I am looking for a reference with a proof for the following fact:


If a right-continuous martingale $(X\_r)\_{ r \geq 0}$ is such that $X\_0=0,(X^2\_r-r)\_r,(X\_r^3-3rX\_r)\_r,(X\_r^4-6rX\_r^2+3r^2)\_r$ are martingales then $(X\_r)\_{r}$ is a Brownian motion.



Characterization of Brownian motion: processes with right-continuous paths
It suffices to prove that the paths are continuous, and then the result follows from
<https://almostsuremath.com/2010/04/13/levys-characterization-of-brownian-motion/>


A version of this question was also asked on math stack exchange, for convenience I include the proof also here.


**Proposition**
Let $(X\_u)\_{u}$ be right-continuous martingale with $X\_0=0$, such that $(X^2\_u-u)\_u,(X\_u^3-3uX\_u)\_u,(X\_u^4-6uX\_u^2+3u^2)\_u$ are martingales. Then for every integer $M \ge 1$, the path $(X\_u)\_{u}$ is a.s. continuous in $[0,M]$.


**Proof:** Let $\{\mathcal F\_t\}$ be the canonical filtration of $\{X\_t\}$. Fix $u \ge 0$ and write
$E^u[\,\cdot\,]:=E[\,\cdot \,| \mathcal F\_u]$. Denote $\mathcal G\_t:=\mathcal F\_{u+t}$ for $t \ge 0$. Observe that the process $\{Y\_t\}\_{t \ge 0}$ defined by $$Y\_t:=X\_{u+t}-X\_u \tag{1}$$ is a
$\{\mathcal G\_t\}$-martingale.


**Claim:** For any bounded $\{\mathcal G\_t\}$-stopping time $\tau$, we have
$$E^u[Y\_\tau^4]=3E^u[\tau^2]\,.$$


The claim is proved below. Now we will use it to complete the proof of the proposition.


Fix $\epsilon>0$ and $\delta>0$. Let $P^u[\,\cdot\,]:=P[\,\cdot \, | \mathcal F\_u]$. Applying the claim to
$$\tau:=\delta \wedge \min\{t\ge 0: |Y\_t| \ge \epsilon\}$$
gives
$$P^u[|Y\_\tau| \ge \epsilon]\cdot\epsilon^4 \le E^u[Y\_\tau^4] \le 3\delta^2\,,$$
so
$$P[|Y\_\tau| \ge \epsilon] \le 3\delta^2 \epsilon^{-4}\,. \tag{\*}$$
For $k \ge 1$, we will use $(\*)$ for $\delta\_k={32}^{-k}$ and $\epsilon\_k=2^{-k}$ to bound the probability of
$$A\_k:=\bigcup\_{j=0}^{{32}^k M-1} \Big\{\max\_{0 \le t \le \delta\_k} |X\_{j\delta\_k+t}-X\_{j\delta\_k}| \ge \epsilon\_k\Big\} \,.$$
We obtain
$$P(A\_k) \le 32^k M \cdot 3\delta\_k^2 \epsilon\_k^{-4}=3M\epsilon\_k\,.$$
By the Borel-Cantelli lemma, almost surely only finitely many of the events $A\_k$ occur. This implies that $\{X\_t\}$ is a.s. continuous in $[0,M]$.
$\hspace{6.6in} \Box$


**Proof of Claim**


$$X\_u-u^2= E^u[(X\_u+Y\_\tau)^2-(u+\tau)]\,, \quad \text{so} \quad E^u[Y\_\tau^2-\tau]=0 \,. \tag{2}$$
Similarly,
$$X\_u^3-3uX\_u=E^u[(X\_u+Y\_\tau)^3-3(u+\tau)(X\_u+Y\_\tau)]\,, \quad \text{so by} \; (2),$$
$$E^u[Y\_\tau^3-3\tau Y\_\tau]=0 \,. \tag{3}$$
Also,
$$X\_u^4-6uX\_u^2+3u^2=E^u[(X\_u+Y\_\tau)^4-6(u+\tau)(X\_u+Y\_\tau)^2+3(u+\tau)^2]\,, $$
so
$$0=E^u[Y\_\tau^4+4X\_u Y\_\tau^3+6X\_u^2 Y\_\tau^2-6u Y\_\tau^2-12\tau X\_u Y\_\tau-6\tau(X\_u^2+Y\_\tau^2)+6u\tau+3\tau^2] \,. $$
Therefore,
$$E^u[Y\_\tau^4]=4X\_u E^u[Y\_\tau^3-3\tau Y\_\tau]+6(X\_u^2-u) E^u[Y\_\tau^2-\tau] +3E^u[\tau^2]=3E^u[\tau^2]\,. \tag{4}$$
$\hspace{6.6in} \Box$


