MHB Can this system of equations be solved analytically?

[Siron]

Hello!

I have the following system of equations:

$$\left \{ \begin{array}{rcr} \mu^4+6 \mu^3 \delta^2 + 3 \delta^4& = &\frac{(k-3)}{s} \sqrt{v}(\mu^3+3\mu \delta^2)\\ \mu^4+10\mu^2\delta^2+15\delta^4 & = & \frac{v}{s}(w-10s)(\mu^2+3\delta^2)\end{array}\right.$$

The goal is to find $\mu$ and $\delta$. All the other parameters are constants.
Would it be possible to obtain an analytical solution? Either way, I tried to run it in Maple with no success.

Thanks in advance!
Cheers.

 

[Ackbach]

Let
\begin{align*}
a&=\frac{k-3}{2} \, \sqrt{v} \\
b&=\frac{v}{s} \, (w-10s);
\end{align*}
then Mathematica gives you one of those fairly difficult-to-work-with solutions using the Root function, and lots of expressions like $\#1$ that I've never been able to make heads or tails of. I will mention that $\mu=\delta=0$ is a solution, which you can see by inspection. The other solutions are quite complicated, but Mathematica does give you an expression for them. I recommend using a native Mathematica install to do this, and not Wolfram Development Platform, as the WDP is right at its limits of computation time (roughly 20 to 30 seconds per command), and is tending to get pages that Chrome, at least, wants to kill.

I've usually found Mathematica to be the absolute best at symbolic manipulations, and MATLAB rules the numerical world.

What's the context in which such a beast as this system arises?

 

[Siron]

Thank you for the response Ackbach! It's a problem which I have to solve for my thesis in the area of financial engineering (calibration method: matching moments). If you want, I state the complete system. It consist of four equations and four unknowns $\lambda, \mu, \delta$ and $\sigma$:
$$\left \{ \begin{array}{rcl} \displaystyle v & = & T(\sigma^2+\lambda(\mu^2+\delta^2)) \\ s & = & \displaystyle \frac{\lambda (\mu^3+3\mu \delta^2)}{\sqrt{T}(\sigma^2+\lambda(\sigma^2+\delta^2))^{3/2}} \\
k & = & \displaystyle 3+\frac{\lambda(3\delta^4+6\delta^2\mu^4+\mu^4)}{T(\sigma^2+\lambda(\mu^4+\delta^2))^2} \\
w & = & \displaystyle \frac{\lambda (15 \delta^4 \mu + 10 \delta^2 \mu^3 + \mu^5)+10T\lambda(\mu^3+3\delta^2 \mu)(\sigma^2+\lambda(\mu^2+\sigma^2)}{T^{3/2}(\lambda(\delta^2+\mu^2)+\sigma^2)^{5/2}}
\end{array}\right. $$

where $v,s,k$ and $w$ are constants, more precisely they are measurements which I have. The unknowns $\mu \in \mathbb{R}$ and $\delta > 0$ are the parameters of a Normal distribution, $N(\mu, \delta^2)$. So as you can see in the second equation, the numerator is the third central moment of $N(\mu,\delta^2)$. Similarly, in the third equation the numerator is the fourth central moment. In the last equation the numerator also consist of the fifth central moment, a third central moment and an additional second central moment where $\sigma^2$ is added.

Moreover, all numerators are of the same form, using the first equation of the system they all can be written in function of $v/T$ (which is known). In that case my system reduces to:
$$\left \{ \begin{array}{rcl} \displaystyle v & = & T(\sigma^2+\lambda(\mu^2+\delta^2)) \\ \frac{s}{T} v^{3/2} & = & \displaystyle \lambda (\mu^3+3\mu \delta^2) \\
\frac{(k-3)v^2}{T} & = & \lambda(3\delta^4+6\delta^2\mu^4+\mu^4)\\
\frac{v^{5/2}}{T}(w-10s) & = & \displaystyle \lambda (15 \delta^4 \mu + 10 \delta^2 \mu^3 + \mu^5)
\end{array}\right. $$
and where I also used the first and second equation to further simplify the last equation. The system is now simplified. To summarize: $\mu \in \mathbb{R}, \sigma>0, \delta>0, \lambda>0$.

I know it's pretty messed up to solve. My idea was to solve the last two equations simultaneously so that I get an expression for $\mu$ and $\sigma$. From there I can use equation 2 to obtain an expression for $\lambda$. Finally equation one will then give me an expression for $\sigma$.

I will look for a free download/trial of Mathematica to try and solve with the given constraints on the parameters.

 

[Ackbach]

Here's the code I ran on WDP as best as I can reconstruct it. Because WDP is having difficulties in evaluating the cell, for some reason it hangs up when I just try to reload the page.

Solve[{m^4+6m^3 d^2+3d^4==a(m^3+3 m d^2),m^4+10m^2 d^2+15d^4==b(m^2+3d^2)},{m,d}]