Originally Posted by

**Juju** Hi,

is the following sequence of functions $\displaystyle (f_h)_{h \in [0,1]}$ uniformly convergent for $\displaystyle h \to 0$?

$\displaystyle f_h(a):=\left(1+a(\mu-r)h+\frac{1}{2}a^2 \gamma(1-\gamma)\sigma^2h+Ch^\frac{3}{2})\right)^\frac{1}{h }$

whereas

$\displaystyle \gamma \in (0,1)$ is a constant

$\displaystyle C, \mu \in \mathbb{R}$ are constants

$\displaystyle \sigma,r >0 $ are constants

I think the pointwise limit is given by

$\displaystyle \lim\limits_{h \to 0}f_h(a)=e^{a(\mu-r)+ \frac{1}{2}a^2 \gamma(1-\gamma)\sigma^2}.$

Is this correct? Yes.

Does $\displaystyle (f_h)_{h \in [0,1]}$ also converge uniformly to $\displaystyle f$? Uniform convergence is only defined if the domain of the functions is specified. If you want the variable a to run through the whole real line, then the convergence is not uniform. But I think that the functions will converge uniformly on any bounded interval.