If $W_t$ is a Wiener process, what is $ \operatorname{Var}(W_t+W_s)$?

I was staring at this for a while before realizing your solutions are equivalent. For example, if $s < t$, then $$ \begin{aligned} t + s + 2 \min(s, t) &= \max(s, t) + \min(s, t) + 2 \min(s, t) \\ &= \max(s, t) + 3 \min(s, t). \end{aligned} $$


Your solution is correct - and so is theirs. This is because $$t+s+2\mathrm{min}(s,t) = \mathrm{max}(s,t) + 3\mathrm{min}(s,t).$$

What I'm wondering about is how they got to that form.


Nothing is wrong -- since $t+s = \max(s,t) + \min(s,t)$ your solution equals the book's.

It may simplify the calculation to assume WLOG that $t<s$ -- then $W_t+W_s=2W_t+(W_s-W_t)$, and $W_t$ and $W_s-W_t$ are independent.

Tags:

Stochastic Calculus

Stochastic Processes

Brownian Motion

Related

How to prove that $\int_{0}^{\infty}\ln^2(x)\sin(x^2)dx=\frac{1}{32}\sqrt{\frac{\pi}{2}}(2\gamma-\pi+\ln16)^2$ Why is the $C^\infty(M)$-module of smooth sections of a vector bundle $E$ not free? If $p,q$ are distinct primes and $a$ is not divisible by $p$ or $q$, then $\gcd(a, pq)=1$ Number of ways to arrange word 'KBCKBCKBC' $ I(r) = \int_0^{2\pi}\frac{\cos(t) - r}{1 - 2r\cos t + r^2}\,dt$ is always zero for $r\in[0,1)$. Why? Is there a possible geometric method to find length of this equilateral triangle? Finding sum of none arithmetic series Intuition for the Stone-Čech compactification via ultrafilters On the evalution of an infinite sum On $A$ algebra homomorphisms $A[[X_1,...,X_n]]\to Q(A)$, where $A$ is a complete DVR If $f(x)=\int_{0}^{x}\sqrt{f(t)}\,dt$ then $f(6)$ is? $\sum_{n=1}^\infty a_n \cos nx$ unbounded near $0$ if $\sum a_n$ diverges?

Recent Posts