# 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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### System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
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### Compute $\int_1^2 B_t \; dB_t$

I have to compute the following Ito integral: $$\int_1^2 B_t \; dB_t$$ where $(B_t)_{t \geq 0}$ is the 1-dimensional Brownian Motion. In the definition of Ito integral, the integral is taken from $0$ ...
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Let $B_t$ be a brownian motion. and let $\{W_t=B_t-tB_1:0\le t\le 1\}$ be a brownian bridge. Now let $Y_t=(1+t)W_{t\over 1+t}$. Proof that $Y_t$ is a brownian motion in $[0, \infty)$ My attempt: 1) $... 1answer 37 views ### Verifying a Brownian motion through the Laplace transform Let$X(t)$be a continuous stochastic process and$\mathcal G(t)$be the$\sigma$-algebra generated by$\{X(\tau) : \tau\leq t \}$. Suppose that for any$0\leq s\leq t$and$\lambda\in\mathbb C$... 0answers 25 views ### Is$B_{t\wedge H_a}$bounded in$L^2$? Let$a >0$,$(B_t)_{t\geq0}$be a standard Brownian motion. Define the stopping time $$H_a := \inf\{t \geq 0 : B_t \geq a\}.$$ Then is the martingale$M_t$where$M_t: = B_{t\wedge H_a}bounded ... 0answers 23 views ### Brownian Motion maximum process intuition I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. ... 0answers 32 views ### Brownian motion independent RVs Let(W_t)_{t\in\lbrack 0,T\rbrack}$be a standard Brownian motion. Does there hold that$W_s(W_t-W_s)$and$W_k(W_l-W_k)$for$0\leq s<t\leq k<l\leq T$are independent RVs? 1answer 30 views ### What is the difference between these two formulas that price a stock? [closed] What is the difference between these two formulas? They are both related to the price of a stock in the black-scholes model. The fact that the second one uses$tas a subscript which means it's not a ... 0answers 24 views ### Brownian motion - absolute value I'm having some trouble integrating the equation in 8.2.5 (I'm trying to do 8.2.6). I need to do some form of u-substitution but I'm unsure of u=?. Also, once I've done the integration, to show that ... 0answers 26 views ### Reflected Brownian Motion probability So I know that R(t) = |5 + B(t)| and that B(25) ~ N(0,25). I was told that P{R(t)>=10} = P{|5+B(25)|>=10} = P{B(25)>=5)+P{B(25)<=-15} but I'm not entirely sure how to get that. And I've been ... 0answers 41 views ### Brownian Moment Generating Function and Hitting Times Here is my question. I've done the first part, but I'm stuck on the second. If I can work out (/be advised) how to do the second, then I hope to be able to do the third similarly. Please note: While ... 2answers 91 views ### Crossing of Brownian Motion Sample Paths I would like to ask for a more rigorous statement and proof of Lemma on page 5 of this paper. In essence, it states that two distinct sample paths of a Brownian motion does not strictly cross (meaning ... 0answers 28 views ### What is the general Taylor Expansion for the following function of a function. guys. I am stuck with a general form of Taylor Expansion of following function, which is defined as a function of a function: $$F(X(t+h))-F(X(t))=[X(t+h)-X(t)]\frac{dF}{dX}(X(t))+\\\frac{1}{2}[X(t+... 0answers 39 views ### Gaussian processes and bias I would like to simulate two Gaussian processes along a time grid. Ideally, I would like to see the average of the samples, for each grid point, to be close to the mean. Using the antithetic method, I ... 2answers 63 views ### How to prove that the stochastic integral process is gaussian? I would like to prove that for a C^1-function f and a Wiener process W, the integral process defined by$$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$Is a centered gaussian ... 0answers 37 views ### Survival probability of a biased random walker A random walker moves to +1 with probability p and moves to -1 with probability q=1-p. If he starts at point m, what is the probability that he doesn't hit the point zero after k steps, ... 1answer 38 views ### How to find the standard deviation from the given information and what is B(0) equal to? Assume that the risk free rate is 0 and that the stock price is given by the equation S(t)=6e^{2t+2B(t)} where B(t) is the standard Brownian motion. Determine the price at time 0 of the ... 1answer 52 views ### Maximum process of Brownian motion Consider the linear standard Brownian motion (B_t)_{t \ge 0}. We define the maximum process (M_t)_{t \ge 0} of (B_t)_{t \ge 0} to be such that M_t = \max_{0\le s \le t} B_s. Prove that the ... 0answers 38 views ### How it is shown by the following integral? Example: Ornstein-Uhlenbeck Process. Let dx=-\eta xdt+\sigma dz be an Ornstein-Uhlenbeck Process Write the moment-generating function for x(t) as$$ M(θ,t)≡E(e^{-θx})=∫_\infty^∞ ϕ(x_0,t_0;x,... 