\begin{algorithm}[H]
\caption{Fast and General MC for OU processes}\label{alg:simul1}
\begin{algorithmic}
\Require $X_{r}$ value of the process at time $r$, $\Delta t$ simulation horizon.
\State 1. Compute the characteristic function $\phi(\cdot)$ of the integral process.
\State 2. Retrieve the CDF $P(x)$ on the $x$-grid by FFT inversion of 
$\phi(\cdot)$.
\State 3. Interpolate the obtained CDF.
\State 4. Draw a uniform random variable $U\sim\mathcal{U}(0,1)$.
\State 5. Compute $X_{t} = X_r e^{-b \Delta t} + P^{-1}(U)$.
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
[H]
\caption{Fast and General MC for OU processes}\begin{algorithmic}
\Require $X_{r}$ value of the process at time $r$, $\Delta t$ simulation horizon.
\State 1. Compute the characteristic function $\phi(\cdot)$ of the integral process.
\State 2. Retrieve the CDF $P(x)$ on the $x$-grid by FFT inversion of
$\phi(\cdot)$.
\State 3. Interpolate the obtained CDF.
\State 4. Draw a uniform random variable $U\sim\mathcal{U}(0,1)$.
\State 5. Compute $X_{t} = X_r e^{-b \Delta t} + P^{-1}(U)$.
\end{algorithmic}
\end{algorithm}
{
    "title": "fast and general simulation of l\\'evy-driven ou processes for energy   derivatives",
    "id": "2401.15483",
    "abstract": "l\\'evy-driven ornstein-uhlenbeck (ou) processes represent an intriguing class of stochastic processes that have garnered interest in the energy sector for their ability to capture typical features of market dynamics. however, in the current state-of-the-art, monte carlo simulations of these processes are not straightforward for two main reasons: i) algorithms are available only for some particular processes within this class; ii) they are often computationally expensive. in this paper, we introduce a new simulation technique designed to address both challenges. it relies on the numerical inversion of the characteristic function, offering a general methodology applicable to all l\\'evy-driven ou processes. moreover, leveraging fft, the proposed methodology ensures fast and accurate simulations, providing a solid basis for the widespread adoption of these processes in the energy sector. lastly, the algorithm allows an optimal control of the numerical error. we apply the technique to the pricing of energy derivatives, comparing the results with existing benchmarks. our findings indicate that the proposed methodology is at least one order of magnitude faster than existing algorithms, all while maintaining an equivalent level of accuracy.",
    "categories": "q-fin.cp q-fin.mf q-fin.pr",
    "doi": "",
    "created": "2024-01-27",
    "updated": "",
    "authors": [
        "roberto baviera",
        "pietro manzoni"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.15483"
}
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