First, the following lemma establishes the oscillation properties of the diffusion (martingale) through a subinterval or slab. The expected number of ‘upcrossings’ of the density function diffusion across a subinterval or “slab”

is finite for all and the number of upcrossings on a semi-infinite interval is always zero. This is essentially a Doob upcrossing inequality. Let be the density function diffusion or martingale solution of the SDE (-) for initial data so that . Let be a finite interval with . Now let denote the number of ‘upcrossings’ of the interval or slab , which is the number of times that the density function diffusion has passed from below to above at some . Then if is a martingale

[1] The number of upcrossings through the slab is finite for all

[2] The limit gives a semi-infinite

interval or slab so that the number of upcrossings from below to infinity at any , is zero

so that for the initial data, the density diffusion never blows up for any finite . Define the ‘hitting times’

with and . Construct a stochastic process

Beginning with the first time that , the process

evolves via increments of until the the first time . The process repeats if again the martingale diffusion falls below or at time and so on. All terms in the summation vanish for sufficiently large that so that there are at most terms that are non-zero. Then

so that

Since is also a martingale then

. It follows that

\

Taking the limit as .

This then gives for any so a blowup never occurs.

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