Finding sequences such that function of sum of r.v's is martingale

Question

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!

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Answer 1

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 $\mathcal F_n$.

Answer 2

Compute

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

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

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

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