I have the following problem.
- $(X_n)_{n\geq0}, n\in\mathrm{R}$, is a family of iid r.v., normally distributed $\mathcal{N}(0,1)$
- $\mathcal{F_n} := \sigma((X_i)_{1\leq i\leq n})$
- $x\in\mathrm{R}, \phi(x)$ is the repartition function of a the normal distribution ($\phi(x):=\mathrm{P}[X_1\leq x]$)
if $b$ is fixed and $\in \mathrm{R}$, show that $(M_k)_{0\leq k < n}$, where
- $M_k = \phi\left(\frac{b-S_k}{\sqrt{n-k}}\right)$, $\forall k \in \left\{ {1,..,n-1}\right\} $, $S_k = \sum_{i=1}^{k} X_i$, $S_0=0$
is a martingale.
Please help me
Factlet: For every $(u,v)$, if $X$ is standard normal, then $$E(\phi(uX+v))=\int_\mathbb R \phi(ux+v)\phi(x)\,\mathrm dx.$$ Simple algebraic manipulations show that $$\phi(ux+v)\phi(x)=\phi(uvw+w^{-1}x)\phi(vw),\qquad \color{red}{w=(u^2+1)^{-1/2}},$$ hence, using the change of variable $z=uvw+w^{-1}x$, one gets the key-identity $$E(\phi(uX+v))=\phi(vw)\int_\mathbb R \phi(uvw+w^{-1}x)\,\mathrm dx=\phi(vw)\int_\mathbb R \phi(z)\,(w\,\mathrm dz),$$ that is, $$\color{red}{E(\phi(uX+v))=w\cdot\phi(vw).}$$
Application: If $M_{k+1}=C_k\phi(u_kX_{k+1}+Y_k)$ for some constants $C_k$ and $u_k$ and some random variable $Y_k$ $\mathcal F_k$-measurable then $$E(M_{k+1}\mid\mathcal F_k)=C_kw_k\cdot\phi(w_kY_k).$$ where, naturally, $w_k=(u_k^2+1)^{-1/2}$. Identifying $u_k$, $C_k$ and $Y_k$ should show (once the text of the exercise is corrected) that the RHS is $M_k$.