# Tagged Questions

Questions about stochastic analysis or stochastic calculus, for example the Ito integral. See https://en.wikipedia.org/wiki/Stochastic_calculus

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### Stochastic calculus with normal distribution

For $l=1,2......$ prove that $E[W^{2l+1} (t)]=0$ I am trying to find the ways of solving the task from Stochastic calculus, but it seems to be very difficult to start with. I am really appreciate ...
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### bromnian motion and use of Lebesgue's differentiation theorem

Let $M$ be a Brownian motion with $M_0=0$ and $V\in L(M)$. Use Lebesgue's differentiation theorem to prove that there exists a predictable process $H\in L(M)$ such that $V\cdot M$ and $H\cdot M$ are ...
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### show that a function with brownian motion is a martingale

Let $B=(B^1,B^2)$ be a two-dimensional Brownian motion w.r.t. the Filtration $\mathcal{F}^B$. Show that $(M_t^2)_{t\in \mathbb{R}_{+}}:=(e^{B_t^1} \cos(B_t^2))_{t\in\mathbb{R}_{+}}$ I've tried it ...
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### Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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### Integral Representation of Brownian Motion [duplicate]

B is a Brownian motion with values in $\mathbb{R}$. I have to find a process $(F_t)_{t\in[0,T]}$ such that $X=E[X]+\int_0^T F_s dB_s$, for $X=B_T$, $X=\int_0^T B_tdt$, $X=B^2_T$, $X=B^3_T$ and find a ...
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### Question regarding Brownian motion

Hello I have two questions regarding the construction of Ito's integral in Øksendals book from here: http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf On page 25 he lists these 3 properties ...
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### Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
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### The Stratonovich Integral and its meaning as the limit in mean square of a sum?

I am studying the Stratonovich Integral and on wikipedia, Stratonovich Integral, it states that the integral, for a process $X:[0,T] \times\Omega \to \mathbb{R}$, as: $$\int_0^T X_t \circ dW_t$$ ...
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### Exact definition of time correlation function

Given a stochastic process $X(t)\in \Omega=\mathbb{R}$, markovian, ergodic and with a unique stationary distribution (e.g. a Boltzmann distribution), the time correlation function is usually defined ...
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### Conditional expectation on Function space

This question is from a notation in section 13.4 of the book "Linear and Nonlinear Filtering for Scientists and Engineers" By N U Ahmed In this section, the author is deriving the Zakai ...
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### What is meant by local time of BM on the boundary $\partial D$?

I'm familiar with local time $L_t^a$ at level $a$ for a 1-D Brownian motion $B$. I'm reading this paper which talks about a 2D Brownian motion $B$ in a bounded domain $D$ that gets reflected when it ...
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### Indicator function minus probability on an event is a martingale

Define \begin{align} \epsilon_j = \mathbb{1}_{A_j} - \mathbb{P}(A_j) = \begin{cases} 1- \mathbb{P}(A_j) \qquad &\text{if } \omega \in A_j\\ - \mathbb{P}(A_j) \qquad &\text{otherwise }, \end{...
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### Hitting times for Brownian Motion (2)

In this post there is shown that for a standard Brownian motion $\mathbb{E}[\tau^p]<\infty$ for all $p \geq 1$, where \begin{align} \tau = \inf\{t \geq 0 : B_t = \alpha \ \ \text{or}\ \ B_t=-\beta\...
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### Show the existence of a right-continuous modification

Suppose ($X_t$)$_{t \geq 0}$ is a stochastic process with independent increments and the function $t \rightarrow \mathbb{E}X_t$ is continuous. Prove that $(X_t)$ has a right-continuous modification. ...
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### definition of stochastic process on countable space

I found the following definition of stochastic process: A stochastic process on a countable state space V is a sequence of random variables $(S_n)_{n \in \mathbb{N}}$ taking values in V and defined on ...
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### Measurability From countable to Uncountable sequences family of functions

I am studying Protter Stochastic Integration and Differential Equations. In a theorem on hitting times Protter has the following theorem. Let $X$ be an adapted cadlag stochastic process, and $\Lambda$...
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### Geometrical Brownian motion Passage time

Recently I have been self-studying stochastic analysis. One of the exercises was to calculate the probabilty of Brownian motion reaching certain level before time T given that W(t)=x. This wasn't that ...
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### Is knowledge of PDE useful for SDE?

I am a stochastic analysis student and am particularly interested in stochastic differential equations. What always struck me as odd is how little PDE (or even ODE for that matter) seems to have ...
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### Solution for SDE: $dF_t= \beta_t\left(F_t - \alpha\right)dW_t$

I am trying to derive the solution for the following stochastic differential equation, but I must be doing something wrong in my calculations because I can't arrive to the correct solution. The SDE ...
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### Showing the Haar wavelet is a complete and orthonormal sequence within $L_2[0,1]$

I define the mother Haar wavelet to be: \begin{align} \phi(t) = \begin{cases} 1 &\mbox{if } 0 \leq t < 1/2 \\ -1 & \mbox{if } 1/2 \leq t \leq 1 \\ 0 &\mbox{otherwise}. \end{...
Consider the Brownian $W_u$. Suppose you are only considering realizations of this brownian that verify both $W_0=0$ and, for a specific (given) $t$, $W_t=a$. Under these specific conditions, what is ...
Consider $\begin{split} \newcommand{\d}{\mathrm d} \d x &= -yx \d t + x^2 \d B\\ \d y &= -2 y^2 \d t + 2xy \d B. \end{split}$ This is a system of two SDEs driven by the same standard ...