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

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What does this mean in the context of Stochastic Calculus?

I've reading into some Stochastic Calculus books and I've been stumped by two concepts used recurringly in the book. The first is a subscripted 1 which appears in the definition of a simple process ...
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20 views

2 2-dimensional Brownian motions are close to each other

Suppose $B^1$ is a standard 2 dimensional Brownian motion and $B^2$ is a 2 dimensional Brownian motion with mean zero and covariance matrix $\Gamma = \begin{pmatrix} a & b \\ b & a \\ ...
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1answer
60 views

volume of some stochastic processes

for a continuous and differentiable curve $\vec{x}_t$ in $\mathcal{R}^n$ parameterized by a single variable $t$, there is a well defined way of computing the volume of this one-dimensional manifold ...
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337 views

Expectation of a Poisson Process

Cars pass a certain street location according to a Poisson Process with rate $\lambda$. An old lady and her trusty boyscout want to cross the street at this location. They wait until they can ensure ...
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49 views

Conditional expectation and coupled set of ODEs

How to find a coupled set of ODEs and initial conditions for the deterministic functions $a$ and $b$ such that $$\mathbb{E}\left[e^{-\int_{t}^{T} W^2(u)du} | \mathcal{F(t)}\right] = e^{-a(T-t) - ...
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1answer
49 views

Addition corresponds to convolution and subtraction?

We know that if two random variables have proper densities, than the density of the sum of them is given by the convolution. But what can we say about the difference of two random variables? $X-Y$ ...
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120 views

Geometric Brownian motion - Volatility Interpretation

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
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41 views

Proving $(\int_0^t f(X_s) dW_s)_{t \in [0T]}$, $f$ a $k$-Lipschitz function, is a continuous martingale

Consider $X =(X_t)_{t \in [0T]}$ progressively measurable with $X_t \in \mathbb L^p, \forall t \in [0,T]$ for $p\geq 1$ and $f$ a $k$-Lipschitz function. I would like to show that $(\int_0^t f(X_s) ...
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1answer
100 views

The pure jump part of Levy process and Martingale

Assume $X_{t}$ be a Levy process with generating triplet $(\sigma, \gamma, \nu)$. Here $\nu$ is the measure on $R$ satisfying $$ \int_{R}\min( 1,y^{2})\nu(dy)<\infty $$ According to the standard ...
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1answer
50 views

Clarification about a very simple stochastic integral

I'm studying stochastic integrals right now and I feel like this question is incredibly easy but I'm not sure. I want to evaluate $\int_0^t sdB_s$. Using Ito's formula I get $tB_t$ by setting ...
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55 views

Stopping time problem - Show that T is bounded

Let $a< 0 < b$ and $W_t$ is Brownian motion $T_a$=inf{$t\ge$0|$W_t\le a$} $T_b$=inf{$t\ge$0|$W_t\ge b$} T=min{$T_a$,$T_b$} $1)$ Show that $T$ $<$ $\infty$ My attempt : ...
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70 views

How to calculate the value of $E[X^4], E[X^6],E[X^8] $…?

I learned that when X is a normal random variable , $X$~ $N(0,1)$ , $E[X^2]=1$ $E[X^4]=1.3=3$ $E[X^6]=1.3.5=15$ $E[X^8]=1.3.5.7=105$ For the general case , when variance is s , how do you do ...
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1answer
122 views

Simple integral with stochastic Brownian motion integrand

Consider $$\int_0^t \sin(B_s) ds$$ where $B_s$ is standard Brownian motion, I was wondering can I write $$\int_0^t \sin(B_s) ds = - ( \cos(B_t) - \cos(B_0)) = - \cos(B_t) ? $$ by using the ...
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1answer
39 views

Can I make the substitution of dP when using the CDF?

Random variable $X \geq 0$ with parameter $\lambda>0$ and $X$ has the c.d.f. $$ F (a) = P{(X ≤ a)} = 1 − \exp(−λa)$$ for $a \geq 0$. Consider $Z = (λ'/λ)\exp(-(λ'-λ)X)$ Show that $E[Z]=1$ thus ...
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96 views

Stochastic Integration and Ito Calculus

Before reading this I must not I think I am a little behind on some of the prereq for this topic but I really want to be able to understand it in a relatively meaningful way. I am having trouble ...
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1answer
21 views

Non-interacting particles

Suppose $X(t)$ is a Markov chain taking values in $\{0,1\}^2$. Suppose $q$ is the q matrix whose positive valued entries are $q((0,0),(1,0)) = \beta_{0}$ $q((1,0),(0,0)) = \delta_{0}$ ...
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1answer
20 views

