# justification of square root process

In finance, many stochastic processes $X(t)$ are defined via $$dX = \text{(some drift term)} dt + \sigma X^\gamma dW_t$$ with $\gamma = 1/2$ (for instance the Heston model or the CIR process). Generally, this is called a square-root process. My question is: How does one justify the choice of $\gamma = 1/2$. I am aware that it is convenient to chose $0 < \gamma < 1$ since for $\gamma > 1$, no unique Martingale measure exists. But why exactly $\gamma = 1/2$ and not, say $\gamma = 6/7$. (I have found one related question here Why square root of volatility in Heston model? but no satisfying answer has been given.)

1. C.I.R Process belongs the class of affine diffusion processes.For processes within this class, a closed form solution of the characteristic function exists(Duffie,et al). For more details, Suppose we have given a scalar SDEs, i.e., $$dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dW_t$$ this process ($\{X_t\}_{0\leq t\leq T}$) is said to be of the affine form if \begin{align} &&\mu(X_t,t)=\alpha_0+\alpha_1X_t\\ &&\sigma^2(X_t,t)=\beta_0+\beta_1X_t \end{align} where $\alpha_j,\beta_j\in R$. We claim C.I.R Process belongs the class of affine diffusion processes,because
\begin{align} & \mu (t,{{r}_{t}})\,\,=\kappa (\theta -{{r}_{t}})=\underbrace{\kappa \theta }_{{{\alpha }_{0}}}+\underbrace{(-\kappa )}_{{{\alpha }_{1}}}\,{{r}_{t}} \\ & {{\sigma }^{2}}(t,{{r}_{t}})=\sigma^2r_t=\underbrace{0}_{\beta_0}+\underbrace{{{\sigma }^{2}}\,}_{\beta {{}_{1}}}\,{{r}_{t}} \\ \end{align} Now, if $\gamma\ne\frac{1}{2}$,then C.I.R Process doesn't belong the class of affine diffusion processes(please check yourself) and the discounted characteristic function is not of the following form $$\Phi(\phi,r_t,t,T)=e^{A(\phi,\tau)+B(\phi,\tau)r_t}$$ Consequently, conditional distribution of on $r_t$ doesn't follow a non-central chi-square distribution.
1. If $\gamma>\frac{1}{2}$ e.g $\gamma=\frac{6}{7}$ then Feller's Condition holds for any value of $\kappa$ and $\theta$ (we know $\kappa,\theta>0$)
$$\underset{{{r}_{t}}\to 0}{\mathop{\lim }}\,\,\left( \kappa (\theta -{{r}_{t}})-\frac{1}{2}\frac{\partial }{\partial r}(\sigma\,r_{t}^{\,\gamma})^2 \right)=\kappa \theta>0$$ In other words, $r_t$ is always positive and this is inconsistent with financial Modeling. Also,if $\gamma<\frac{1}{2}$ then
$$\underset{{{r}_{t}}\to 0}{\mathop{\lim }}\,\,\left( \kappa (\theta -{{r}_{t}})-\frac{1}{2}\frac{\partial }{\partial r}(\sigma\,r_{t}^{\,\gamma})^2 \right)\rightarrow-\infty$$ In other words, $r_t$ is always negative and this is inconsistent with reality.