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README.md

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@@ -26,7 +26,7 @@ You can find the documentation [here](https://jumpdiff.readthedocs.io/).
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# Jump-diffusion processes
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## The theory
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Jump-diffusion processes<sup>1</sup>, as the name suggest, are a mixed type of stochastic processes with a diffusive and a jump term.
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One for of these processes which is mathematically traceable is given by the [Stochastic Differential Equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation)
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One form of these processes which is mathematically traceable is given by the [Stochastic Differential Equation](https://en.wikipedia.org/wiki/Stochastic_differential_equation)
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<img src="/Others/SDE_1.png" title="A jump diffusion process" height="25"/>
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docs/source/jd_processes.rst

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The theory
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----------
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Jump-diffusion processes\ :sup:`1`, as the name suggest, are a mixed type of stochastic processes with a diffusive and a jump term.
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One for of these processes which is mathematically traceable is given by the `Stochastic Differential Equation <https://en.wikipedia.org/wiki/Stochastic_differential_equation>`_
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One form of these processes which is mathematically traceable is given by the `Stochastic Differential Equation <https://en.wikipedia.org/wiki/Stochastic_differential_equation>`_
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.. math::
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\mathrm{d} X(t) = a(x,t)\;\mathrm{d} t + b(x,t)\;\mathrm{d} W(t) + \xi\;\mathrm{d} J(t),
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which has four main elements: a drift term :math:`a(x,t)`, a diffusion term :math:`b(x,t)`, linked with a Wiener process :math:`W(t)`, a jump amplitude term :math:`\xi(x,t)`, which is given by a Gaussian distribution :math:`\mathcal{N}(0,\sigma_\xi^2)`, and finally a jump rate :math:`\lambda`, which is the rate of the Poissonian jumps :math:`J(t)`.
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which has four main elements: a drift term :math:`a(x,t)`, a diffusion term :math:`b(x,t)`, linked with a Wiener process :math:`W(t)`, a jump amplitude term :math:`\xi(x,t)`, which is given by a Gaussian distribution :math:`\mathcal{N}(0,\sigma_\xi^2)` coupled with a jump rate :math:`\lambda`, which is the rate of the Poissonian jumps :math:`J(t)`.
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You can find a good review on this topic in Ref. 2.
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Integrating a jump-diffusion process
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:align: center
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:alt: The 2nd Kramers---Moyal coefficient
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You have this stored in `moments[2,...]`.
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You have this stored in :code:`moments[2,...]`.
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Retrieving the jump-related terms
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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

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