Please explain $E[S_{min(n,T)} ]= E [S_{0}]=0$

If $S_{n}$ is a simple random walk i.e $X_{k}= +/- 1$ with prob = 0.5 T = inf {n > = 0 |$S_{n}$ = 1} is a stopping time. T is finite almost surely. .Explain $E[S_{min(n,T)} ]= E [S_{0}]=0$ I know ...
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1answer
46 views

Integrating the difference of brownian motion

I'm reading the solutions to an exercise where it is stated that $$\int_t^T\Big(W(u) - W(t)\Big)du = \int_t^T (T-u)dW(u).$$ But can someone enlighten me to what theorem/rule can be used to show this? ...
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125 views

Approximation of stochastic processes in Protter

I'm reading Stochast integration and stochastic differential equation by Protter. In particular I have a question about Theorem 10 in chapter 2.4. Here Protter defines a simple predictable processes ...
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1answer
93 views

Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...
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57 views

Stochastic Differential Equation- When martingale?

Suppose I'd like to check the martingale property for some SDE. What do I have to require for it to be martingale? I know that no drift is one requirement, but what are the others?
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36 views

Strictly increasing maps

For $p\ge n$, how many strictly increasing maps from $N^*_n$ to $N^*_p$ do exist, where $N^*_n = \{1, 2, \dots, n\}$ is the set of the first $n$ integers greater than 0 ? My answer: uncountable many. ...
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59 views

If two stochastic integrands are equal on some measurable set, will the stochastic integrals be equal on that set?

Given a $X$ semi-martingale on a filtered probability space $(\Omega,\{\mathcal F_t\}_{t\le\infty},P)$ I am trying to prove: For any $B\in\mathcal F_\infty$ and processes $a_1,a_2$ such that ...
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3answers
72 views

The uniqueness of solution for stochastic differential equation involved with sign function.

When I read a paper about Levy distribution thoerem (http://www.maphysto.dk/publications/MPS-RR/1998/22.pdf). In the first page, the author mentioned the following: There is a unique strong solution ...
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1answer
50 views

Independence random variables

I found two theorems in my notes and they seem to be somewhat complementary which made me doubt that both of them are true: a) Let $X,Y: \Omega \rightarrow \mathbb{R}$ be a measurable function and ...
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1answer
105 views

Diffusion processes

I am trying to work out a problem to which I have not found similar solutions on the website. Perhaps you can help me out. Let $X = (X_t)_{t\geq0}$ be a non-negative diffusion process which solves ...
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0answers
18 views

Reference or Intuition on a Stochastic Equation (Klyatski-Tatarski formula)

I am working through an already not so easy to find paper from the 70s, which in turn uses an even older result that i can not find at all. Im refering to ...
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57 views

Martingale and Stochastic equation

Using the Ito formula, I can show that the martingale $$ Z_{t}=\frac{1}{\sqrt{1-t}}\exp -\frac{B_t^2}{2(1-t)}\qquad 0\leq t<1 $$ admits the following differential $$ dZ_t=-\frac{B_t}{1-t}Z_tdB_t. ...
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1answer
147 views

Distribution of Stopped Brownian motion at hitting time of another Brownian motion.

Suppose $B_t$ and $W_t$ are two independent Brownian motions and $\tau$ is the first hitting time of $B_t$ to some $a >0$. Compute the distribution of $W_{\tau}$. We can try the characteristic ...
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1answer
63 views

A few questions about Stochastic Processes and Numerical Methods

I am having a few problems understanding the Ornstein Uhlenbeck solutions, on wikipedia under solution (http://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck_process) it described using variation of ...
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1answer
204 views

Distribution of ceiling function and absolute value of random variable

Given a distribution function $f_X$, where $X$ is some random variable. I want to get the distribution functions of $|X|$ and $\lceil X \rceil$( the last one may only have an easy form if $X$ is ...
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1answer
39 views

Unbiased estimate $\lambda^2$

Given a Poisson distribution I want to figure out whether $d:(x_1,...,x_n) \mapsto x_1^2$ and $d':(x_1,...,x_n) \mapsto x_1x_2$ are unbiased estimations for $\lambda^2$ ? I mean it would sound ...
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1answer
173 views

Proof that a median minimizes 1-norm. [duplicate]

I was wondering whether there is an easy way to show the following: We have a data set $x_1,...,x_n$ and $m$ is a median if for at least half of the n data points we have that $x_i \le m$ and for ...
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150 views

Determine if this is a Martingale

I am trying to check if the process $S_t$ is a martingale, where $\mathrm dS_t = \frac{I_{S_t > 0}}{S_t} \mathrm dW_t$, $S_0 = 1$. We know that $S_t$ is a local martingale because if we stop it ...
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1answer
574 views

Sum of two Markov processes another Markov process?

