# Tagged Questions

Questions related to Brownian motion, a continuous stochastic process denoted by $W_t$, $t\geq 0$, with independent increments, such that $W(t)-W(s)$ is normally distributed, with $0$ mean and variance $t-s$.

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### How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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### derive integration by parts for a stochastic integral

The question is to show the following identity: $\int_{0}^{T}tdW(t) = TW(T)-\int_{0}^{T}W(t)dt$ This can be done quite easily with ito's however the question explicitly says to show the identity ...
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Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\lambda$ be the Lebesgue measure on $[0,\infty)$ $\mathcal D:=C_c^\infty([0,\infty))$ and $\mathcal D'$ be the dual space of $\... 2answers 45 views ### How can we prove that the generalized stochastic process induced by a real-valued Brownian motion is Gaussian? Let$(B_t)_{t\ge 0}$be a real-valued Brownian motion on a probability space$(\Omega,\mathcal A,\operatorname P)$,$\lambda$be the Lebesgue measure on$[0,\infty)$and $$\langle W,\phi\rangle:=\int\... 0answers 16 views ### Covariance functional of a generalized real-valued Brownian motion Let (B_t)_{t\ge 0} be a real-valued Brownian motion on a probability space (\Omega,\mathcal A,\operatorname P), \lambda be the Lebesgue measure on [0,\infty) and$$\langle W,\phi\rangle:=\int\... 0answers 34 views ### Distribution of “range” of a process Let$X_t$be a stochastic process, for example a brownian motion (i.e.$X_{t+h} - X_t \sim \mathcal{N}(0,\sqrt{h}^2)$). The difference between now's value$X_t$and a past value$X_{t-100}$is $$M_t ... 0answers 47 views ### Why do people all the time exploiting almost sure properties of a stochastic process as if they were sure properties? All the time, I see people working with a given Brownian motion (B_t)_{t\ge 0} on a fixed probability space (\Omega,\mathcal A,\operatorname P) and suddenly exploiting its almost sure properties ... 1answer 31 views ### Expectation of a generalized real-valued Brownian motion Let (B_t)_{t\ge 0} be a real-valued Brownian motion on a probability space (\Omega,\mathcal A,\operatorname P), \lambda be the Lebesgue measure on \mathbb R and$$\langle W,\phi\rangle:=\int_{[... 1answer 27 views ### Is$\phi B(\omega,\;\cdot\;)$Lebesgue integrable over$[0,\infty)$for a real-valued Brownian motion$B$and$\phi\in C_c^\infty(\mathbb R)$? Let$(B_t)_{t\ge 0}$be a real-valued Brownian motion on a probability space$(\Omega,\mathcal A,\operatorname P)$and$\lambda$be the Lebesgue measure on$\mathbb R$. Is $$W(\phi):=\int_{[0,\infty)}\... 0answers 46 views ### Distance between Brownian Motion and scaled Gaussian random walk I'm currently reading this paper: http://user.math.uzh.ch/barbour/pub/Barbour/SteinDiffusion.pdf and in equation (2.26) the author uses the following fact: If Z(t) is a standard Brownian Motion and ... 1answer 25 views ### how to show that definition for stochastic process in continuous time applies to stock prices I know that the formal definition of a stochastic process is: {X(t,\omega)\,\,t\ge0} is a stochastic process if: For any fixed t\ge0, X(t,\omega) is a random variable For any fixed \omega ... 0answers 38 views ### Why is E(X_t|B_t)=\frac{E(X_tB_t)}{E(B_t^2)}B_t? Why is E(X_t|B_t)=\frac{E(X_tB_t)}{E(B_t^2)}B_t ? Does this always hold In an exercise I have to show that E(X_t|B_t)\neq X_t, where X_t=\int_0^t B_s ds, I think the definition of X_t does ... 