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SRKX
SRKX
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Are two combined Is the average of independent BMsBrownian Motions still a BMBrownian Motion?

If W$W$ and B$B$ are independent Brownian MotionsBrownian Motions (BM thereafter), then the average of W$W$ and B$B$ is X(t)=(1/2)(W(t)+B(t))$X_t=\frac{1}{2}(W_t+B_t)$. 

Where do I begin to show that indeed it is still a BM, or believe it is.?

Also, if both are martingales, then X(t)$X_t$ must be a martingale also. How would I prove this considering it has the two random variables.?

Are two combined independent BMs still a BM?

If W and B are independent Brownian Motions then the average of W and B is X(t)=(1/2)(W(t)+B(t)). Where do I begin to show that indeed it is still a BM, or believe it is.

Also, if both are martingales, then X(t) must be a martingale also. How would I prove this considering it has the two random variables.

Is the average of independent Brownian Motions still a Brownian Motion?

If $W$ and $B$ are independent Brownian Motions (BM thereafter), then the average of $W$ and $B$ is $X_t=\frac{1}{2}(W_t+B_t)$. 

Where do I begin to show that indeed it is still a BM?

Also, if both are martingales, then $X_t$ must be a martingale also. How would I prove this considering it has the two random variables?

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user17854
user17854
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Are two combined independent BMs still a BM?

If W and B are independent Brownian Motions then the average of W and B is X(t)=(1/2)(W(t)+B(t)). Where do I begin to show that indeed it is still a BM, or believe it is.

Also, if both are martingales, then X(t) must be a martingale also. How would I prove this considering it has the two random variables.