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Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite Let $(W_s)_s geq 0$ be a Wiener process and $tau$ be a random variable with an exponential distribution with parameter $lambda$. Suppose that $W$ and $tau$ are independent. In this question, we see that the distribution of the stopped Wiener process $W_tau$ corresponds to a Laplace distribution with scale parameter $fracsigmasqrt2lambda$ where $sigma$ is the instantaneous variance of the Wiener process. Suppose now, that the variance $sigma^2$ also follows a stochastic process such that we have: $$dS = sigma SdW_s \ dsigma^2 = alphasigma^2dt + xisigma^2dW_sigma$$ where $alpha$ and $xi$ are independent of $S$ and $dW_s$ and $dW_sigma$ are independent Wiener processes. My aim is to derive the distribution of $log S_tau$. First, if $barV_T$ denotes the mean variance over some time interval $[0,T]$ defined by $$barV_T = frac1T intlimits_0^Tsigma^2(t)dt,$$ it is easy to show (Lemma of Îto) that $$log S_T = log S_0 -

Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite Let $G$ be a group of order $p^n$. I have to show that there are normal subgroups of $G$ of orders $1,p,...,p^n-1,p^n$. My attempt : The trivial subgroup is a normal subgroup of order $1$. Assume $G$ has a normal subgroup $N$ of order $p^k,k<n$. To show: $G$ has a normal subgroup of order $p^k+1$. Let $gamma : G rightarrow G/N$ be the natural homomorphism. The kernel is $N$, whose size is $p^k$. So $gamma$ is $p^k-textto-1$. Since $k<n$, $G/N$ is also a $p$-group. Hence, $G/N$ has a non-trivial centre. $Z(G/N)$ is then also a $p$-group. So by Cauchy's theorem, it has an element $a$ of order $p$. The subgroup $langle arangle$ generated by this element is a subgroup of $Z(G/N)$. Therefore, it is normal in $G/N$. Then the pullback of $langle arangle$ is normal in $G$. Since the order of $langle arangle$ is $p$, the order of $gamma^-1langle arangle$ is $p^k×p=p^k+1$. Thus we have found a normal subgroup o

Clash Royale CLAN TAG #URR8PPP up vote 0 down vote favorite I am interested in solving a definite double integral of the following form: beginalign f(a,b) &= int_0^infty expBig(frac-x^22aBig)int_x^infty expBig(frac-y^22bBig) dydx\ &= int_0^infty expBig(frac-x^22aBig)texterfcBig(fracxsqrt2bBig)dx, endalign for $a,b>0$, where erfc is the complementary error function. One potential way to go would be to use a power-series expansion (e.g., see answer by robjohn to this question or this paper), but I'm finding that a bit difficult to follow. I'm wondering if anybody has any ideas about ways to get an approximate answer. For now, I'm just trying to see if I can fit the numerical solution with a function of $a$ and $b$. definite-integrals normal-distribution gaussian-integral error-function share | cite | improve this question asked Aug 21 at 14:09 funtoast 16 2 add a comment  |  up vote 0 down vote favorite I