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Zbl 1042.60034
Anh, V. V.; McVinish, R.
Fractional differential equations driven by Lévy noise.
(English)
[J] J. Appl. Math. Stochastic Anal. 16, No. 2, 97-119 (2003). ISSN 1048-9533; ISSN 1687-2177/e

Let for $f(t)$ be defined its Riemann-Liouville fractional derivative by $$D^\alpha f(t)= {1\over\Gamma(n- \alpha)}{d^n\over dt^n} \int^t_0 (t-\tau)^{n-\alpha-1} f(\tau)\,d\tau,\quad \alpha\in [n-1,n),\ n= 1,2,\dots$$ and its Riemann-Liouville fractional integral by $$I^\alpha f(t)= {1\over\Gamma(\alpha)} \int^t_0 (t-\tau)^{\alpha- 1}f(\tau)\,d\tau,\quad \alpha> 0.$$ The authors consider the fractional differential equations $$(A_n D^{\beta_n}+\cdots+ A_1 D^{\beta_1}+ A_0 D^{\beta_0})X(t)= \dot L(t),\quad \beta_n> \beta_{n-1}>\cdots> \beta_1>\beta_0,\ n\ge 1,\tag1$$ and the fractional integral equation $$X(t)+ {A_{n-1}\over A_n} I^{\beta_n- \beta_{n-1}} X(t)+\cdots+ {A_0\over A_n} I^{\beta_n- \beta_0} X(t)= {1\over A_n} I^{\beta_n- 1}L(t),\quad \beta_n\ge 1,\tag2$$ driven by Lévy noise $\dot L$. The singularity spectrum of solution $X(t)$ of (2) is obtained. They also study conditions under which this solution is a semi-martingale.\par The authors give a numerical scheme to approximate the sample paths of equations of the form (1). This scheme is almost surely uniformly convergent. Using this numeric algorithm the authors present the sample paths of some fractional differential equation.
[Maria Stolarczyk (Katowice)]
MSC 2000:
*60H10 Stochastic ordinary differential equations
60H35 Computational methods for stochastic equations

Keywords: stochastic fractional differential and integral equation; semi-martingale representation

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