scholarly journals Asymptotics in small time for the density of a stochastic differential equation driven by a stable Lévy process

2018 ◽  
Vol 22 ◽  
pp. 58-95
Author(s):  
Emmanuelle Clément ◽  
Arnaud Gloter ◽  
Huong Nguyen
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Asymptotic Behavior ◽  
Stochastic Differential Equation ◽  
Malliavin Calculus ◽  
Stable Process ◽  
Small Time ◽  
Stable Lévy Process ◽  
Asymptotic Mixed Normality ◽  
Drift Coefficient

This work focuses on the asymptotic behavior of the density in small time of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). We assume that the process depends on a parameter β = (θ, σ)T and we study the sensitivity of the density with respect to this parameter. This extends the results of [E. Clément and A. Gloter, Local asymptotic mixed normality property for discretely observed stochastic dierential equations driven by stable Lévy processes. Stochastic Process. Appl. 125 (2015) 2316–2352.] which was restricted to the index α ∈ (1, 2) and considered only the sensitivity with respect to the drift coefficient. By using Malliavin calculus, we obtain the representation of the density and its derivative as an expectation and a conditional expectation. This permits to analyze the asymptotic behavior in small time of the density, using the time rescaling property of the stable process.

scholarly journals LAMN property for the drift and volatility parameters of a sde driven by a stable Lévy process

2019 ◽  
Vol 23 ◽  
pp. 136-175 ◽  
Author(s):  
Emmanuelle Clément ◽  
Arnaud Gloter ◽  
Huong Nguyen
Keyword(s):  
Stable Process ◽  
Transition Density ◽  
Fixed Time ◽  
Small Time ◽  
Time Interval ◽  
Lévy Measure ◽  
Stable Lévy Process ◽  
Continuous Time Process ◽  
Asymptotic Mixed Normality ◽  
Drift Coefficient

This work focuses on the local asymptotic mixed normality (LAMN) property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a truncated α-stable process with index α ∈ (0, 2). The process is observed on the fixed time interval [0,1] and the parameters appear in both the drift coefficient and scale coefficient. This extends the results of Clément and Gloter [Stoch. Process. Appl. 125 (2015) 2316–2352] where the index α ∈ (1, 2) and the parameter appears only in the drift coefficient. We compute the asymptotic Fisher information and find that the rate in the LAMN property depends on the behavior of the Lévy measure near zero. The proof relies on the small time asymptotic behavior of the transition density of the process obtained in Clément et al. [Preprint HAL-01410989v2 (2017)].

scholarly journals Homogenization for deterministic maps and multiplicative noise

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130201 ◽  
Author(s):  
Georg A. Gottwald ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Multiplicative Noise ◽  
Stochastic Integrals ◽  
Deterministic System ◽  
One Dimensional ◽  
Stable Lévy Process ◽  
Fast Flow

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.

Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (02) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density. In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

scholarly journals Monte-Carlo Galerkin Approximation of Fractional Stochastic Integro-Differential Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Abdallah Ali Badr ◽  
Hanan Salem El-Hoety
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Monte Carlo ◽  
Stochastic Differential Equation ◽  
Galerkin Method ◽  
Existence And Uniqueness ◽  
Galerkin Approximation ◽  
Integro Differential Equation ◽  
Uniqueness Of The Solution ◽  
Time Continuum

A stochastic differential equation, SDE, describes the dynamics of a stochastic process defined on a space-time continuum. This paper reformulates the fractional stochastic integro-differential equation as a SDE. Existence and uniqueness of the solution to this equation is discussed. A numerical method for solving SDEs based on the Monte-Carlo Galerkin method is presented.

scholarly journals Superdiffusive limits for deterministic fast–slow dynamical systems

2020 ◽  
Vol 178 (3-4) ◽  
pp. 735-770
Author(s):  
Ilya Chevyrev ◽  
Peter K. Friz ◽  
Alexey Korepanov ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Stochastic Differential Equation ◽  
Lévy Process ◽  
Stochastic Integral ◽  
Levy Process ◽  
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.

Dynamic Approach for Financial Asset Price by Feynman –Kac Formula

2018 ◽  
Vol 13 (4) ◽  
pp. 290-299
Author(s):  
Prashanta kumar Behera
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Partial Differential Equation ◽  
Stochastic Differential Equation ◽  
Asset Price ◽  
Selling Price ◽  
Financial Asset ◽  
Dynamic Approach ◽  
Expected Payoff ◽  
The Moment

 This paper has obtains the partial differential equation that describes the expected price of a financial asset whose price is a stochastic process given by a stochastic differential equation. We tried finding the expected selling price of an asset and exiting time by using of Feynman –Kac Formula. We assume that the asset is sold at the moment when its price rises above or falls below a certain limit, and thus the solution v has to satisfy x - v = 0 at the boundary points x. The expected selling price depends nearly linearly on the price at time t, and also weakly on t and the expected payoff of an asset for which a limit sales order has been placed and the same asset without sales order over a time span T, as a function of t

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