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

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1answer
291 views

proof of Feynman–Kac formula

the article given by wikipedia http://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula#Proof states at some point of the proof that: (line 7) ''the third term is o(dtdu) and can be dropped'' Can ...
1
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1answer
199 views

Inequality for Euclidean norm

Let:| | be Euclidean norm on $\mathbb{R}^{n}$ and $b : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n}$ and $\sigma : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n\times m}$ two continuous functions. ...
1
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2answers
141 views

Variance of sum of two ito integrals

I don't really understand how to solve the following problem: Var(X) where X = $\int_0^2 2t dW(t) + \int_4^6 W(t) dW(t)$ If I use $E [(A+B)^2] = E(A^2) + E(B^2) + 2E(AB)$ I get to the point where I ...
3
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0answers
218 views

Essential supremum of a conditional expectation

Given the function \begin{equation} P(x,t) := \sup\limits_{t \le \tau \le T} E\left( g(X^{t,x}_{\tau}) \right) \end{equation} where $X^{t,x}$ is the unique solution to the SDE \begin{equation} X_u ...
2
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0answers
106 views

interchange stochastic and deterministic integration

If $f$ is a function in $L^2([0,1]^m)$, W is one-dimensional Brownian motion, $a,b \in [0,1]$, are the following two integrals equal? $$\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2} ...
1
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1answer
85 views

$\mathbb{E} \int_a^b W^3(t)\,dW(t)=?$

Is it true that $\mathbb{E} \int_a^b W^3(t)\,dW(t)=0$, for $a < b \in \mathbb{R}$ I know that for an adapted process $\Delta(t), t\geq 0$, the integral $\int_0^t \Delta(u)dW(u)$ is a ...
1
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1answer
42 views

Definition of a random variable in the context of a hypergeometric distribution

We defined a random variable in a probability space $(\Omega, E, P)$ as a map $X: \Omega \rightarrow \mathbb{R}$. Unfortunately, I somehow have the impression that this term "random variable is used ...
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0answers
130 views

Please rate this stochastic processes course. What should my next course be?

I am currently doing a course in Stochastic Processes that uses the book "Adventures in Stochastic Processes" by Sidney I. Resnick. The topics covered in the book are as follows: Discrete Index Sets/ ...
1
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2answers
87 views

How to show that $\mathbb{E}(\int_0^T t\mathrm \, dW_t) = 0 $?

I just want to know why $\mathbb{E}\left(\int_0^T t \,\mathrm dW_t\right)=0$. I know it's got something to do with the Gaussian distribution but I don't really know what.
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0answers
90 views

Joint probability distributions of variables satisfying a certain constraints

Here is my question, given a set of random variables - {x_i}, i=1,2, ...n. And the corresponding pdfs are given by {PDF_i}, i=1,2, ...n. Now if I were it impose a certain set of constraints on ...
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2answers
111 views

Expectation of absolute value of stationary time series

Let $Y_t$ be a stochastic process (time series). We consider stationarity as follows: $Y_t$ is said to stationary if the mean $\mu_t = \mathbb{E}(Y_t)$ is constant (given $\mathbb{E}|Y_t|<\infty$) ...
-1
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0answers
64 views

Doob-Meyer Decomposition [closed]

I would like to find the Doob-Meyer Decomposition on the form $X[k] = X_{DM}[k] + E[k]$ for the binomial counting sequence $X[k]$. The binomial counting sequencer is given by $X[k] = ...
1
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1answer
200 views

Application of Optional Sampling Theorem

Lets assume that Brownian Motion starts from some point $x$ for which $a<x<b$ holds. Let $\tau=\inf\{t:B_t\not\in [a,b]\}$ be a stopping time. Now I want to prove that for $\theta>0$ ,an ...
1
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1answer
91 views

Changing Brownian motion and law

If I have some variable that depends on Brownian motion, how do I see clearly that replacing that Brownian motion with a different Brownian motion won't affect the law of my variable? To make this ...
1
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0answers
115 views

Expectation of $n$-dimensional Inverse Bessel Process

I think the main problem for me is to calculate the integral of $$\int_{0}^{\infty}\frac{e^{-\frac{r^2}{2t}}}{\sqrt{x^2+r^2}}r^{n-1}dr,n\geq2$$ For n=2, change of variable $y=\sqrt{x^2+r^2}$ would ...
1
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1answer
125 views

