Questions on the calculus of stochastic processes, or processes that have a random component.

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1answer
17 views

probability of two successive random numbers has the same starting number

Question/problem(subtask b): What is the probability of two successive random numbers has the same starting number? What we do know is that a random number generator randomizes numbers of 6-digits ...
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9 views

Pushforward measure

We define $X := \{0,1\}, \mu := \frac{1}{2} (\delta_0 + \delta_1)$ and $(\Omega, \mathcal{F},\mathbb{P}) : = \bigotimes_{n=1}^{\infty} \left( X, 2^X,\mu \right)$. For $\omega \in \Omega$ we denote the ...
4
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0answers
31 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
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1answer
25 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
2
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1answer
29 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
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17 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
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1answer
41 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...
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25 views

Almost sure convergence of stochastic integral

Consider a probability space $(\Omega, \mathcal{F}, P)$ equipped with a Brownian motion $W$. Let $(\xi_n)_{n=1}^\infty$ be a sequence of adapted $\mathcal{F}(t)$-progressively measurable processes ...
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10 views

Wasserstein distance and maximization covariance

My question deals with the second order wasserstein distance $W_2$ on the set of measures, which is defined by: $W_2(\nu_1,\nu_2)^2= inf_{\Pi(X,Y)} E_{\Pi} (X-Y)^2$ where $\Pi$ is chosen such that ...
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1answer
21 views

Expected Value of the exponential of a stochastic integral

What is the expected value of the following process: $$ e^{\int_0^t B_u\, du} $$ Thanks.
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2answers
21 views

Find the value of the real $\alpha$ for which exp($2B(t) - \alpha t$) is a martingale.

I tried to answer using the three conditions to be a martingale (measurability, integrability, and martingality), validating the integrability condition, which is $$ E |e^{2B(t) - \alpha t} | < ...
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0answers
29 views

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion

Is $W^3(t)$ a martingale if $W(t)$ is a Brownian motion? The answer seems like no to me. Using Ito's lemma I can write $$W^3(t)=\frac{3}{2}W^2(t)+\int_0^t3W(u)dW(u)$$ The second piece on the LHS is an ...
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0answers
19 views

Deterministic integrals involving a Brownian motion [duplicate]

I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$. The first expression is bewildering me a bit, since it seems like somehow the Itô ...
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1answer
29 views

Prove directly from the definition of the Ito's integral

I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question. Prove directly from the ...
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0answers
28 views

Fail of reverse implication of completeness.

Let $\mathcal{P}'\subset \mathcal{P}$ be two equivalent classes of probability measures on a measure space $(\mathcal{X},\mathcal{B})$, e.g. $\mathcal{P}:=\{P_{\theta} : \theta \in \Theta\}$. Let $T$ ...
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0answers
25 views

Solve the stochastic differential equation

I have to solve the following SDE: $$dX_t=X_t dt+2W_tdW_t$$ Let $Y_t=X_t e^{-t}$. By Ito formula we have: $$dY_t=-X_te^{-t}dt+e^{-t}(X_t dt+2W_tdW_t)=2e^{-t}W_tdW_t$$ Thus ...
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18 views

Definition of Simple Predictable Process

I am reading Protter's book "Stochastic Integration and Differential Equations". He (page 51) defines $H$ to be a simple predictable processes if it has a representation ...
2
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1answer
48 views

Completeness of a statistic. Implication of equivalent probability classes [duplicate]

Let $\mathcal{P}'\subset \mathcal{P}$ be two equivalent classes of probability measures on a measure space $(\mathcal{X},\mathcal{B})$, e.g. $\mathcal{P}:=\{P_{\theta} : \theta \in \Theta\}$. Let $T$ ...
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24 views

Sufficency condition of a statistic [closed]

Consider the random variable $(X_{1},Y_{1}),\ldots (X_{n},Y_{n})$ i.i.d with density \begin{align} f_{\theta}(x,y):=e^{-\theta x-\frac{y}{\theta}} \end{align} The maximum likelihood estimator for ...
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0answers
17 views

Inverse Bessel Process

Is there any reference on this process? For example, analytical derivations for the hitting times, density, etc? Im studying local martingales and am interested in the density of hitting times for ...
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0answers
70 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
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1answer
10 views

What's the covariance of $B_t$ and $B_{t^2}$, where $B_t$ is the standard Brownian Motion?

