# 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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I am stuck on an exercise in my book: The question relies on the following fact: Let $M$ be a continuous, non-negative local martingale such that $M_0=1$ and $M_t \rightarrow 0$ almost surely as $... 0answers 54 views ### What is the resulting stochastic process of divided Geometric Brownian motions Let$W_{1,t},W_{2,t},...,W_{n,t}$be$n$independent geometric Brownian motions. Now let's say I construct the following processes: $$X_1 = \frac{W_1}{\sum_i^n W_{i,t}}$$ $$X_2 = \frac{W_2}{\... 2answers 112 views ### A variation of Lévy's characterization of Brownian motion It is shown here, without using stochastic calculus, that if W_t is a standard Brownian motion, then$$ f(W_t)-\frac{1}{2}\int_0^t f''(W_s)ds $$is a martingale, where f\in C^2 and compactly ... 2answers 189 views ### integral of exponential of Brownian motion I am currently reading a proof that uses the following fact without proof: If B is a scalar standard Brownian motion, then \int_0^\infty e^{B_s} \,ds = + \infty a.s.. How can we justify this ... 0answers 40 views ### Extension of Law of Iterated Logarithms Suppose I have a stochastic differential equation (X_t is a vector) dX_t = f(X_t) dt + \sigma g(X_t) d\eta(t) and define V = \sum_{i=1}^{n} x_i. Here, \eta(t) is an Ornstein-Uhlenbeck process. ... 1answer 217 views ### Geometric brownian motion - Ito's lemma I have a question about geometric brownian motion. dS = uSdt + /sigma/SdW and then we do log(S) and we want to found dlog(S). So we use Ito's lemma en I get the dt part of the lemma but I don't see ... 1answer 73 views ### Examples of Wiener Martingales (X_t,\mathcal{F}_t) is called a Weiner martignale if i) X_t is a Wiener Process ii) (X_t,\mathcal{F}_t) is a martingale. (Here \mathcal{F}_t is an increasing \sigma-field family). Let (\... 1answer 56 views ### Finding b such that e^{5B_t - bt} is a martingale I have X_t = e^{5B_t} and Where B_t is brownian motion at time t. M_t = X_t \cdot e^{-bt} I need to find a value for b such that M_t is a martingale. I am encountering difficulty, ... 0answers 24 views ### Scaled distribution of Brownian motion If I have X = 5(B_t - B_s) Does this have a distribution of \sim \text{N}(0,25(t-s)) ? Since B_t - B_s has distribution \sim \text{N}(0,t-s) Then X = \mu \cdot 0 + \sigma_1 Z where Z \... 2answers 62 views ### Differential and Differential Equation - Difference in meaning? I am a little confused, an exercise by a teacher has been set which says: For X_t = 2e^{B_t} Where B_t is brownian motion at time t. a) Find the stochastic differential d(X_t) b) Find the ... 1answer 179 views ### Show that f(W_t)-\frac{1}{2} \int_0^t f''(W_s) \, ds is a martingale without using Itô's formula I'm learning the basics about Brownian motion (I know nothing about stochastic calculus), and I've shown that if W(t) is a standard Brownian motion, then W(t)^2-t is a martingale. Now I'm trying ... 1answer 101 views ### How to compute stochastic integral: \int_0^t d(B_s^2) Here, B_t is Brownian motion at time t What property is used to compute the integreal \int_0^t d(B_s^2)? Shouldn't there be some other variable attached with the differential d ? 1answer 83 views ### Is Brownian Motion increasing? Given a process Y_t = e^{B_t} We know that since Brownian motion is continuous for t \geq 0. Since B_t is a completely random motion, it is true that we cannot say whether it is monotone ... 1answer 68 views ### How to calculate \mathbb{E}((B_3-B_2)(B_4-B_{\pi}) \mid B_1) for a Brownian motion (B_t)_{t \geq 0} I want to find the best predictor of (B_3-B_2)(B_4-B_{\pi}) given an observation of B_1 Where B_t is brownian motion for time t \geq 0. I am not sure how to approach this. I know it will be ... 0answers 26 views ### Best predictor of Brownian motion Let B_t be brownian motion at time t \geq 0. Then I want to find the best predictor of B_8 + 4 given that there are observations of brownian motion up to time t = 1. Approach: Essentially, ... 