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

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stochastic calculus, stopping time, ito integral vector brownian motion

I'm referring to chapter 4, question 7 in Harrison's book 'Brownian Motion and Stochastic Flow Systems.' Problem In the setting of (9) let $f_{n}(x)=E_{x}[\int_{0}^{T}X_{t}^{n}dt]$. Use Ito's ...
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0answers
39 views

Poisson process and Heaviside function

Show that Poisson process $p(t)$ of intensity $\lambda$ can be written as $$p(t)=\sum_{t>t_n}\delta(t-t_n),$$ where function $\delta:\mathbb{R}\rightarrow\mathbb{R}$ is Heaviside's function: ...
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27 views

Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
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1answer
34 views

Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as ...
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0answers
21 views

Conditional probability of geometric brownian motion [on hold]

I created paths using GBM to implement The stochastic mesh method. But the method requires the conditional distribution, given some S(t) the probability of S(t+1). I've searched and can't find this ...
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13 views

Moments of random variable distributed as the product of normal cdf and pdf [on hold]

How do I find the first and second moments of a random variable whose pdf is $c\Phi(-Y-a)\phi(Y-a)$ Where $\Phi$ is the standard normal cdf, $\phi$ is the standard normal pdf, c is a normalising ...
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1answer
31 views

Ito formula when g(t,x) is an integral

Suppose we have a stochastic process which is written as an Ito process. $$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$. If $Y_t$ is defined as a stochastic process as a function of $X_t$, then we can find $dY_t$ ...
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45 views

what does this integral stand for?

i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ...
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1answer
41 views

Word Problem: Probability of Y books Fitting in Book Case

Problem: You have $4600$ cm of book case. The thickness of the books are independently distributed with $X \sim N(1.8$ cm$,0.7^2)$. Approximately determine what the probability of ...
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19 views

Should I ask about math books? I.e. scan of index page or references page? [on hold]

I mean it's not harmful to anyone if I ask about certain page of certain book? I need page 263 from this book ...
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19 views

How should I calculate the MLE based on a random sample from $PAR(\theta,2)$

Consider a random sample of size $n$ from a Pareto distribution, $X_i \sim PAR(\theta, \kappa =2)$. I have to compute the MLE, $\hat \theta$, to three decimale places. So I started doing the ...
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43 views

Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
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0answers
17 views

System of SDEs and independence

I am recently reading a paper that seems to claim the following fact without justification: $Y^1_t, \ldots, Y^n_t$ are stochastic processes defined on $\mathbb{R}$. Let $b: \mathbb{R}^2 ...
2
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1answer
44 views

Why Are Semimartingales the Largest Possible Class of Stochastic Integrators?

I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ...
3
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0answers
42 views

Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
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1answer
25 views

Do Optional and Progressive Processes Have Counterparts in Discrete Time?

We know that predictable $\implies$ optional $\implies$ progressively measurable. Source Predictable processes have obvious/simple counterparts in discrete time. Do optional processes and ...
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22 views

Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv ...
4
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2answers
27 views

Solving a nonlinear scalar Ito SDE

I need to solve the SDE: $$ dX_t = (X_t)^3 dt + (X_t)^2 dW_t ; X(0)=1 $$ Now what I found is this is an SDE of the form: $$dXt =a(X_t)dt+b(X_t)dW_t$$ where $a(x) = \frac{1}{2} b(x)b′(x)$ Using the ...
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1answer
35 views

Equivalence of two Ito formulae

Let $X$ and $Y$ be two $1$-dimensional Ito processes. There are two Ito formulae for the product $X_tY_t$ given by $d\left(X_tY_t\right)=X_tdY_t+Y_tdX_t+d\left[X_t,Y_t\right]$ ...
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1answer
35 views

What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...
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1answer
21 views

Determining distribution and therefrom probability

The problem is as follows: Assume that $V_1$ and $V_2$ are independent random variables with $V_1 \sim \chi^2(5), V_2\sim\chi^2(9)$. Find the value of $b$ such that: $$P[\frac {V_1}{V_1 + V_2} \lt b] ...
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23 views

Esscher-Transform/ Levy-Process: Measure induced by trajectory

For a Levy-process $X_t$ w.r.t. to a measure P we define $\Theta$ as the set, for which $E[exp(\theta X_t)]$ is defined and finite. Note $\Theta$ is independent of $X_t$. Define ...
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1answer
36 views

To test whether a process is a Martingale (Stochastic calculus)?

If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to ...
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calculus of a derivative (stochastic calculus) [closed]

I have a doubt : I know that if $x_{t}=\int_{0}^{t}\gamma(s)dW_{s}$ (with $W_{s}$ a brownian motion), we have : $dx_{t}=\gamma(t)dW_{t}$ What about if $x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}$. Do I have ...
3
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69 views

stochastic exponential uniformly integrable martingale

$N$ is a continuous local martingale and $T_c:=\inf\{t>0:[N]_t>c\}$, $c>0$ . I need to show that the stochastic exponential $\mathcal{E}(-N)$ is a uniformly integrable martingale if and only ...
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12 views

Solving the following SDE with a constant

Given is the stochastic differential equation: $\frac{dX(t)}{X(t)}=\mu+\sigma \theta dt+ \sigma dW(t)$, where $W(t)$ is the standard Wiener process and $X(0)=x_0$ I try to solve this by the Itos ...
2
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32 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, ...
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1answer
55 views

A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that ...
2
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1answer
72 views

Solution to General Linear SDE

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
5
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0answers
42 views

Can Stochastic Integration be Further Generalized?

Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ...
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1answer
36 views

Existence and uniqueness of SDE, is the independence requirement needed?

In Bernt Øksendals Stochastic differential equations he has this theorem in chapter 5: $\\\\\\$ However, in the proof I can not see where he uses the independence condition I marked in red. Do you ...
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33 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^∞ ...
3
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44 views

Question about “Stochastic Analysis on Manifolds”

After Definition 2.3.1 Hsu says that if $M$ is a closed submanifold of $\mathbb{R}^N$ then a semimartingale $X$ on $M\subseteq\mathbb{R}^N$ should satisfy $$X_t=X_0+\int_0^tP\left(X_s\right)\circ ...
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29 views

stochastic differential equation exact solution

whats (is there) exact solution of (for) this sde? $dX_{t}=\mu X_{t}dt+\sqrt{\sigma X_{t}} dW_{t}$ and what's the distribution of that? thanks
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33 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} ...
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2answers
29 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 ...
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49 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 ...
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26 views

Why an optional process could not be predictable?

We know that a predictable process is also optional (*). Why an optional process could not be predictable ? Why we cannot use the same arguments as the proof for (*) ?
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1answer
43 views

Quadratic Variation Brownian motion martingale (2)

Let $B_t$ be a standard Brownian motion and $M_t = B_t^2 -t$. From here we are aware of the identity \begin{align} [M]=[B^2]. \end{align} Now, I want to apply Itô's formula to $B_t^2$ and from that ...
2
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1answer
29 views

discretized Brownian motion

These are the definitions I'm working with: A (standard) Brownian motion in $\mathbb{R}$ is a stochastic process $W(t)$ $(t \geq 0)$ such that the following properties hold: $W(0) = 0$ almost ...
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26 views

A stochastic volatility model

An example of stochastic volatility model: $$\begin{cases} \frac{dX_t}{X_t} &= g_t dW_t \\ dg_t &= - k g_t dt + \sigma dZ_t \end{cases} $$ where $Z_t$ and $W_t$ are Brownian motions and ...
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12 views

Spectral Density of an ARMA process.

For an upcoming Stochastic Processes exam, we have had a sudden brief email about Spectral Density as the lecturer had forgotten to mention it in classes. He states, For an ARMA process with ...
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1answer
19 views

Application of Ito's rule

I have that $\sigma$ is a piecewise continuous function on $[0,t]$, $W$ is Brownian motion, $X(t)=\int_0^t\sigma(s)dW(s)$, and $Z(t)= e^{iuX(t)},$ for some fixed $u\in\mathbb{R}$. It is then stated ...
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58 views

Can this approximation result for stochastic processes be modified.(p=1 instead of p=2)?

In McKeans stochastic integrals from 1969 he proves this: You have a filtered probability space $(\Omega,\mathcal{F},P)$, where the filtration is based on a Brownian motion. Assume that $X_t$ is ...
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31 views

Ratio distribution of independent exponentially distributed variables

first things first: I am not a studied mathematician and therefore lack thorough knowledge of the topic - please consider this, even though I will of course try to express myself as accurately as ...
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2answers
62 views

Discounted price process in Black-Scholes model is a martingale with respect to Q.

I have been presented a proof that the discounted price process in the Black and Scholes formula is a martingale, but there is something important omitted, and I am not able to fill in the gap. I will ...
2
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1answer
15 views

Definition of Cylindrical Brownian Motion and Spatial Correlation

From Gawarecki and Mandrekar, Stochastic Differential Equations in Infinite Dimensions: We call a family $\{ W_t \}_{t\geq 0}$ defined on a filtered probability space $(\Omega, \mathcal{F}, ...
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13 views

Partial Integration for Semimartingales

Let $X,Y$ be 2 continuous semimartingales. It could be shown that for every $t>0$, \begin{align} X_tY_t = X_0Y_0 + \int_0^t X_s dY_s + \int_0^t Y_s dX_s + \langle X, Y \rangle _t. \end{align} Let ...
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22 views

Geometric Brownian motion hitting time

Let $X$ be a geometric Brownian motion $dX_t = \mu X_t dt + \sigma X_t dW_t, X_0 > 0$ and ${\cal F}$ its natural filtration. Let $\tau_a$ be the first hitting time of $a$ by $X$. How can we relate ...
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27 views

Ito formula proof

Is there a simple way to prove $$x=f(t,x_t)\\df(t,x_{t})=\frac{\partial f}{\partial t}dt+\frac{\partial f}{\partial x}dB_t+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}(dx_t)^2$$? can we prove it by ...