Let X X be a BES 3 3 starting from 0 Prove that 1 X 1 X is a local martingale, but not a martingale Establish that, for u 1 u 1 , E ( 1 X 1 R u ) 1 X u 2 ( X u 1 u ) E ( 1 X 1 R u ) 1 X u 2 ( X u 1 u ) where ( a ) a 0 d y e y 2 2 ( a ) 0 a d y e y 2 2 Such a formula measures the non martingale prop...

SOLUTION 1 Xt is a 3D BES process starting from 0 Then 1Xt is well defined for all t0 sin ...

Let X X be a BES 3 3 starting from 0 . Prove that 1 / X

Question:

Let $X$ be a BES $^{3}$ starting from 0 . Prove that $1 / X$ is a local martingale, but not a martingale. Establish that, for $u < 1$ ,

$E (\frac{1}{X_{1}} | R_{u}) = \frac{1}{X_{u}} \sqrt{\frac{2}{π}} Φ (\frac{X_{u}}{1 - u})$

where $Φ (a) = \int_{0}^{a} d y e^{- y^{2} / 2}$ . Such a formula " measures" the non-martingale property of the local martingale $(1 / X_{t}, t \leq 1)$ . In general, the quantity $E (Y_{t} ∣ F_{s}) / Y_{s}$ for $s < t$ , or even its mean $E (Y_{t} / Y_{s})$ , could be considered as a measure of the non-martingale property of $Y$ .