From 8395332eadffeb3ccf787e9312facbf297f26f65 Mon Sep 17 00:00:00 2001 From: "Ricardo M. S. Rosa" Date: Sun, 9 Jul 2023 21:18:56 -0300 Subject: [PATCH] check parameters --- latex/rode_conv_em.pdf | Bin 876015 -> 876418 bytes latex/rode_conv_em.tex | 14 +++++++------- latex/rode_conv_em_short.pdf | Bin 466224 -> 466563 bytes latex/rode_conv_em_short.tex | 6 ++++-- 4 files changed, 11 insertions(+), 9 deletions(-) diff --git a/latex/rode_conv_em.pdf b/latex/rode_conv_em.pdf index 6de94bb7d73cac46b45b625668cdaecbd2f4c732..f0f66aba5e2825a7741c39eead9c5e1bf59d4c60 100644 GIT binary patch delta 43497 zcmV)KK)S#0yETHpHGs4Mo@fCxlaU-Ne_Ks+<2Dk#&#&N7t{F4<2AI?wlFdxEvKuFx zxN1|G%t2$(wz9J1mE<^^U%#&bkRm9El0BM_+MSC@ks#1$biaPxfLU=>u;QC1U!6bs z?4_;>S60TT;{39(qA;#1UFC}NN%3kV%*iQN+{n=nC(ew@lT)eW=!L(E(ck|5fBlQE zUz~-9U!Hv(p8gnHeHUB(6o3DCBC65%KYe-f`uy7`EQ;{iOHmb^mzCCn1YpIfkY&Zg z@aqLW6=w7+<7z&6MnCN6a=Ex^9&6@%y zdJP4zt^h`s+REaTn^IU6#P}sNe^k1#Wn~$scHxuaRF)ia@GsB%?JSH)oK-%>pQnPlN*?wH<&`)gfJl6*#R-$ckpv(_aR4S3gf6`FfMoW#& 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\sim \mathcal{N}(0, 1)$, on the interval $[0, T]$. \cref{figlinearhomogeneousrode} illustrates the order of convergence. +\cref{tablinearhomogeneousrode} shows the estimated strong error obtained from the $M = 200$ Monte Carlo simulations for each chosen time step $N_i = 2^4, \ldots, 2^{14}$, with initial condition $X_0 \sim \mathcal{N}(0, 1)$, on the time interval $[0, T] = [0.0, 1.0]$. The target resolution is set to $N_{\textrm{tgt}} = 2^{16}$, since we have the exact distribution for it. \cref{figlinearhomogeneousrode} illustrates the order of convergence. \subsection{Non-homogeneous linear system of RODEs with different types of noises} @@ -1101,9 +1101,9 @@ \subsection{Non-homogeneous linear system of RODEs with different types of noise \label{figallnoises} \end{figure} -\cref{taballnoises} shows the estimated strong error obtained from the Monte Carlo simulations. \cref{figallnoises} illustrates the order of convergence. +\cref{taballnoises} shows the estimated strong error obtained from the $M = 80$ Monte Carlo simulations for each chosen time step $N_i = 2^6, \ldots, 2^9$, on the time interval $[0, T] = [0.0, 1.0]$. All coordinates of the initial condition are normally distributed with mean zero and variance 1. The target resolution is set to $N_{\textrm{tgt}} = 2^{18}$. -Finally, \cref{figsamplepathsallnoises} illustrates sample paths of all the noises used in this system. +\cref{figallnoises} illustrates the obtained order of convergence, while \cref{figsamplepathsallnoises} illustrates some sample paths of all the noises used in this system. \begin{figure}[htb] \includegraphics[scale=0.6]{img/noisepath_allnoises.png} @@ -1268,7 +1268,7 @@ \subsection{Fractional Brownian motion noise} \end{equation} computing a distributionally exact solution of this form is a delicate process. Thus we check the convergence numerically by comparing the approximations with another Euler approximation on a much finer mesh. -More precisely, the Euler approximation is implemented for \eqref{linearnonhomogeneousfbm} with several values of $H$. We fix the time interval as $[0, T] = [0.0, 1.0]$, set the resolution for the target approximation to $N_{\textrm{tgt}} = 2^{18}$, choose the time steps for the convergence test as $\Delta t = 1/N$, $N = 64,$ $128,$ $256,$ and $512,$ and use $M = 200$ samples for the Monte-Carlo estimate of the strong error. The fBm noise term is generated with the $\mathcal{O}(N)$ fast Fourier transform (FFT) method of Davies and Harte, as presented in \cite{DiekerMandjes2003} (see also \cite[Section 14.4]{HanKloeden2017}). \cref{taborderdepHfBm} shows the obtained convergence estimates, for a number of Hurst parameters, which is illustrated in \cref{figorderdepHfBm}, matching reasonably well the theoretical estimate of $p = \min\{H+1/2, 1\}.