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Apply the Itô formula to $f(t,x)=x_0{e}^{\sigma x+(\mu -{\sigma }^2/2)t}$ and the semimartingale vector $(t,B_t)$. Its derivatives giveThereforeThis is adapted to the given Brownian filtration and is consequently a strong solution; it is geometric Brownian motion.

A power diffusion has a power of its positive state as noise coefficient. Here the normalized zero-drift family is considered on $U=(0,\infty )$ with $X_0>0$. Its coefficients are locally Lipschitz continuous on that domain, so a maximal local solution of a stochastic differential equation is pathwise unique before its lifetime. The Lamperti transform reduces its noise coefficient to a constant; the case $\alpha =1$ is geometric Brownian motion. This normalized family is part of the constant-elasticity-of-variance models, but is not the whole model with arbitrary drift and scale.

The difference of the terminal power-call and power-put payoffs is $S_T^{\beta }-K^{\beta }$. Risk-neutral pricing and the geometric Brownian motion moment formula therefore give the displayed parity, with ${\lambda }_{\beta }=\beta (r-q)+{\sigma }^2\beta (\beta -1)/2$. At $\beta =1$ this reduces to dividend-adjusted put-call parity.

For zero cash interest, constant excess drift $\mu$, volatility $\sigma$, and both terminal and consumption utility $2\sqrt{x}$, set $A=\sigma {\sigma }^T$ and assume $\mu$ belongs to the image of a linear map $A$. Let $A^{+}$ be the Moore-Penrose inverse and $q={\mu }^T{A}^{+}\mu$. Then $h^{\prime }+qh+1=0$, $h(T)=1$, so $h(t)=e^{q(T-t)}+(e^{q(T-t)}-1)/q$ for $q>0$, and $h(t)=1+T-t$ for $q=0$. The Hamilton-Jacobi-Bellman equation has the displayed solution. Maximizing dollar investment and consumption are ${\theta }_t=2X_t{A}^{+}\mu$ and $C_t=X_t/h(t)$. The resulting positive portfolio wealth is a geometric Brownian motion with a deterministic time-dependent drift, and finite-horizon moment bounds justify equality in verification.

For a square-root stock claim priced by $C(t,T)=\mathbb{E}[\sqrt{S_T}\mid {\mathcal{F}}_t]$, the constant-volatility geometric Brownian motion price is $\sqrt{S_t}\exp (-{\sigma }^2(T-t)/8)$. If $t<T$, $S_t>0$ and $0<C(t,T)\le \sqrt{S_t}$, its unique nonnegative inverse volatility is the displayed $\Sigma (t,T)$. For bounded predictable volatility in $dS=S\sigma dW$, bounds $0\le a\le \sigma \le b$ give $a\le \Sigma \le b$, without independence from the Brownian motion. At zero stock price the claim price is also zero and no volatility is identified by this equation; a separate convention is needed.