A remarkable $\sigma$-finite measure unifying supremum penalisations for a stable Lévy process

Abstract We consider deterministic fast–slow dynamical systems on $$\mathbb {R}^m\times Y$$ R m × Y of the form $$\begin{aligned} {\left\{ \begin{array}{ll} x_{k+1}^{(n)} = x_k^{(n)} + n^{-1} a\big (x_k^{(n)}\big ) + n^{-1/\alpha } b\big (x_k^{(n)}\big ) v(y_k), \\ y_{k+1} = f(y_k), \end{array}\right. } \end{aligned}$$ x k + 1 ( n ) = x k ( n ) + n - 1 a ( x k ( n ) ) + n - 1 / α b ( x k ( n ) ) v ( y k ) , y k + 1 = f ( y k ) , where $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) . Under certain assumptions we prove convergence of the m-dimensional process $$X_n(t)= x_{\lfloor nt \rfloor }^{(n)}$$ X n ( t ) = x ⌊ n t ⌋ ( n ) to the solution of the stochastic differential equation $$\begin{aligned} \mathrm {d} X = a(X)\mathrm {d} t + b(X) \diamond \mathrm {d} L_\alpha , \end{aligned}$$ d X = a ( X ) d t + b ( X ) ⋄ d L α , where $$L_\alpha $$ L α is an $$\alpha $$ α -stable Lévy process and $$\diamond $$ ⋄ indicates that the stochastic integral is in the Marcus sense. In addition, we show that our assumptions are satisfied for intermittent maps f of Pomeau–Manneville type.

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Conditioned stable Lévy processes and the Lamperti representation

Journal of Applied Probability ◽

10.1017/s0021900200002369 ◽

2006 ◽

Vol 43 (04) ◽

pp. 967-983 ◽

Cited By ~ 10

Author(s):

M. E. Caballero ◽

L. Chaumont

Keyword(s):

Ruin Probability ◽

Lévy Process ◽

Lévy Processes ◽

First Passage Time ◽

Passage Time ◽

Levy Processes ◽

Levy Process ◽

Stable Lévy Process ◽

Self Similar ◽

Positive Half

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

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Transition density of a reflected symmetric stable Lévy process in an orthant

Institute of Mathematical Statistics Lecture Notes - Monograph Series - Probability, statistics and their applications: papers in honor of Rabi Bhattacharya ◽

10.1214/lnms/1215091661 ◽

2003 ◽

pp. 117-131

Author(s):

Amites Dasgupta ◽

S. Ramasubramanian

Keyword(s):

Lévy Process ◽

Transition Density ◽

Levy Process ◽

Stable Lévy Process

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The Term Structure Model under the α-Stable Levy Process

Journal of Derivatives and Quantitative Studies ◽

10.1108/jdqs-01-2010-b0003 ◽

2010 ◽

Vol 18 (1) ◽

pp. 77-100

Author(s):

Joon Hee Rhee ◽

Soo Chun Park

Keyword(s):

Term Structure ◽

Generating Function ◽

Lévy Process ◽

Stable Process ◽

Moment Generating Function ◽

Levy Process ◽

Bond Price ◽

Term Structure Model ◽

Stable Lévy Process ◽

The Moment

This paper derives the analytic solutions of the pure discount bond price under the various types of -stable Levy process. It is well-known that only a few cases in-stable Levy process have the moment generating function. This paper extends the model to damped-stable Levy processes, which have artificial stable process with the moment generating function. This paper also extends models to stochastic volatility by time change method of Levy process.

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