$ +More precisely, the Euler approximation is implemented for \eqref{linearnonhomogeneousfbm} with several values of $H$. We fix the time interval as $[0, T] = [0.0, 1.0]$, the initial condition as $X_0 \sim \mathcal{N}(0, 1),$ set the resolution for the target approximation to $N_{\textrm{tgt}} = 2^{18}$, choose the time steps for the convergence test as $\Delta t = 1/N$, with $N = 2^6, \ldots, 2^9,$ and use $M = 200$ samples for the Monte-Carlo estimate of the strong error. The fBm noise term is generated with the $\mathcal{O}(N)$ fast Fourier transform (FFT) method of Davies and Harte, as presented in \cite{DiekerMandjes2003} (see also \cite[Section 14.4]{HanKloeden2017}). \cref{taborderdepHfBm} shows the obtained convergence estimates, for a number of Hurst parameters, which is illustrated in \cref{figorderdepHfBm}, matching the theoretical estimate of $p = \min\{H+1/2, 1\}.$ \begin{table} \begin{tabular}[htb]{|c|c|c|c|} @@ -1345,7 +1345,7 @@ \subsection{Population dynamics with harvest} The function $f=f(t, x, y)$ is continuously differentiable infinitely many times and with bounded derivatives within the positively invariant region. Hence, within the region of interest, all the conditions of \cref{thmmixedcasepractical} hold and the Euler method is of strong order 1. -Below, we simulate numerically the solutions of the above problem, with $\gamma = 0.8,$ $\varepsilon = 0.3,$ $r = 1.0,$ and $\alpha = \gamma r = 0.64$ The geometric Brownian motion process $\{G_t\}_{t\geq 0}$ is taken with drift coefficient $\mu = 1.0,$ diffusion coefficient $\sigma = 0.8,$ and initial condition $y_0 = 1.0.$ The Poisson process $\{N_t\}_{t \geq 0}$ is taken with rate $\lambda = 15.0$. And the step process $\{H_t\}_{t \geq 0}$ is taken with steps following a Beta distribution with shape parameters $\alpha = 5.0$ and $\beta = 7.0$. The initial condition $X_0$ is taken to be a Beta distribution with shape parameters $\alpha = 7.0$ and $\beta = 5.0$, hence we can take $R = 1$. We take $M = 200$ samples for the Monte-Carlo estimate of the strong error of convergence. For the target solution, we solve the equation with a time mesh with $N_{\mathrm{tgt}} = 2^{18}$ points, while for the approximations we take $N = 2^i$, for $i=4, \ldots, 9$. +Below, we simulate numerically the solutions of the above problem, with $\gamma = 0.8,$ $\varepsilon = 0.3,$ $r = 1.0,$ and $\alpha = \gamma r = 0.64$ The geometric Brownian motion process $\{G_t\}_{t\geq 0}$ is taken with drift coefficient $\mu = 1.0,$ diffusion coefficient $\sigma = 0.8,$ and initial condition $y_0 = 1.0.