1answer 36 views ### Compute \mathbb{E} [W(t_1)W(t_1 + t_2)W(t_1 + t_2 + t_3)] when W is a Brownian motion Let (W(t))_{t \geq 0} be standard Brownian motion, and let t_1, t_2, t_3 \in \mathbb{R}_{> 0} with t_1 < t_2 < t_3 be arbitrary. Compute:$$ \mathbb{E} [W(t_1) * W(t_1 + t_2) * W(t_1 +... 1answer 50 views ### Computing expectation of brownian motion I need to compute the following:$E\left[ B_t \int_0^tB_s^2 \, ds \right]$for$t≥0$Where$B_t$is a standard Brownian motion. I'm thinking this is really obvious, But I cannot get my head round ... 0answers 26 views ### Wiener's construction of the Wiener Measure I am writing an essay about Norbert Wiener and I already have sufficient info about him in general and his history, but now I would like to know how he constructed the Wiener measure. I found some ... 0answers 42 views ### Brownian motion, harmonic functions and the Dirichlet problem I am having trouble understanding one detail of the standard use of Brownian motion to solve the Dirichlet problem, I will write the statement and proof and then point to the detail I don't ... 1answer 31 views ### An issue of dependent and independent random variables involving geometric Brownian motion. Let$X(t)=X(0)e^{\mu t + \sigma Z(t)}$be a geometric Brownian motion (GBM) where$Z(t)$is the standard Brownian motion with drift$0$and the variance rate per unit of time is$1$. Now, let$s<t$... 1answer 54 views ### Iterated logarithm law for difference (supremum(W) - infimum(W) ) is it 2srt(2/pi) sqrt(t loglog(t))? Law of iterated logarithm says that $$\sup(W(t)) \sim \sqrt{2 t \log(\log(t))}.$$ Consider$\sup(W(t)) - \inf(W(t))$my guess based on numerics that it should be $$2\sqrt{\dfrac 2\pi} \sqrt{t \... 0answers 54 views ### Simulation of brownian motion and fractional brownian motion It's easy to simulate a path of a brownian motion with the method explained in Wiener process as a limit of random walk: ... 1answer 81 views ### Exercise 8.12 Introduction to stochastic processes Gregory Lawler [closed] Let X_t be a standard Brownian motion starting at 0 and let T=min \{t:|X_t|=1\} and \hat{T}=min \{t:X_t=1\} (a) Show that there exist positive constants c, \beta such that for all t>0,... 0answers 30 views ### Construction of Wiener Process using integral of covariance multiplied by a function I read in the notes of Stochastic Processes that there is a construction of Wiener Process (knowing that Cov(W_s, W_t)=min(s,t) ) which going like this: consider operator Q on C([0,1])$$Qf(t)= ... 0answers 27 views ### How to evaluate the expectation of the exponential of reflected brownian motion How do you compute this expectation$\mathbb{E} \left [ e^{\varepsilon|W_t|} \right] $where$W_t$is a Brownian Motion Do I need to expand the absolute value? Can I use the standard Taylor series ... 0answers 63 views ### Showing that$P(W_{t}/\sqrt{t \log(t)}>1+\epsilon)\to0$when$t\to\infty$, where$(W_t)$is a Wiener process I have a question about the martingales$\dfrac{e^{W_{t}^2/(1+2t)}}{\sqrt{1+2t}}$. With use of this martingale I want to show that$P(\dfrac{W_{t}}{\sqrt{t log(t)}}>1+\epsilon)$goes to$0$if$t$... 1answer 23 views ### Mean time for the trajectory. Find mean What is the mean of time when the trajectory of the wiener process,$W_t$, is over the line$y=t$? We need to find$\Bbb{E}\tau$, where$\tau=\sum\limits_{a,b:\forall t\in(a,b) ; W_t>t}\left(b-a\...
$X_t = e^{B_t-\frac{1}{2}t^2}$ I need to find $[X]_t$, the quadratic variation process. I have tried to solve the problem and my main question is whether this approach is correct or not. Given ...