Expectation value quantum mechanics momentum operator

What is the random variable that belongs to the expectation value of momentum in quantum mechanics. Or in general: Is there any way we can define the expectation values that occur in quantum mechanics ...
4
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3answers
698 views

Distribution of stochastic integral

Assume that $\mathrm{d}S = \sigma \, \mathrm{d}W$ with initial level $S(0)$ and where $\mathrm{d}W$ is usual Brownian motion. Now $$A(T) = \frac{1}{T} \int_0^T S(t) \, \mathrm{d}t.$$ ...
0
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1answer
37 views

Probability Space and proof of existence for my specific problem involving stochastic differential equations

I have a question regarding the probability space for my problem. This deals with radiation therapy. If X(t) and Y(t) represent the number of two types of cancer cells. X(t) and Y(t) satisfy the ...
0
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1answer
64 views

Cadlag process integration

Let $A,B$ be non-decreasing cadlag processes such that $A_0 = B_0 = 0$ and limits $A_\infty = \lim_{t \to \infty} A_t$ and $B_\infty = \lim_{t \to \infty} B_t$ are finite. I am trying to prove that ...
2
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2answers
149 views
2
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1answer
211 views

determine Fisher information of $N(0,\sigma^{2})$ over $\sigma^{2}$

So far i've got that $I(\sigma^{2}) = E_{\sigma^{2}}[\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(X)]^{2}$. And i got that $\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(x) = ...
1
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1answer
27 views

How to show $F(x_1,\ldots,x_n) = \int_0^{x_1}\cdots\int_0^{x_n}\prod p(t_k,t_{k-1},y_{k-1},y_k)dy_1\cdots dy_n$?

Suppose $B_t$ a Brownian motion, and I fix $0=t_0<t_1<\cdots<t_n$. I want to show that $$P(B_{t_1} \leq x_1, \ldots, B_{t_n} \leq x_n) = F(x_1,\ldots,x_n) = ...
2
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1answer
383 views

Ornstein - Uhlenbeck Process

I'm considering the Ornstein - Uhlenbeck process $$ V_t = e^{\lambda t} v_o + \int_0^t e^{-\lambda (t-s)} dB_s $$ with $ \lambda > 0$, $v_0 \in \mathbb{R}$, and $B$ a brownian motion. I want to ...
1
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1answer
57 views

covariation of martingales

For a bivariate brownian motion $(B_t,W_t)_t$ we have that the covariation is given by $\langle W,B\rangle _t = t\rho$ where $\rho$ is the (constant) correlation between $W$ and $B$. Does this hold ...
2
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1answer
195 views

$L^1$ convergence of a sequence of stochastic integrals and convergence of their quadratic variations

On a filtered probability space $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$ containing a Brownian motion $W_t$. Let $\sigma^n_t$ be a sequence of square intergable adapted processes and consider: ...
1
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1answer
78 views

Weird equality of expectations involving stochastic integral

First of all, sry for the title. I just couldn't figure out any better description for this weird problem: Let $X$ be a bounded real r.v. and $(A_t)_{t\geq0}$ an increasing bounded process (hence ...
1
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1answer
40 views

Special adapted process

I want to construct a process $(Z_n)$ adpated to a filtration $\mathcal{F}_n$ such that $$E[Z_{n+1}\mid Z_n]=Z_n, E[Z_{n+1}\mid\mathcal{F}_n]\not=Z_n$$ I start by taking the three values $1,2,3$; ...
0
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1answer
92 views

Simple random walk two properties

First suppose that $X_1,...,X_t$ are IID random variables and $P(X_1=1)=p, P(X_1=-1)=1-p=q$ for $p\in (0,1)$, then $S_t=X_1+...+X_t, S_0=0$ is the simple random walk. I have two questions: (1) Why ...
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1answer
97 views

Stochastic differential for general semimartingale

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: "$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
2
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1answer
83 views

Can an Itō integral be $\infty$?

