$ , is dense in $L^2(\mu_s)$ ???

I want to draw your attention on the fact that the expecation of $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ is 1 under $\mu_s$, but not those of $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s^2}dt}$.

Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $W_t$ the coordinate functionnal defined by $W_t(w)=w_t$, denote by $F_t$ the borel sigma field of $W_t$,and define the stochastic intégral as usual.It is a classical result that the linear span of the set $e^{\int u_v dW_t -\int \frac{u^2}{2} dt}$, u in $L^2[0,1]$ is dense in $L^2(\mu)$. My question is :

Is that true that the linear span f the set $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ , u in $L^2[0,1]$ , is dense in $L^2(\mu_s)$ ???

I want to draw your attention on the fact that the expecation of $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ is 1 under $\mu_s$, but not those of $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s^2}dt}$, moreover $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ is a weight that enables to use the Cameron -Martin theorem (or even the Girsanov theorem) on the space $(W,F,\mu_s)$.

Assume that W is the classical Wiener space C([0,1],R) note \mu the Wiener measure, and call \mu_s the image of \mu under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Let notes the coordinate functionnal :$W_t(w)=w_t$, and define the stochastic intégral as usual.It is a classical result that the linear span of the set $e^{\int u_v dW_t -\int \frac{u^2}{2} dt}$, u in $L^2[0,1]$ is dense in $L^2(\mu)$. My question :

Is that true that $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ , u in $L^2[0,1]$ , is dense in $L^2(\mu_s)$ ???

I want to drop your attention on the fact that the expecation of $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ is 1 under $\mu_s$. It is a martingale under $\mu_s$, but not under $\mu$- under $\mu$ only $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s^2}dt}$ is martingale.

Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $W_t$ the coordinate functionnal defined by $W_t(w)=w_t$, and define the stochastic intégral as usual.It is a classical result that the linear span of the set $e^{\int u_v dW_t -\int \frac{u^2}{2} dt}$, u in $L^2[0,1]$ is dense in $L^2(\mu)$. My question is :

Is that true that the linear span f the set $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ , u in $L^2[0,1]$ , is dense in $L^2(\mu_s)$ ???

I want to draw your attention on the fact that the expecation of $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s}dt}$ is 1 under $\mu_s$, but not those of $e^{\int \frac{u_v}{s} dW_t -\int \frac{u^2}{2s^2}dt}$.