0answers 27 views ### Verifying data came from a Wiener Process From the Wiki article a Wiener Process has the properties that $$E[W_t] = 0$$ $$Var[W_t] = t$$ According to A Standard Wiener Process the Wiener Process is given by: W(t) - W(s) \tilde{} \sqrt{t-... 1answer 42 views ### Limit Brownian Bridge Integral As a solution of the Brownian Bridge SDE, we arrive at the solution \begin{align} X_t = (1-t) \int_0^t \frac{1}{1-s}\ dB_S \end{align} defined for 0 \leq t <1. In order to show that for any g \... 1answer 34 views ### Wiener process and stochastic int Let h:[0,1] \rightarrow \left\{-1,1 \right\}. How to show that X_t=(\int_0^th(s)dW_s)_{t \in [0,1]} is a Wiener process? I know from the lecture that for every h process \int h \ dW_s is ... 0answers 37 views ### n times integrated Brownian motion martingale process According to this post, we found that a n times integrated Brownian motion could be expressed as, \begin{align} V_n(t) = \int_0^t V_{n-1}(s)\ ds = \frac{1}{n!} \int_0^t (t-s)^n\ dB_s, \end{align} ... 0answers 49 views ### Find P(B_3>0,B_6>0) where (B_t) is a Brownian motion Suppose that B_{t} is a standard Brownian Motion. What is the probability that both B_{3} and B_{6} take positive values? This is what I've tried but then I get stuck and I'm not sure how to ... 2answers 32 views ### Integration by parts - Brownian motion and non-random function Let B be a standard one-dimensional Brownian motion. I want to show for a continuously differentiable non-random function \phi that, \begin{align} \int_0^t \phi(s) dB_s = \phi(t) B_t - \int_0^t ... 1answer 22 views ### Derivation of a property of standard Wiener processes I am reading A Standard Wiener Process and am struggling to piece together how they arrived at their conclusion. The major properties of any Wiener Process are: W(t) = 0 W(t) - W(s) \sim N(0, t-... 2answers 173 views ### Is it Possible to Construct all Proofs in Complex Analysis using Brownian Motion? (First, I am very aware of the fact that Brownian motion is actually probably more difficult to understand than at least basic complex analysis, so the pedagogical merits of such an approach would be ... 0answers 60 views ### Brownian Motion Third Power Martingale using Ito Integral Let (B_t)_{t \geq 0} be a standard Brownian motion and M_t = B_t^2 - t. According to this and this posts we know that \begin{align} [M] = [B^2] = 2 \int_0^t B_s^2\ ds. \end{align} Now, without ... 0answers 24 views ### covariance and expectional in proccess Show that the process X=(W_{\sqrt{t}}I_{(1,2)}(t))_{t \ge 0} \in \mathcal{L}_3^2. (W- Wiener) Additionally calculate, for t,s \in [1,2], EX_t and Cov(X_t,X_s) I have no idea how to start ... 2answers 82 views ### Integral of Wiener Squared process I don't have a background of stochastic calculus. It is known fact that definite integral of standard Wiener process from 0 to t results in another Gaussian process with slice distribution that ... 1answer 38 views ### Marginally Gaussian not Bivariate Gaussian - Ito Integral Let (W_t)_{0\leq t\leq 1} be a Wiener process defined up to time 1 on some probability space. Consider the random vector\left(W_{1},\int_0^1 \operatorname{sgn}(W_s) \, dW_s\right)=:(W_1,X_1)$$... 0answers 26 views ### Compute the Mean of the Following Process Given the following process: \Delta \ln(St+1)= \mu - (\sigma2/2) + \sigma(\varepsilon(t+1)) (where both \mu and \sigma squared are of S) How does one calculate the mean of S(t+1)/S(t)? (... 1answer 27 views ### Expected location of Brownian motion on the circle Intuitively it seems likely that the expected whereabouts of Brownian motion on the unit circle would be the origin \left(0,0\right), at least in the limit as t\to\infty. Is this right? Are there ... 1answer 65 views ### Covariance of stochastic integral I have a big problem with such a task: Calculate \text{Cov} \, (X_t,X_r) where X_t=\int_0^ts^3W_s \, dW_s, t \ge 0. I've tried to do this in this way: setting up t \le r$$\text{Cov} \, (... 0answers 41 views ### Probability that Brownian motion falls between two piecewise constant functions I'll first present the problem, and then describe my motivation: Supposea_j \in \mathbb{R}$,$b_j \ge 0$, and$0 = t_0 < t_1 < \cdots < t_J$are time points. Let$W_t$be a standard ... 1answer 14 views ### Show that$(B_t)$and$(tB_{1_t})$has the same distribution where ($B_t)_t$is a brownian motion Let$(B_t)_{t\geq 0}$a brownian motion s.t.$B_0=0$. I want to show that$B_t$and$tB_{1/t}$has the same law. $$p\{tB_{1/t}\leq x\}\underset{u=1/t}{=}p\{\frac{1}{u}B_u\leq x\}=p\{B_u\leq ux\}$$ ... 2answers 44 views ### Mean and Variance Geometric Brownian Motion with not constant drift and volatility I have to derive the Geometric Brownian motion (with not constant drift and volatility), and to find the mean and variance of the solution.$\quad \left\{\begin{aligned} & d X_t = \mu(t) X_t d t +...
How to show that the process $X_t=tW_t - \int_0^t W_s ds$ is a martingale? I guess I have to use the definition of martingale and properties of Wiener process, but I stack with this integral. Please,...