The asymptotic behavior of densities related to the supremum of a stable process

By R. Doney, Mladen Savov. 2010

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P(S1 > x) v A� −1 x −� as x ! 1 and P(S1 � x) v B� −1 � −1 x �� as x # 0. [Here � = P(X1 > 0) and A and B are known constants.] It is also known that S1 has a continuous density, m say. The main point of this note is to show that m(x) v Ax −(�+1) as x ! 1 and m(x) v Bx ��−1 as x # 0. Similar results are obtained for related densities.

Subfields: Applied Mathematics, Mathematical Physics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/7cf173c8f4d483b9bd3d81c2c4b0b20292982439

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Citations in the corpus

Bernstein-gamma functions and exponential functionals of Levy Processes (2016, methodology): “The latter for example immediately recovers the asymptotic behaviour of the density of the supremum of a stable Lévy process as investigated in [8, 25, 40, 56].”
Local behaviour of first passage probabilities (2010, methodology): “(44) Since it is known (see (109) in [21] or Proposition 10 in [16]) that k11 := ∫∞ 0 zp̃(z)dz < ∞, and we can assume Kε ↑ ∞ as ε ↓ 0 we see that, provided lim ε↓0 lim sup n→∞ 1 U(x)P (τ = n) ∑ y∈[0,εcn]∪[Kεcn,∞) P (Sn > x− y, Tx > n)F (y) = 0, (45) it will follow from (43) that P (Tx = n) ∽ U(x)k11”
On extrema of stable processes. (2010, background): “In the case of a general stable process Doney and Savov [13] obtain the first term of asymptotic expansion of p(x) as x→ 0+ or x→ +∞, and they also mention that it is possible to obtain higher order asymptotic terms as x→+∞ (see Section 5 in their paper).”