0
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1answer
24 views

(Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...
0
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1answer
22 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
0
votes
0answers
14 views

Convergence of the distribution of a GBM at a random time when time converges in probability

I have got the following question. Let $(S_t)_{t\in[0,T] }$ be a geometric Browninan motion. Consider a sequence of bounded random variables $(\tau_n)_{n\in\mathbb N}$ such that $\tau_n\downarrow ...
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votes
1answer
26 views

convergence of Ito integral

Suppose there is a deterministic process $\phi$ in $L^2(R)$. Need to prove that $\int_0^n \phi_u dW_u$ converges in $L^2(P)$ to some $X\in L^2(P)$ as $n\rightarrow\infty$. Also need to show that ...
0
votes
1answer
97 views

On the quadratic variation

I understand that the Quadratic Variation of Brownian Motion $B_t$ is $[B_t,B_t]=t$ and I know that the equality is under the meaning of $\mathcal{L}^2$ convergence. Yet I saw in some book saying that ...
1
vote
1answer
144 views

Laplace Transform of a Brownian motion

If $v(\omega,t) : \Omega \times [0,\infty) \to \mathbb{R}$ is a Standard Brownian motion, then for what values of $s,\omega$ does the Laplace transform $l(\omega,s) = \int_0^\infty e^{-st} v(\omega,t) ...
1
vote
1answer
99 views

Convergence to Brownian motion integral

Let $X_i$ be i.i.d with $\mathbb{E}(X_i) = 0$ and $Var(X_i) =1, \, S_n = \sum_{i=1}^n X_i$. I would like to show that $\sum_{i=1}^n \frac{f(S_i/\sqrt{n})}{n}$ converges to $\int_0^1 f(B_t)dt$ in ...
1
vote
1answer
478 views

Convergence of the exponential martingale

How can we show that this martingale $$ e^{aW_{t} - \frac{1}{2}a^2t}$$ converges to $0$ as $ t \rightarrow \infty$ using law of iterated logarithm, for $a \neq 0$.