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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Ito's formula for this stochastic differential - please explain this step?

Referring to those two lines, can someone please explain how those results were obtained? My understanding is, the following formula is being referenced: $$dV_t = dV(S_t,t) = \frac{\partial ...
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1answer
53 views

Itô integral of an elementary process

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}=(\mathcal{F}_t,t\ge 0)$ be a filtration on $(\Omega,\mathcal{A})$ $H=(H_t,t\ge 0)$ be a stochastic process on ...
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2answers
252 views

Is the reflected Brownian Motion a Markov process

Let $W$ be a Brownian Motion (BM). The reflected BM is defined by $X=|X_0+W|$. We need to show that this process is a Markov process w.r.t. its natural filtration and we need to compute its ...
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46 views

Is this a self-financing portfolio?

I have $S_t = 10 + B_t$, $\beta_t = 1$, $a_t = 2B_t$, $b_t = -t - B_t^2 - 20B_t$ Then the value, $V = a_t S_t + b_t \beta_t$ Is this a self financing portfolio? Note, $B_t$ is brownian motion I am ...
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59 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even though its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
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1answer
21 views

Stopped brownian motion

Assume $B_t$ is a standard complex (or 2D if you wish) brownian motion and $\tau$ is a stopping time relative to $B_t$. I want to know if it is possible to construct another brownian motion $W_t$ such ...
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18 views

Liminf Brownian Motion question

For this assignment I'm working on, I was able to prove that: $$\limsup_{t \rightarrow \infty} \frac{B_t}{\sqrt t} = \infty$$ where $B_t$ is a Brownian Motion. I'd like to be able to prove: ...
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48 views

Relationship of SDE and Feynman-Kac PDE

I am struggling with this problem: Given a stochastic differential equation $$ dX_t = b(X_t) dt + \sigma (X_t) \,dW_t $$ where $W$ is a Brownian motion and the functions $b$ and $\sigma$ are ...
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2answers
46 views

Standard Brownian Conditional expectation

(Given the process $(B(t))t≥0$ of Brownian motion, define the random variables $$Y=\int_0^{1}B(s)\,ds $$ $$X=B(1) $$ Determine the quantities $E(Y|X)$, $Var(Y−E(Y|X))$ and the conditioned density ...
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1answer
41 views

Continuity of $x \mapsto E_{x}[F]$, Brownian motion

I have a question about Brownian motion. Let $(\Omega,\mathcal{F},P)$ be a Probability space and $(B_{t})_{t \in [0,\infty[}$ be a standard $1$-dimensional Brownian motion defined on ...
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1answer
84 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
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1answer
89 views

Local martingale but not martingale

On wikipedia there is an example of a local martingale which is not a martingale, but I do not understand why it is a local martingale. We have the process $ X_t = \begin{cases} ...
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1answer
98 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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1answer
47 views

Strong Markov Property and Product of Expectations

Let $(B_{t})_{t\geq0}$ be a Brownian motion and let $\tau=\inf\left\{ t\geq0:B_{t}\leq-4\right\} $ be a stopping time. Then the strong Markov property ensures that e.g. $A:=\left\{ ...
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0answers
36 views

If two Brownian motion starts and end at the same points, can we say something about there difference?

Let $X$ and $Y$ be two standard Brownian motions with mean $0$ and variance $1$, both started at zero. If we know that \begin{align} X_n &= Y_n, \end{align} for some $n>0$, can we say ...
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1answer
28 views

If $B=(B_t,t\ge 0)$ is a Brownian motion and $(\mathcal{F}_t,t\ge 0$ is its generated filtration, then $X_t-X_s$ are independent of $\mathcal{A}_s$

A real-valued stochastic process $B=(B_t,t\ge 0)$ is called Brownian motion $:\Leftrightarrow$ $B_0=0$ $B$ has independent and stationary increments, i.e. ...
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1answer
18 views

Brownian Motion Finding M(t)

If I have that {$B(t); t >=0$} is a standard Brownian motion, with $B(0)=0$, and I let $M(t)$ = max{$B(u) ; 0 \leq u \leq t$} and I am supposed to: a) Evaluate Pr{$M(4) \leq 2$} and b) Find the ...
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1answer
49 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
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0answers
41 views

Augmentation of a Filtration

In class, we showed that Brownian Motion is a martingale with respect to the filtration $F_t = \sigma(B(s): 0\leq s \leq t) $. For a HW assignment, I need to show it's a martingale with respect to a ...
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1answer
21 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
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1answer
51 views

Pathwise solution of a stochastic integral equation, without stochastic calculus

Let $f$ be a Lipschitz continuous function from $\mathbb{R}$ to $\mathbb{R}$ and $W$ be a standard Brownian motion. I don't know any stochastic calculus (nothing about stochastic integrals, nothing ...
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60 views

Density of running supremum of Brownian motion until a stopping time

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 ...
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0answers
41 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 = ...
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2answers
68 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 ...
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0answers
25 views

Levy process of argument in the complex plane

I am stuck on this question: Let $B$ be a Brownian motion in $\mathbb{C}$ started at $1$. Let $\theta_t$ be a continuous determination of the argument of $B_t$, i.e. $B_t = |B_t| e^{i \theta_t}.$ ...
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2answers
51 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 ...
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30 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. ...
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1answer
74 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 ...
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32 views

Variance estimation of a diffusion process

The framework of this question is a 1 dimensional diffusion process, defined ny the following equation: $dx_t=adt+bdw_t$ Where $w_t$ is a standard berownian motion and and $a$ is a constant drif ...
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1answer
40 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 ...
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1answer
31 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, ...
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19 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 ...
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2answers
56 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 ...
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110 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 ...
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21 views

Showing that $X_t = \int^{1/[X]_t}_0 f_u dW_u$ is a Brownian motion

Assume we have an Ito process $$ X_t = \int^t_0 f_u d W_u $$ where $f_u$ is a deterministic function of $u$ and $W_u$ is a Brownian motion adapted to $\lbrace \mathcal F_t \rbrace$. I want to show ...
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1answer
68 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 $d$ ?
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1answer
48 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 ...
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1answer
59 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 ...
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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, ...
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1answer
22 views

Conditional expectation and brownian motion - check my answer please

$X = \frac{ B_1+ B_3 - B_2}{\sqrt{2}}$ 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, ...
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1answer
23 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 ...
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1answer
38 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 ...
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1answer
89 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 ...
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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 ...
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2answers
88 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 $X = \exp(2W_{T/2}+W_{T-T/2})$ ...
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1answer
32 views

proving independence of stochastic integrals

Does anyone know how to show that the stochastic integrals \begin{equation} \bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}} \end{equation} are ...
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2answers
33 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 ...
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1answer
36 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, ...
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1answer
83 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 ...
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0answers
18 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 $1$.