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37
TimeChanged Lévy Processes and Option Pricing
, 2002
"... As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return ..."
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Cited by 89 (12 self)
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As is well known, the classic BlackScholes option pricing model assumes that returns follow Brownian motion. It is widely recognized that return processes differ from this benchmark in at least three important ways. First, asset prices jump, leading to nonnormal return innovations. Second, return volatilities vary stochastically over time. Third, returns and their volatilities are correlated, often negatively for equities. We propose that timechanged Lévy processes be used to simultaneously address these three facets of the underlying asset return process. We show that our framework encompasses almost all of the models proposed in the option pricing literature. Despite the generality of our approach, we show that it is straightforward to select and test a particular option pricing model through the use of characteristic function technology.
Affine processes and applications in finance
 Annals of Applied Probability
, 2003
"... Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and OrnsteinUhlenbeck type processes. We show, and provide foundations for, a wide range of financial ap ..."
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Cited by 39 (5 self)
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Abstract. We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and OrnsteinUhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.
The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest
, 1997
"... Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solu ..."
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Cited by 25 (7 self)
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Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solution X of the Ito SDE dXv = n 4 \Gamma X 2 v \Gamma t \Gamma R v 0 Xudu \Delta \Gamma1 o dv + 2 p XvdBv on the interval [0; V t (X)), where V t (X) := inffv : R v 0 Xudu = tg, and Xv = 0 for all v V t (X). This conditioned form of the RayKnight description of Brownian local times arises from study of the asymptotic distribution as n !1 and 2k= p n ! ` of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points. The SDE is the continuous analog of a simple description of a GaltonWatson branching process conditioned on its total progeny....
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 18 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Skew convolution semigroups and affine Markov processes
 Ann. Probab
, 2006
"... A general affine Markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup. We provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup. The corresponding affine Markov process is con ..."
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Cited by 10 (0 self)
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A general affine Markov semigroup is formulated as the convolution of a homogeneous one with a skew convolution semigroup. We provide some sufficient conditions for the regularities of the homogeneous affine semigroup and the skew convolution semigroup. The corresponding affine Markov process is constructed as the strong solution of a system of stochastic equations with nonLipschitz coefficients and Poissontype integrals over some random sets. Based on this characterization, it is proved that the affine process arises naturally in a limit theorem for the difference of a pair of reactant processes in a catalytic branching system with immigration.
SelfInteracting Random Motions  A Survey
"... . We present a survey of results concerning selfinteracting random walks and selfrepelling continuous random motions. A selfinteracting random walk (SIRW) is a nearest neighbour walk on the onedimensional integer lattice Z which starts from the origin and at each step jumps to a neighbouring sit ..."
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Cited by 10 (2 self)
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. We present a survey of results concerning selfinteracting random walks and selfrepelling continuous random motions. A selfinteracting random walk (SIRW) is a nearest neighbour walk on the onedimensional integer lattice Z which starts from the origin and at each step jumps to a neighbouring site, the probability of jumping along a bond being proportional to w(number of previous jumps along that lattice bond), where w : N ! R+ is a monotone weight function. We consider exponential [ w(n) = expfng, > 0 ], subexponential [ w(n) = expfn g, > 0, 0 < < 1 ], polynomially decaying [ w(n) n , > 0 ], asymptotically constant [ w(n) 1 ] and weakly increasing [ w(n) n , 0 < < 1 ] weight functions. These weight functions dene variants of the socalled `myopic selfrepelling' and `reinforced' random walk. We present functional limit theorems for the local time processes of these random walks and limit theorems for the position of the random walker at late times. A...
CHANGING THE BRANCHING MECHANISM OF A CONTINUOUS STATE BRANCHING PROCESS USING IMMIGRATION
, 2007
"... Abstract. We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population ..."
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Cited by 10 (8 self)
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Abstract. We consider an initial population whose size evolves according to a continuous state branching process. Then we add to this process an immigration (with the same branching mechanism as the initial population), in such a way that the immigration rate is proportional to the whole population size. We prove this continuous state branching process with immigration proportional to its own size is itself a continuous state branching process. By considering the immigration as the apparition of a new type, this construction is a natural way to model neutral mutation. It also provides in some sense a dual construction of the particular pruning at nodes of continuous state branching process introduced by the authors in a previous paper. For a critical or subcritical quadratic branching mechanism, it is possible to explicitly compute some quantities of interest. For example, we compute the Laplace transform of the size of the initial population conditionally on the non extinction of the whole population with immigration. We also derive the probability of simultaneous extinction of the initial population and the whole population with immigration. 1.
Skew convolution semigroups and related immigration processes. Theory Probab
 Appl
"... Abstract. A special type of immigration associated with measurevalued branching processes is formulated by using skew convolution semigroups. We give characterization for a general inhomogeneous skew convolution semigroup in terms of probability entrance laws. The related immigration process is con ..."
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Cited by 10 (7 self)
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Abstract. A special type of immigration associated with measurevalued branching processes is formulated by using skew convolution semigroups. We give characterization for a general inhomogeneous skew convolution semigroup in terms of probability entrance laws. The related immigration process is constructed by summing up measurevalued paths in the Kuznetsov process determined by an entrance rule. The behavior of the Kuznetsov process is then studied, which provides insights into trajectory structures of the immigration process. Some wellknown results on excessive measures are formulated in terms of stationary immigration processes. Key words: measurevalued branching process; superprocess; immigration process; skew convolution semigroup; entrance law; entrance rule; excessive measure; Kuznetsov measure
Cauchy's principal value of local times of Lévy processes with no negative jumps via continuous branching processes
, 1997
"... this paper is devoted to preliminaries on L'evy processes with no negative jumps, continuous state branching processes and the Lamperti connection between the two. The framework is more general than that needed for our main purpose, but is relevant for other applications we have in mind. The result ..."
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Cited by 6 (1 self)
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this paper is devoted to preliminaries on L'evy processes with no negative jumps, continuous state branching processes and the Lamperti connection between the two. The framework is more general than that needed for our main purpose, but is relevant for other applications we have in mind. The result on the joint distribution of (C