1
$\begingroup$

Let $\left(X_n\right)_{n\geq 1}$ be independent such that $\mathbb E\left(X_i\right)=m_i$ and $\mathrm{Var}(X_i)=\sigma_{i}^{2}$ for $i\geq 1$. Let $\displaystyle S_{n}=\sum_{i=1}^{n}X_i$ and $\mathcal F=\sigma\left(X_i,1\leq i \leq n\right)$. Find sequences $\left(b_n\right)_{n\geq 1}$, $(c_{n})_{n\geq 1}$ of real numbers such that $$\left(S_n^2+b_nS_n+c_n\right)_{n\geq 1}$$ is a $(\mathcal F_{n})_{n\geq 1}$martingale.

I'd really appreciate someone letting me know where to start with this. Thanks!


$\endgroup$
1
$\begingroup$

A good start would consist in computing the conditional expectation of $S_{n+1}^2+b_{n+1}S_{n+1}+c_{n+1}$ with respect to $\mathcal F_n$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ Am I on the right track with computing the conditional expectation? $\endgroup$ – Chris Apr 17 '13 at 7:54
  • $\begingroup$ As the definition of martingales involves conditional expectation, yes. You can try the computations and post them as an answer. $\endgroup$ – Davide Giraudo Apr 17 '13 at 9:18
  • $\begingroup$ Ah I had added the computation to my question, now posted it as an answer. $\endgroup$ – Chris Apr 17 '13 at 16:27
0
$\begingroup$

Compute

$$\mathbb E\left[S_{n+1}^2+b_{n+1}S_{n+1}+c_{n+1}|\mathcal{F}_{n}\right]=$$

$$\mathbb E\left[\left(\sum_{i=1}^{n+1}X_i\right)^2+b_{n+1}\left(\sum_{i=1}^{n+1}X_i\right)+c_{n+1}\left\vert\vphantom{\frac{1}{1}}\right.\mathcal{F}_{n}\right]=$$

$$(n+1)\mathrm{Var}\left[X_i\right]+((n+1)\mathbb E[X_i])^2+b_{n+1}(n+1)\mathbb E[X_i]+c_{n+1}=$$

$$(n+1)\sigma_{i}^2+((n+1)m_i)^2+b_{n+1}(n+1)m_i+c_{n+1}$$

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.