Let $dX_{t} = m_1(l_1-X_{t})dt+\sigma_1 dW_{t}$ and $dY_{t} = m_2(l_2-Y_{t})dt+\sigma_2(\rho dW_{t}+\sqrt{1-\rho^2}dW_{t}^{1})$ where the $m_i$'s, $l_i$'s and $\sigma_i$'s are constants, $\rho \in ...
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1answer
70 views

Quadratic variation - Semimartingale

We know that any Semimartingale has Quadratic variation. I am interested to know if the converse is also true i.e. if a process has quadratic variation then it is semimartingale. Can some one ...
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1answer
576 views

Correlation between two stochastic processes [closed]

Let $$dX_t = k_1 X_t \, dt + \sigma_1 \, dW_t$$ and $$dY_t = k_2 Y_t \, dt + \sigma_2 \left( \rho \, dW_t + \sqrt{1-\rho^{2}} \, dW_t^1\right)$$ where $W_t$ and $W_t^1$ are independent. What is ...
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100 views

Log normal stock prices - Steps after Ito

When we specify a GBM stock price: $$dS = \mu S dt + \sigma S dW$$ And then we change it to: $$\frac{dS}{S} = \mu dt + \sigma dW$$ The we assume: let $Z_t = f(S_t) = \log S_t$, where $f(x) = \log ...
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1answer
272 views

Convergence in distribution ( Two equivalent definitions)

I read that for convergence in distribution it is equivalent to have that either the characteristic functions of the random variables convergence pointwise or we have that $F_{X_n} \rightarrow F_{X}$ ...
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49 views

Differential of the integral of a stochastic process

In the HJM model one considers the forward rates to be on the form $$\mathrm df(t,T) = \alpha(t,T)\,\mathrm dt + \sigma(t,T)\,\mathrm dW(t)$$ In the proof of showing the drift condition on $\alpha$ ...
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74 views

First hitting time in the one-dimensional case by solving a boundary value problem

If have a question about section 3.1 in the paper Kramers' law: Validity, derivations and generalisations by Nils Berglund. (See http://arxiv.org/abs/1106.5799 page 7 - 9) On page 8 it says, that ...
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1answer
56 views

Computation of a simple stochastic integral

For $t \in [0,T]$. consider two stochastic integrals with a nonnegative constant integrand $c$ $$\mathbb{E} \left[ \int_0^{t(\omega)^* \wedge T} c \cdot dW_t \right]$$ where $t^*$ is random ...
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1answer
81 views

Is any FV-Process a special Semimartingale?

Any FV-Process can be represented as the difference of two increasing (or decreasing) processes and so any FV-Process is a quasimartingale. Due to Raos Theorem any FV-Process is a special ...
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1answer
64 views

lower bounds for a stochastic integral

for all $t \in [0,T]$, consider a stochastic integral as follows: $\int_0^{min \{t^*,T \}} f(t,\omega) dt$ where $f \geq 0$ is a nonnegative stochastic process and $t^*$ is a random stopping time. I ...
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1answer
81 views

Distribution of integral wrt. to a Levy process

Assume that a stochastic process is given by $X_{t} = \int_0^t e^{-k(t-s)}dY_{s}$ where $Y_{s}$ is a Levy process. Is there any way I can use the knowledge about the Levy measure of $Y_{t}$ in ...
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127 views

Clarification in stochastic integration

In the book "Stochastic Processes" by Bass R.F. when he constructs the Stochastic Integral, at some point he defines for $Y$ predictable $$||Y||_2= \left(\mathbb E \int_0^{\infty}Y_t^2\text{d} \langle ...
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1answer
431 views

Expected value of Stock Price, Poisson Process

I would appreciate a hint regarding the following question (taken from Durret, Essentials of Stochastic Processes, questions 2.38 "Let $S_t$ be the price of stock at time t and suppose that at times ...
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1answer
53 views

Why $\int _0 ^t \phi_s ^2 ds < \infty \ \mathbb P \text{-a.e.}$ do not implies $\mathbb E [\int _0 ^t \phi_s ^2 ds] < \infty$?

Why $\phi =(\phi_t)_{t \in [0,T]}$ is a progressive mesurable stochastic process do not implies $\mathbb E [\int _0 ^t \phi_s ^2 ds] < \infty$? I know that if $X$ is a positive random variable ...
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2answers
1k views

Intuition for random variable being $\sigma$-algebra measurable?

Is there some sort of intuition or a good ilustrative example for random variables being $\sigma$-algebra measurable? I understand the definition, but when looking at martingales, the meaning of ...
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1answer
18 views

Ito's lemma for a boolean

If I have a stochastic process defined as usual by $dx=f(x,t)dt+g(t,x)dW$, how can I compute the Ito's formula for a process $n=\phi(t,x):=(x/t>a)$, i.e., $dn = (\ldots)dt + _\ldots$ ? I have ...