$ The Poisson process $\{N_t\}_{t \geq 0}$ is taken with rate $\lambda = 15.0$. And the step process $\{H_t\}_{t \geq 0}$ is taken with steps following a Beta distribution with shape parameters $\alpha = 5.0$ and $\beta = 7.0$. The initial condition $X_0$ is taken to be a Beta distribution with shape parameters $\alpha = 7.0$ and $\beta = 5.0$, hence we can take $R = 1$. We take $M = 200$ samples for the Monte-Carlo estimate of the strong error of convergence. For the target solution, we solve the equation with a time mesh with $N_{\mathrm{tgt}} = 2^{18}$, while for the approximations we take $N = 2^i$, for $i=4, \ldots, 9$. Notice that we can write \[ @@ -1490,7 +1490,7 @@ \subsection{A toggle-switch model for gene expression} We do not have an explicit solution for the equation so we use as target for the convergence an approximate solution via Euler method at a much higher resolution. -For the mesh parameters, we set $N_{\textrm{tgt}} = 2^{18}$ and $N_i = 2^i$, for $i=4, \ldots, 9$. For the Monte-Carlo estimate of the strong error, we choose $M = 200.$ \cref{tabletoggleswitch} shows the estimated strong error obtained with this setup, while \cref{figtoggleswitch} illustrates the order of convergence. \cref{figtoggleswitchevolution} shows a sample solution, while \cref{figtoggleswitchnoise} illustrates the two components $(A_t, B_t)$ of a sample noise. +For the mesh parameters, we set $N_{\textrm{tgt}} = 2^{18}$ and $N_i = 2^i$, for $i=5, \ldots, 9$. For the Monte-Carlo estimate of the strong error, we choose $M = 100.$ \cref{tabletoggleswitch} shows the estimated strong error obtained with this setup, while \cref{figtoggleswitch} illustrates the order of convergence. \cref{figtoggleswitchevolution} shows a sample solution, while \cref{figtoggleswitchnoise} illustrates the two components $(A_t, B_t)$ of a sample noise. \begin{table} \begin{tabular}[htb]{|r|l|l|l|} @@ -1602,7 +1602,7 @@ \subsection{An actuarial risk model} For the numerical simulations, we use $O_0 = 0$, $\nu = 5$ and $\varepsilon = 0.8$, for the Ornstein-Uhlenbeck process $\{O_t\}_t$; $\lambda = 8.0$ and $C_i \sim \mathrm{Uniform}(0, 0.2)$, for the compound Poisson process $\{C_t\}$; $R_0 = 0.2$, $\mu = 0.02$ and $\sigma = 0.4$, for the interest rate process $\{R_t\}_t$; and we take $X_0 = 1.0$, so that $U_0 = X_0 + O_0 + R_0 = 1.2$. We set $T = 3.0,$ as the final time. -For the mesh parameters, we set $N_{\textrm{tgt}} = 2^{18}$ and $N_i = 2^i$, for $i=6, \ldots, 9$. For the Monte-Carlo estimate of the strong error, we choose $M = 100.$ \cref{tableriskmodel} shows the estimated strong error obtained with this setup, while \cref{figriskmodel} illustrates the order of convergence. \cref{figriskmodelnoise} shows a sample path of the noise, which is composed of three processes, while \cref{figriskmodelsurplus} shows a sample path of the surplus. +For the mesh parameters, we set $N_{\textrm{tgt}} = 2^{18}$ and $N_i = 2^i$, for $i=6, \ldots, 9$. For the Monte-Carlo estimate of the strong error, we choose $M = 400.$ \cref{tableriskmodel} shows the estimated strong error obtained with this setup, while \cref{figriskmodel} illustrates the order of convergence. \cref{figriskmodelnoise} shows a sample path of the noise, which is composed of three processes, while \cref{figriskmodelsurplus} shows a sample path of the surplus. \begin{table} \begin{tabular}[htb]{|r|l|l|l|} diff --git a/latex/rode_conv_em_short.pdf b/latex/rode_conv_em_short.pdf index 551cb612659caf0e8a974c5363021e9788d63659..740f64bd9b8d3f08a0bce2bc0295f003b7fd173c 100644 GIT binary patch delta 40588 zcmV(@K-Ryox*UVO9DuX|`C$S$IFpeaDSzEsOLN=E5x(nJU=OJRE6dq=Ki1{L?pl6q zt);BZq>{3IfJ8__g(PHvTHE>c>7E&Y05qhmn3byJA`Ax2>7JgipCg)GXEb|p^7G>4 zhv&*=mN>0twz$d|Gel(CI%1v87R&5i{&w+Z#$A4X_F^`dQs=*%pZ)FBPrW!j`+wyW zCsls_JbYZdj1#AS{SAox@b&59<@vKPem>(iKaU^Y{B-v0{B>O5+38ytQLb{XW*-*6 zp3o>{_dH#kd^}+=q#4UN<-{ekHNQx?0CPNvIAIsG~)z91S5jkZ&VsJM8hvM9Gap$NYA)EfRUrH?JzQA 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