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19
Smoothness of scale functions for spectrally negative Lévy processes
, 2006
"... Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that stan ..."
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Cited by 25 (8 self)
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Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often complicated to establish requiring excursion theory in favour of Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic.
Boundary Harnack principle for subordinate Brownian motions
"... We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κfat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the mini ..."
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Cited by 21 (18 self)
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We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κfat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded κfat open sets with respect to these processes with their Euclidean boundaries.
On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes
, 2007
"... We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries ..."
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Cited by 16 (5 self)
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We consider the classical optimal dividend control problem which was proposed by de Finetti [Trans. XVth Internat. Congress Actuaries
Potential theory of subordinate Brownian motions revisited’, Stochastic analysis and applications to finance–essays
 in honour of Jiaan Yan, (eds
, 2012
"... The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger ..."
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Cited by 13 (10 self)
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The paper discusses and surveys some aspects of the potential theory of subordinate Brownian motion under the assumption that the Laplace exponent of the corresponding subordinator is comparable to a regularly varying function at infinity. This extends some results previously obtained under stronger conditions.
On the potential theory of onedimensional subordinate Brownian motions with continuous components
, 2008
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Twosided Green function estimates for killed subordinate Brownian motions
, 2010
"... A subordinate Brownian motion is a Lévy process which can obtained by replacing the time of Brownian motion by an independent increasing Lévy process. The infinitesimal generator of a subordinate Brownian motion is −φ(−∆), where φ is the Laplace exponent of the subordinator. In this paper, we consid ..."
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Cited by 9 (7 self)
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A subordinate Brownian motion is a Lévy process which can obtained by replacing the time of Brownian motion by an independent increasing Lévy process. The infinitesimal generator of a subordinate Brownian motion is −φ(−∆), where φ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion term. This class of processes include symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes, and much more. We give sharp twosided estimates on the Green functions of these subordinate Brownian motions in any bounded κfat open set D. When D is a bounded C 1,1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain the boundary Harnack principle in C 1,1 open set with explicit decay rate.
Convexity and smoothness of scale functions and de Finetti’s control problem
, 2008
"... Motivated by a classical control problem from actuarial mathematics, we study smoothness and convexity properties of qscale functions for spectrally negative Lévy processes. Continuing from the very recent work of [2] and [24] we strengthen their collective conclusions by showing, amongst other res ..."
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Cited by 8 (5 self)
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Motivated by a classical control problem from actuarial mathematics, we study smoothness and convexity properties of qscale functions for spectrally negative Lévy processes. Continuing from the very recent work of [2] and [24] we strengthen their collective conclusions by showing, amongst other results, that whenever the Lévy measure has a nonincreasing density which is log convex then for q> 0 the scale function W (q) is convex on some half line (a ∗ , ∞) where a ∗ is the largest value at which W (q)′ attains its global minimum. As a consequence we deduce that de Finetti’s classical actuarial control problem is solved by a barrier strategy where the barrier is positioned at height a ∗.
Potential theory of geometric stable processes
 PROBAB. THEORY RELATED FIELDS
, 2006
"... In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable pr ..."
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Cited by 6 (3 self)
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In this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate Brownian motions with geometric stable subordinators. More precisely, we establish the asymptotic behaviors of the Green function and the Lévy density of symmetric geometric stable processes. The asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes. The Green function behaves near zero as 1/(x  d log 2 x), while the Lévy density behaves like 1/x  d. We also study the asymptotic behaviors of the Green function and Lévy density of subordinate Brownian motions with iterated geometric stable subordinators. As an application, we establish estimates on the capacity of small balls for these processes, as well as mean exit time estimates from small balls and a Harnack inequality for these processes.
Global Heat Kernel Estimates for ∆ + ∆ α/2 in Halfspacelike domains
, 2011
"... Suppose that d ≥ 1 and α ∈ (0, 2). In this paper, we establish by using probabilistic methods sharp twosided pointwise estimates for the Dirichlet heat kernels of { ∆ + a α ∆ α/2; a ∈ (0, 1]} on halfspacelike C1,1 domains for all time t> 0. The large time estimates for halfspacelike domains are ..."
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Cited by 4 (4 self)
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Suppose that d ≥ 1 and α ∈ (0, 2). In this paper, we establish by using probabilistic methods sharp twosided pointwise estimates for the Dirichlet heat kernels of { ∆ + a α ∆ α/2; a ∈ (0, 1]} on halfspacelike C1,1 domains for all time t> 0. The large time estimates for halfspacelike domains are very different from those for bounded domains. Our estimates are uniform in a ∈ (0, 1] in the sense that the constants in the estimates are independent of a ∈ (0, 1]. Thus they yield the Dirichlet heat kernel estimates for Brownian motion in halfspacelike domains by taking a → 0. Integrating the heat kernel estimates with respect to the time variable t, we obtain uniform sharp twosided estimates for the Green functions of { ∆ + aα∆α/2; a ∈ (0, 1]} in halfspacelike C1,1 domains in Rd.
Dirichlet Heat Kernel Estimates for ∆ α/2 + ∆ β/2
, 2009
"... For d ≥ 1 and 0 < β < α < 2, consider a family of pseudo differential operators { ∆ α + a β ∆ β/2; a ∈ [0, 1]} on R d that evolves continuously from ∆ α/2 to ∆ α/2 + ∆ β/2. It gives arise to a family of Lévy processes {X a, a ∈ [0, 1]} on R d, where each X a is the independent sum of a symmetric αs ..."
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Cited by 1 (1 self)
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For d ≥ 1 and 0 < β < α < 2, consider a family of pseudo differential operators { ∆ α + a β ∆ β/2; a ∈ [0, 1]} on R d that evolves continuously from ∆ α/2 to ∆ α/2 + ∆ β/2. It gives arise to a family of Lévy processes {X a, a ∈ [0, 1]} on R d, where each X a is the independent sum of a symmetric αstable process and a symmetric βstable process with weight a. For any C 1,1 open set D ⊂ R d, we establish explicit sharp twosided estimates (uniform in a ∈ [0, 1]) for the transition density function of the subprocess X a,D of X a killed upon leaving the open set D. The infinitesimal generator of X a,D is the nonlocal operator ∆ α + a β ∆ β/2 with zero exterior condition on D c. As consequences of these sharp heat kernel estimates, we obtain uniform sharp Green function estimates for X a,D and uniform boundary Harnack principle for X a in D with explicit decay rate.