Continuity of the stochastic process $X_t=\int_{0}^{t}(a+b\frac{u^n}{t^n} ) dW_{u} $ I am wondering about the continuity of the stochastic process 
$$X_t=\int_{0}^{t}(a+b\frac{u^n}{t^n} ) dW_{u} $$
Where n=1,2,..
At t=0, there seems to be a discontinuity except for b=0 . Is there a discontinuity? 
 A: This problem comes down to showing the continuity of
$$\frac{1} {t^n}\int_0^t u^n dW_u $$
From  Itô's formula, we have
$$\frac{1} {t^n}\int_0^t u^n dW_u =W_t-\frac{n}{t^n}\int_{0}^{t}W_u u^{n-1}du$$
So essentially we have to evaluate the limit 
$$\lim_{t\to 0} \frac{1}{t^n}\int_{0}^{t}W_u u^{n-1}du$$
Where n is any natural number 
Since $W_u u^{n-1}$ is continuous a.s thus from the fundamental theorem of calculus and L'Hôpital's rule, we have 
$$\lim_{t\to 0} \frac{1}{t^n}\int_{0}^{t}W_u u^{n-1}du=\lim_{t\to 0}\frac{W_{t}t^{n-1}}{nt^{n-1}}=\lim_{t\to 0}\frac{W_{t}} {n} =0$$
a.s
Thus $$X_t$$ is continuous a.s
More generally for the stochastic process
$$Y_t=\int_{0}^{t}f(u,t)dW_{u}$$
Where f is of the form 
$$f(u, t) =\frac{b(u)} {g(t)} $$
From  Itô's formula, we have
$$f(t, t)W_{t} - f(0,t)W_{0}-\frac{1}{g(t) }\int_{0}^{t}\frac{db} {du} W_{u} du$$
If $\frac{db} {du}$  is continuous then
$$\lim_{t\to 0} \frac{1}{g(t) }\int_{0}^{t}\frac{db} {du} W_{u} du=\lim_{t\to 0}\frac{b' (t)} {g'(t)} W_{t} $$
Thus sufficient(but not necessary)  conditions for continuity at t=0 are 
$$f(t, t)$$ is continuous
$$f(0,t)$$ is continuous 
$$\frac{db} {du}$$  is continuous
$$\lim_{t\to 0}\frac{b' (t)} {g'(t)}$$ exists and is finite 
Bonus:Check this for $$f(u, t) =\frac{\sin(u)} {t}$$ 
For the stochastic process $X_t$ given in the question, certain a,b can make it a Wiener process
We have already established $X_0=0$  a.s.
Note that X_t has mean 0 and variance
$(a^2+\frac{b^2}{2n+1}+\frac{2ab}{n+1})t$
and it is normally distributed since the limit of a sequence of normal random variables is also a normal random variable.
For its variance to be t 
 $(a^2+\frac{b^2}{2n+1}+\frac{2ab}{n+1})=1$
Also note that since $X_t-X_s$ and $X_s$ are jointly normally distributed(any of their linear combination is individually normally distributed)  and their covariance vanishes for
$b=0$
and
$\frac{a} {n+1}+\frac{b}{2n+1}=0$
These equations determine the a, b e.g two of the solutions are
$a^2=1, b=0$
Hence $X_t-X_s$ is independent of $ X_s$ for these a, b. 
Now since 
$|X_t-X_s|=|t-s|^{\frac{1}{2}}Z$
Where Z is the standard normal random variable and the equality is in distribution. Then
$E|X_t-X_s|^{2m}=C_{m} |t-s|^{m} $
Thus from the Kolmogorov continuity theorem $X_{t} $ is a.s $<\frac{1}{2}$ holder continuous for those a, b. 
These conditions make $X_t$ a Wiener process for those a, b. 
