Serving the Quantitative Finance Community
https://forum.wilmott.com/
AI scientists love claiming that they invented this. More word inflatiion. BTW, IHTLAEBMs are 20 years old: https://cs.nyu.edu/~yann/research/ebm/Is physics "better" than maths in this context?
Some believe that PINNs (physics informed NNs) are the new wave.
#include <cmath>
#include <vector>
#include <boost/numeric/odeint.hpp>
using value_type = double;
using state_type = std::vector<value_type>;
struct FiaccoOde
{ // Boost C++ odeint library used
FiaccoOde() {}
void operator()(const state_type& input, state_type& dxdt, value_type t)
{ // ODE based on minimisation problem
value_type x = input[0]; value_type y = input[1]; value_type z = input[2];
value_type lambda1 = input[3]; value_type lambda2 = input[4];
// Transform (0, infinity to (0,1)
double tmp = (1.0 - t);
auto g1 = [](double x, double y, double z) {return x - y * y - z * z;};
auto g2 = [](double x, double y, double z) {return x* x + y * y - 1.0;};
// Compute gradients of f = abs(x-2) + abs(y-2)
double fx = 1.0; double fy = 1.0; double fz = 0.0;
if (x < 2.0) fx = -1.0; if (y < 2.0) fy = -1.0;
// Exact gradient of constraint functions
// g1(x,y,z) = x - y^2 - z^2 = 0 // z is slack variable
double g1x = 1.0; double g1y = -2.0 * y; double g1z = -2.0 * z;
// g2(x, y, z) = x ^ 2 + y ^ 2 - 1 = 0
double g2x = 2.0 * x; double g2y = 2.0 * y; double g2z = 0.0;
// Equation (12) of Barr and Platt; notice the signs
dxdt[0] = -(fx + lambda1 * g1x + lambda2 * g2x) / (tmp * tmp);
dxdt[1] = -(fy + lambda1 * g1y + lambda2 * g2y) / (tmp * tmp);
dxdt[2] = -(fz + lambda1 * g1z + lambda2 * g2z) / (tmp * tmp);
dxdt[3] = g1(x, y, z) / (tmp * tmp);
dxdt[4] = g2(x, y, z) / (tmp * tmp);
}
};
value_type f(value_type x, value_type y)
{ // f() is not a functions of the slack variable z
return (x - 2) * (x - 2) + (y - 2) * (y - 2);
}
int main()
{
namespace Bode = boost::numeric::odeint;
// Initial condition
value_type L = 0.0;
value_type T = 0.9995;
value_type dt = 0.1051001001;
FiaccoOde ode;
// 5-d space: x,y,z and 2 Lagrange multipliers
state_type x = { 1,1,1,1,1 }; // the state, vector of 1 element (scalar problem)
std::size_t steps = Bode::integrate(ode, x, L, T, dt);
std::cout << "Number of steps Cash Karp 54: " << std::setprecision(16) << steps
<< "\napproximate: " << x[0] << ", " << x[1] << ", " << x[2] << '\n';
std::cout << "Function value: " << std::setprecision(8) << f(x[0], x[1]) << '\n';
x = { 1,1,1,1,1 };
Bode::bulirsch_stoer<state_type, value_type> bsStepper; // O(variable), controlled stepper
steps = Bode::integrate_const(bsStepper, ode, x, L, T, dt);
std::cout << "Number of steps Bulirsch-Stoer: " << steps << ", exact: " << ", approximate: " <<
x[0] << ", " << x[1] << "," << x[2] << ", " << x[3] << ", " << x[4] << '\n';
std::cout << "Function value: " << std::setprecision(8) << f(x[0], x[1]) << '\n';
}Look, if I argue with you. I must take a contrary position.You got me fooled here! For a moment I thought you were actually interested in the subject.AI scientists love claiming that they invented this. More word inflatiion. BTW, IHTLA
EBMs are 20 years old: https://cs.nyu.edu/~yann/research/ebm/
Lagrange, Hamilton et al did this stuff for breakfast.
AI Rulez: every so often, come up with a new buzzword.
Not necessarily. I could be arguing in my spare timeFull on contrarianism is boring. So predictable.
I'm sorry, but I'm not allowed to argue anymore.Still boring
So is mine. Due to Covid I developed a minimal epidemic model which, of course, doesn't grow exponentially. i.e. Doesn't break the laws of Maths.My maths background is as a modeller.
I found those complaints informative and had a think. I asked myself "What about affine returns?"And obviously I've complained about the accuracy of these models as well.
Have you ever seen the ugly swan that is the logit-normal distribution? With affine returns I get a mixture distribution that is the sum of the log-normal and, what looks like, a scaled logit-normal. It certainly appears to be fat tailed. Click here for a slide show of growth curves fitted to the noise of S&P 500 stocks and a link to the paper.What maths do we see? Martingales, Monte Carlo simulations, and the same linear diffusion equation, with the occasional trivial integral equation. Ad nauseam.