Solution to Nonlinear Algebraic Equation

Question

I have the following system of 5 equations for 5 unknowns, $p,c_+,c_-,u,v$:

\begin{align} pc_+ - (1-p) c_- &= \mu\sqrt{\delta t}\\ pc_+^2 + (1-p)c_-^2 &= \sigma^2 + \mu^2\delta t\\ \sqrt{\delta t} c_+ &= u - 1\\ \sqrt{\delta t} c_- &= 1 - v\\ uv &= 1 \end{align}

The parameters $\mu,\sigma,\delta t > 0$ are constants. Additionally, the variables $p,c_+,c_-,u,v > 0$. I am trying to solve this system of equations but am having a hard time. I know for sure that it involves creating a new variable to simplify it, but I cannot see what to create. I tried doing $\tilde{u} = (u-1)/\sqrt{\delta t}$ and $\tilde{v} = (1-v)/\sqrt{\delta t}$, which gives the system

\begin{align} p\tilde{u} - (1-p)\tilde{v} &= \mu\sqrt{\delta t}\\ p\tilde{u}^2 + (1-p)\tilde{v}^2 &= \sigma^2 + \mu^2\delta t\\ \sqrt{\delta t} \tilde{u}\tilde{v} &= \tilde{u}-\tilde{v} \end{align}

Then, squaring the first equation gives

$$ p^2\tilde{u}^2 + (1-p)^2\tilde{v}^2 - 2p(1-p)\tilde{u}\tilde{v} = \mu^2\delta t $$

Comparing to the second equation,

$$ p^2\tilde{u}^2 + (1-p)^2\tilde{v}^2 - 2p(1-p)\tilde{u}\tilde{v} = p\tilde{u}^2 + (1-p)\tilde{v}^2 - \sigma^2 $$

Notice that this simplifies greatly. Putting all the variable terms on the right hand side and plugging in Eq 3 gives

$$ \tilde{u}^2 + 2\tilde{u}\tilde{v} + \tilde{v}^2 = (\tilde{u}+\tilde{v})^2 = \frac{\sigma^2}{p(1-p)} \Rightarrow \tilde{u} + \tilde{v} = \frac{\sigma}{\sqrt{p(1-p)}} $$

However, this gives $\tilde{u} + \tilde{v}$ in terms of another variable, so I don't know how to proceed. Any thoughts would be appreciated.

Answer 1

Using the variable substitution $\tilde{u}$ and $\tilde{v}$ as in the original post, we can solve for $p$ from the first two equations. This gives

\begin{align} p &= \frac{\mu\sqrt{\delta t} + \tilde{v}}{\tilde{u}+\tilde{v}}\\ p &= \frac{\mu^2\delta t^2 + \sigma^2 - \tilde{v}^2}{\tilde{u}^2-\tilde{v}^2} \end{align}

Notice that when you equate the two equations, you get

$$ (\mu\sqrt{\delta t} + \tilde{v})(\tilde{u}-\tilde{v}) = \beta^2 - \tilde{v}^2, $$

where $\beta^2 = \sigma^2 + \mu^2 \delta t$. Expanding this out,

$$ \mu\sqrt{\delta t} \tilde{u} - \mu\sqrt{\delta t}\tilde{v} + \tilde{u}\tilde{v} = \beta^2 $$

Lastly, we can use the last equation for $\tilde{u}\tilde{v}$, so that in total we get

$$ \mu\sqrt{\delta t}\tilde{u} - \mu\sqrt{\delta t}\tilde{v} + \frac{1}{\sqrt{\delta t}}(\tilde{u}-\tilde{v}) = \beta^2 \Rightarrow \boxed{\tilde{u} - \tilde{v} = \frac{\sqrt{\delta t}}{1+\mu\delta t}\beta^2} $$

The boxed equation is important because it gives $\tilde{u} - \tilde{v}$ in terms of constants. Notice that in the above, we have $(\mu\sqrt{\delta t} + \tilde{v})(\tilde{u}-\tilde{v}) = \beta^2 - \tilde{v}^2$, so we can plug in the boxed expression into that equation. This leaves one equation for one unknown, $\tilde{v}$, which can be solved using the quadratic formula. The rest is history.

Answer 2

The system of equalities has five CAD components. (Found by cylindrical algebraic decomposition in a CAS using the variable symbol ordering $p \leq c_+ \leq c_- \leq u \leq v$, which is not a constraint on the values of the variables.) Further restricting by the inequalities, we have

