MAXIMAL INEQUALITIES FOR THE ORNSTEIN-UHLENBECK PROCESS

@inproceedings{Graversen1998MAXIMALIF,
  title={MAXIMAL INEQUALITIES FOR THE ORNSTEIN-UHLENBECK PROCESS},
  author={Svend Erik Graversen and G. Peskir},
  year={1998}
}
Let V = (Vt)t≥0 be the Ornstein-Uhlenbeck velocity process solving dVt = −βVtdt + dBt with V0 = 0 , where β > 0 and B = (Bt)t≥0 is a standard Brownian motion. Then there exist universal constants C1 > 0 and C2 > 0 such that C1 √ β E √ log(1 + βτ) ≤ E ( max 0≤t≤τ |Vt| ) ≤ C2 √ β E √ log(1 + βτ) for all stopping times τ of V . In particular, this yields the existence of universal constants D1 > 0 and D2 > 0 such that D1E √ log ( 1 + log(1 + τ) ) ≤ E ( max 0≤t≤τ |Bt| √ 1 + t ) ≤ D2E √ log ( 1… CONTINUE READING

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