What's the covariance of $B_t$ and $B_{t^2}$, where $B_t$ is the standard Brownian Motion? $B_t$ is the standard Brownian Motion, what's $\operatorname{Cov}(B_t,B_{t^2})$?
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1answer
23 views

What is the difference between “filtration for a Brownian motion” and “filtration generated by a Brownian motion”?

I'm reading Shreve's book "Stochastic Calculus for Finance: Vol II". In 5.3.1, after the Theorem 5.3.1 (Martingale representation, one dimension), Shreve explains: "The assumption that the filtration ...
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1answer
18 views

Malliavin Derivative

Motivation : We know that, if the randomness in the system is due to Brownian Motion then any contingent claim with mean zero can be written as Ito integral. (Of course, we need to have boundedness ...
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1answer
63 views

Expected value and variance of a stochastic process

Having trouble finding expected value and variance of a stochastic process defined by SDE: $dX_{t} = a X_{t} dt + b dB_{t}$ $X_0 = x$, $a$ and $b$ are constant values, $B_t$~$N(0,t)$ Thank you for ...
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0answers
16 views

Some sort of Chainrule for measures in statistics [closed]

Lets $X$ be a random variable $X:(\Omega,\mathcal{A})\rightarrow (K,\mathcal{B})$. Suppose that for $T:(K,\mathcal{B})\rightarrow (K',\mathcal{B}')$ (a mapping of the sample space) yields that the ...
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0answers
22 views

How to identify the future distribution of a stochastic variable from its SDE

I would like to know some common practice to identify the future distribution of a random variable modelled by an arbitrary SDE. Would you study it empirically (like generating Monte-Carlo ...
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1answer
21 views

Find a process $f=f(t,W_t)$ such that another process is a martingale

Find a process $f=f(t,W_t)$ such that process: $$X_t=\exp(W_t^2-2tW_t^2)+\int_0^tf(s,W_s)ds$$ is a martingale. Justify the fact that $X_t$ is martingale. I think I should find a process such that ...
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1answer
16 views

A question on integration wr.t to a local martingale

In a lemma in my graduate level course on financial mathematics uses the fact that integral of a progressive portfolio process(which is almost surely lower bounded i.e it is admissible) $\theta_t$ ...
1
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1answer
39 views

Find conditional expectation of $\int_1^2W_t^2dt$ with respect to $F_1$

$$\mathbb{E}(\int_1^2W_t^2dt|F_1)=\int_1^2\mathbb{E}((W_t-W_1+W_1)^2|F_1)dt=\int_1^2\mathbb{E}((W_t-W_1)^2|F_1)dt+\int_1^2 2\mathbb{E}((W_t-W_1)W_1|F_1)dt+\int_1^2 ...
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2answers
32 views

Calculate conditional expectation of integral

I have to calculate $$\mathbb{E}\left(\int_1^2 (t^2W_t+t^3 )\,dt\mid F_1\right)$$ My attempt: $$\int_1^2 (t^2W_t+t^3 )\,dt=\int_1^2t^2W_t\,dt+\frac{15}{4}$$ Now I will focus on: ...
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1answer
28 views

partial derivative of $f(X(t),t)$ with respect to $t$

Suppose that $f(x,t) = x^2$. Clearly, $\frac{\partial f}{\partial t} = 0$. However, let us now consider $f(X(t),t) = X(t)^2$. The book I am reading claims that $\frac{\partial f}{\partial t}(X(t),t) ...
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0answers
73 views

Find the relation between 2 stochastic integral

$g(s,t)(\omega)$ is an adapted stochastic process on $\mathbb R^2$ define: $$X=\int_0^1\int_0^1g(s,t) \,dW_s\,dt$$ $$Y=\int_0^1\int_0^1g(s,t) \,dt\,dW_s$$ Could we conclude that "$X=Y$ a.s"? I ...
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25 views

Regarding proof of converse to Girsanovs theorem

This is regarding an argument from Arbitrage Theory by Thomas Björk - Theorem 11.6, but is attempted self contained. Consider a Wiener process W on probability space ...
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0answers
66 views

How to derive this step in a book called Brownian motion calculus?