Case 1: \begin{align*} p &= - \frac{\mu ^2 \left(2 \delta t \sigma ^2-1\right)+2 \mu \sigma ^2+\sigma ^4}{\mu ^2-2 \mu \sigma ^2-3 \sigma ^4} \\ c_+ &= \frac{\sqrt{-\frac{\left(\mu ^2 (p-3) \sigma -2 \mu (p+1) \sigma ^3-3 (p+1) \sigma ^5\right)^2}{\left(\mu +\sigma ^2\right)^2 \left(\mu ^2 (p-1)-2 \mu (p+1) \sigma ^2-3 (p+1) \sigma ^4\right)}}}{\sqrt{2}} \\ c_- &= \frac{c_+ \left(\mu +\sigma ^2\right) \left(\mu ^2 (p-1)-2 \mu p \sigma ^2-3 (p+1) \sigma ^4\right)}{2 \mu ^2 (p-3) \sigma ^2-4 \mu (p+1) \sigma ^4-6 (p+1) \sigma ^6} \\ u &= \frac{-\mu ^2 (p-1)+2 \mu p \sigma ^2+3 (p+1) \sigma ^4}{2 \sigma ^2 \left(\mu +\sigma ^2\right)} \\ v &= \frac{p}{2} - \frac{(1-3 p) \sigma ^3}{4 \sigma\mu }-\frac{\mu (p-1)}{4 \sigma^2 } \\ \delta t &= \sqrt{\frac{2 \mu ^2+2 \mu \sigma ^2+\sigma ^4}{\mu ^4}} - \frac{\mu +\sigma ^2}{\mu ^2} \\ \end{align*} together with \begin{align*} \mu &\neq (2 \sqrt{3} - 3) \sigma ^2 \\ \mu &\neq 3 \sigma ^2 \text{.} \end{align*} The $\delta t$ equation and the $\mu$ inequalities are constraints on the parameters, so this component is not generic across the entire parameter space.

Case 2: \begin{align*} p &= 1/2 \\ c_+ &= \sigma + \sqrt{\sigma ^2-2 \mu } \\ c_- &= \sigma - \sqrt{\sigma ^2-2 \mu } \\ u &= \frac{c_+ \sigma }{\mu } -1 \\ v &= \frac{c_- \sigma }{\mu } -1 \\ \delta t &= \frac{\sigma ^2 - 2 \mu}{\mu^2 } \\ 0 &< \mu < \sigma ^2/2 \end{align*} The last two relations prevent this solution being generic over the entire parameter space.

Case 3: Two solutions are presented here. To obtain a coherent single solution, of "$\pm$"s and "$\mp$"s, simultaneously choose only the upper or lower sign from all of them. \begin{align*} p &= \frac{\mp \delta t^{3/2} \mu ^2+\sqrt{\left(\delta t \mu ^2+\sigma ^2\right) \left((\delta t \mu +2)^2+\delta t \sigma ^2\right)} \mp 2 \sqrt{\delta t} \mu \pm \sqrt{\delta t} \sigma ^2}{2 \sqrt{\left(\delta t \mu ^2+\sigma ^2\right) \left((\delta t \mu +2)^2+\delta t \sigma ^2\right)}} \\ c_+ &= \frac{\delta t^{3/2} \mu ^2 \mp \sqrt{\left(\delta t \mu ^2+\sigma ^2\right) \left((\delta t \mu +2)^2+\delta t \sigma ^2\right)}+\sqrt{\delta t} \sigma ^2}{2 (\delta t \mu +1)} \\ c_- &= \frac{-\delta t^{3/2} \mu ^2 \mp \sqrt{\left(\delta t \mu ^2+\sigma ^2\right) \left((\delta t \mu +2)^2+\delta t \sigma ^2\right)}-\sqrt{\delta t} \sigma ^2}{2 (\delta t \mu +1)} \\ u &= \frac{\delta t^2 \mu ^2 \mp \sqrt{\delta t} \sqrt{\left(\delta t \mu ^2+\sigma ^2\right) \left((\delta t \mu +2)^2+\delta t \sigma ^2\right)}+2 \delta t \mu +\delta t \sigma ^2+2}{2 (\delta t \mu +1)} \\ v &= \frac{\delta t^2 \mu ^2 \pm \sqrt{\delta t} \sqrt{\left(\delta t \mu ^2+\sigma ^2\right) \left((\delta t \mu +2)^2+\delta t \sigma ^2\right)}+2 \delta t \mu +\delta t \sigma ^2+2}{2 (\delta t \mu +1)} \\ 0 &\neq \delta t (\delta t \mu +1) \left(\delta t^4 \mu ^6+\delta t^3 \mu ^4 \ \left(6 \mu +\sigma ^2\right)+\delta t^2 \mu ^2 \left(12 \mu ^2+4 \mu \ \sigma ^2-\sigma ^4\right)+\delta t \left(8 \mu ^3+8 \mu ^2 \sigma \ ^2-2 \mu \sigma ^4-\sigma ^6\right)+8 \mu \sigma ^2-4 \sigma ^4\right) \left(\delta t^{3/2} \mu ^2 \pm \sqrt{\delta t^3 \mu ^4+2 \delta \ t^2 \mu ^2 \left(2 \mu +\sigma ^2\right)+\delta t \left(2 \mu +\sigma \ ^2\right)^2+4 \sigma ^2}+\sqrt{\delta t} \left(2 \mu +3 \sigma ^2\right)\right) \end{align*} Again, this case has a constraint on the parameters, so is not generic across the entire parameter space.

That there is no solution generic across the entire parameter space is not surprising. In each solution, we have fractions with parameter expressions in the denominators or radical expressions containing parameters. In each case, the parameters must be such that no division by zero occurs and no even root of a negative number occurs.