In other words, can $\int_0^t f(s)dW(s)$ = $\infty$? Thanks!
1
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1answer
102 views

Variance of this probability density

I have the function $\rho(x) = \frac{sin^2(x)}{x^2}$ and I want to calculate its variance on $\mathbb{R}$. Does anybody know how to do this? Cause afaik the integral does not converge.
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2answers
198 views

A mean square derivative

I'm doing an exercise where I have to check some properties about these two stochastical processes: $X(t)=At+B\;\;$ and $\;\;Y(t)=\frac{1}{t}\displaystyle\int_{0}^{t}X(\tau)\;d\tau$, $t>0$. ...
0
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1answer
201 views

Riemann integral of a function of the Wiener process

I'm trying to solve this exercise: $\bullet$ Find mean and variance of the next stochastical process, and prove it is a second order stationary process: ...
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0answers
33 views

A brownian bridge evaluate at a particular random variable

I was wondering of someone could help with the following. I have a random variable given as $\lambda^{*}=\arg \max_{\lambda \in (0,1)} [B(\lambda)-\lambda B(1)]^{2}/\lambda(1-\lambda)$. I am now ...
1
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1answer
23 views

Stochastics with induction

prove that for all $n \in \mathbb{N}$: $\sum_{r=0}^n \binom{n}{r}(-1)^{r} = 0$. The base step is easy, i only get lots of problems when i try to mess with the sum boundries.... so far i've tried: ...
3
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0answers
54 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
1
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1answer
107 views

Markov-chain properties

I have some questions about a Markov-chain $(X_n)$ on a finite state-space $S$ with transition matrix $P$. A function $f:S\rightarrow\mathbb R$ is a columns vector and $Pf$ therefore a matrix ...
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0answers
96 views

Subtraction of Probability Measures

I have just read that apparently the following two conditions are equivalent: $$ \int f dP \geq \int f dQ \Longleftrightarrow \int f d(P-Q) \geq 0$$ for $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ and ...
0
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2answers
110 views

Stochastic differential

Im really new in the stochastic procceses please help me. How can I solve this stochastic differential equation? $$dX = A(t)Xdt$$ $$X(0) = X_0$$ If $A$:[0,$\infty$]$\to$ $R$ is continous and $X$ is ...
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0answers
72 views

Atypical exponential martingale

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} ...
1
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1answer
48 views

Stochastic processes with non-zero higher order variations

I'm under the impression that how non-zero quadratic variation of the Brownian motion results in Itō's lemma or in general, the creation of the Itō's calculus. I'm also aware that stochastic integral ...
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2answers
200 views

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I'm in the last period of the undergraduate course and I have interest in this field, but my university don't have a specialist in ...
0
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1answer
70 views

sup of a submartingale until $t$

My problem: For any submartingale $(X_s)_{s\geq0}$ and for all $t\geq0$ show that $\sup_{s\in[0,t]}\mathbb{E}[|X_s]|]$ is a.s. finite. What I have until now: I know that $\mathbb{E}[X_s]$ is ...
1
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1answer
323 views

About stationary and wide-sense stationary processes

I have just started with stochastical calculus, and I need some help with a pair of problems: $\bullet$If $X(t)$ is a mean square differentiable wide-sense stationary stochastical process then the ...
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1answer
134 views

Basic (continuous) martingale properties

I just learned about martingales in continuous time and solved some basic exercises. But unfortunately there are some seemingly easy and surely basic things I still have problems with. 1) Let ...
2
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0answers
216 views

Difference of two convex functions

This is an exercise from a probability textbook on Ito's formula, basically Ito's formula extends to functions of this type. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f$ is ...
1
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1answer
103 views

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time $$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where ...
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0answers
124 views

Intensity Function of Stochastic process`

I'm fitting some financial data to a model based on a stochastic process and evaluating the fit of it by looking at the compensator. However, I cannot understand well what does it mean to take the ...
1
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0answers
143 views

First hitting time on a element of $\mathcal B ( \mathbb R^d) $ a (right, left) continuous path stochastic process

It's known that, given $\Gamma \in \mathcal B (\mathbb R ^d)$ and $X = > (X_t)_{t\geq 0}$ with right-continuous path, the random time $$T_{\Gamma} = \inf \{ t\geq 0 : X_t (\omega) \in ...
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0answers
37 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...