How to derive the step which result in the magnitude of slope of the path in section 1.8.2? I know it is not the definition itself.
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1answer
48 views

Conditional likelihood of continuously-combounded returns

The simplest possible asset pricing model ist the geometric brownian motion for asset price. Here the price $S_t$ solve the familar $$dS_t = (\mu +0.5 \sigma^2)S_t \, dt + \sigma S_t \, ...
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1answer
16 views

Covariance of Wiener Processes on the same Brownian Motion

I am trying to solve $Cov(Tw_T,\int^{T}_{0}tdw_t)=\mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t]$, my attempt is as below: \begin{split} \mathbb{E}[Tw_T\times\int^{T}_{0}tdw_t] & ...
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0answers
18 views

Clarifications about SDEs, Differentials & Derivatives

A general SDE look like the following: $$ \mathrm{d}\psi=a\mathop{}\!\mathrm{d}t+b\mathop{}\!\mathrm{d}W,\tag{1} $$ where $\psi:t\mapsto y = \psi(t)$ is the solution, while $a$ and $b$ can be both, ...
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0answers
17 views

Stochastic Calculus Research topics

If I just finished taking a Stochastic Calculus course and wanted to do research on it, what are some hot topics currently going on in this area? I know there's a lot but if I wanted to investigate ...
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37 views

Is this stochastic differential equation wrong?

The following is an old exam question I think might be misstated. Consider the SDE $$dX(u)=(a(u)+b(u)X(u))\,du+(\gamma(u)+\sigma(u)X(u))\,dW(u)$$ where $W(u)$ is a brownian motion relative to ...
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0answers
18 views

A problem on Ito integral [duplicate]

Let $W$ be a standard, one-dimensional Brownian motion. Let $T\in(0,+\infty)$. Then $$\lim_{\beta\to+\infty}\sup_{0\le t\le T}\left|e^{-\beta t}\int_0^te^{\beta ...
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1answer
16 views

ratio of two functions of finite variation

According to the Jordan decomposition, a necessary and sufficient condition for a function to have finite variation is that it can be expressed as the difference of two increasing functions. The book ...
3
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1answer
75 views

What is the explicit obstruction to almost sure convergence in stochastic integrals?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
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58 views

What is some reason that there are no book bridge the gap of these three books

I am referring to the (beginner's text- Stochastic Calculus by Mircea Grigoriu and Introduction to Stochastic Calculus by klebaner.) and the advanced texts - stochastic differential equation by ...
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0answers
16 views

Density of this quantity for a Geometric Brownian Motion?

If we define $X_T = X_t e^{(\mu-\frac{1}{2}\sigma^2 ) (T-t) + \sigma W_{T-t}}$ where $W_{T-t}$ is a classical weiner process. How would we go about deriving the density and expectation for $X_{max} - ...
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1answer
64 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [closed]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
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1answer
37 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
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0answers
21 views

How to determine the probability density function, ${f_{\dot X}}\left( {\dot x} \right)$, for the derivative process of a stochastic process?

I would like to calculate the up-crossing rate ($\nu _a^ + $) for a stationary stochastic process, $X(t)$, given by the probability distribution function of its 'intensity', ${f_X}\left( x \right)$, ...
3
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1answer
40 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution ...
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0answers
20 views

Solution of $dX_{t}=(sin(X_{t})+2)dB_{t}$

I am curious if $dX_{t}=(sin(X_{t})+2)dB_{t}$ has a solution i.e $X_{t}$=(stuff in terms of $B_{t}$). What about for $dX_{t}=\sigma(X_{t})dB_{t}$, where $0<\gamma^{-1}\leq \sigma\leq ...