1answer 42 views ### Conditional expectation for linear combinations of Brownian motion X = \frac{ B_1+ B_3 - B_2}{\sqrt{2}} and Y = \frac{B_1 - B_3+ B_2}{\sqrt{2}} Where B_t Is brownian motion at time t\geq0 I want to find \mathbb{E} [Y + 3X | X] It is known to me that X, Y... 1answer 27 views ### Independence of two random variables derived from a Brownian motion If X = B_1 + B_3 - B_2 and Y = B_1 - B_3 + B_2 Where B_t is Brownian Motion for t \geq 0 And I want to state with certainty whether X and Y are indep or not, do I simply just \text{Cov}... 1answer 59 views ### Distribution of Brownian Motion help If X = \frac{B_1 - B_3 + B_2}{\sqrt{2}} Where B_t is brownian motion at time t. And I want to find the the distribution of X, how would I do so? E[X] = 0 is fairly straight forward. For ... 1answer 142 views ### Girsanov's theorem and absolutely continuous restrictions Let W be a Brownian motion on some probability space (\Omega, \mathcal{F}, P). Let \mathbb{F}^W be the filtration generated by W and let X be a process that is progressively measurable w.r.t.... 0answers 28 views ### Reference request for conditional and unconditional covariance of n-times integrated Brownian motion I'm working through an old Diaconis paper on Bayesian numerical analysis, and am currently calculating the details behind his brief comments on using n-times integrated Brownian motion as a function ... 2answers 108 views ### Martingale representation theorem application Let X = \exp(W_{T/2}+W_T). I try to figure the adapted process g(s) such that according to the MRT we have$$X = \mathbb{E}[X]+\int^T_0 g_s dW_s.$$I can figure out X = \exp(2W_{T/2}+W_{T-T/2}) ... 1answer 44 views ### proving independence of stochastic integrals Does anyone know how to show that the stochastic integrals $$\bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}}$$ are ... 2answers 42 views ### Distribution of \int^T_t \sigma (T-u)dW_u where W_t is a Brownian motion I am trying to find the distribution of \int^T_t \sigma (T-u)dW_u where W_t is a Brownian motion. One (very hand-wavey) way is to assume a priori that it is Normally distributed. Then one can ... 1answer 47 views ### Properties of brownian motion I was doing some revision and had an admittedly elementary question. My lecture notes say, the following are properties of Brownian Motion {B_t} (Normal or Gaussian increments) For all s < t, ... 1answer 228 views ### Convergence of exponential Brownian martingale to zero almost surely Define the exponential Brownian martingale as N_t = \exp\left\{a W_t - \frac12 a^2 t \right\} which is a martingale with respect to the natural filtration of W which stands for a standard Brownian ... 0answers 20 views ### Integral of Constant Parameter Martingale What is the \int_{1}^{t}W_1W_sdW_s. This is the question solved by Kuo in his paper an extension of the Ito's Integral (2008) but there limit runs from 0 instead of 1. 1answer 112 views ### Distribution of a transformed Brownian motion Let W be a standard Brownian motion. From an earlier proven result I know that N_t = \exp\left\{a W_t - \frac12 a^2 t \right\} defines a martingale on the natural filtration of W for all a \in \... 1answer 108 views ### n times integrated Brownian motion I have an identity that expresses the n times integrated Brownian motion and I would like to prove that. First, I define what I mean by n times integrated Brownian motion.$$V_1(t) = \int_0^tB_s\, ... 0answers 49 views ### Measurability of the event that Brownian motion hits a given set Let$W$be a Brownian motion in$\mathbb{R}^{2}$on a probability space$\left(\Omega,\mathcal{F},\mathbb{P}\right)$. Let us assume$\mathcal{F}$is the sigma-algebra on the path space$C([0,\...
I am looking for the solution of this problem: Consider a bounded domain $\Omega\subset\mathbb{R}^2$ and let $u(x,y)$ be the probability of exiting $\Omega$ starting at $p=(x,y)$, assuming that the ...