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

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114 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 ...
1
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1answer
249 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 ...
1
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1answer
50 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 ...
4
votes
2answers
476 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 ...
-1
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1answer
14 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 ...
4
votes
1answer
129 views

Strictly stationary exponential Ornstein-Uhlenbeck process?

Can one define the initial value of the exponential Ornstein-Uhlenbeck process $r$, defined by $$r(t) = e^{y(t)}\quad\text{with}\quad dy(t) = k(θ −y(t)) \mathrm dt+\sigma \mathrm dW(t),$$ such that ...
0
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0answers
25 views

Evaluation of $\mathbb E[\int _{t_1} ^{t_2} f(s, X_s^{t,x} )ds \mid \mathcal F _{t_1} ]$ for a markovian SDE solution.

Given a probability space $(\Omega, \mathcal F , \mathbb P)$, a filtration $\mathbb F = (\mathcal F _t )_{t\geq 0}$ and $\mathbb F$-adapted brownian motion $W=(W_t)_{t \geq 0}$, consider $X^{t,x}= ...
0
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1answer
67 views

Solution of Vasicek model driven by infinite activity Levy process

Say that we have the Vasicek model $dY_{t} = \alpha(\beta-Y_{t})dt+\sigma dX_{t}$ where $X_{t}$ is an infinite activity Levy process, $\alpha$,$\beta$ and $\sigma$ are constants. I know that in the ...
1
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1answer
103 views

Expectation of stochastic differential equation

I have solved the nonlinear stochastic equation $dX_t=\frac{1}{2}a(a-1)X_t^{1-2/a}dt+aX_t^{1-1/a}dW_{t}$, by reducing it to a linear one (change of variables $Y_{t}=X_{t}^{1/a}$ and applying Ito ...
0
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1answer
25 views

Autocorrelation of Radial Stochastic Process with Planar Derivatives

I have a random field $h(\vec{r})$ that depends on $\vec{r}=(x,y)$, such that \begin{equation} \langle h(\vec{r})h(\vec{r}+\vec{r}') \rangle \sim \exp(-||\vec{r}-\vec{r}'||/a^2) \end{equation} where ...
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0answers
85 views

Finding dynamics of a dividend paying stock under arbitrary numeraire

Assuming I have a dividend paying asset $S$ with dividend process $D$. Now I would like to use the bank account process $B$ as numeraire and determine the dynamics of $S$ under the the corresponding ...
1
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1answer
39 views

Writing $A(t)=1+S_1S_2^{-1}$ as an Ito diffusion process.

Let $W$ be a Wiener process/Brownian motian and let $$ \begin{align} \mathrm{d}S_1 &= 2S_1(t)dt +3S_1(t) dW\\ \mathrm{d}S_2 &= 4S_2(t)dt +5S_2(t) dW \end{align} $$ Now I'd like to write ...
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0answers
62 views

optimization problem in mathmetical finance using convex duality

I'm interested in the application of stochastic processes and stochastic calculus in mathematical finance. In my lecture I often see a certain optimization problem usually of a convex function. ...
0
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1answer
49 views

Prove that a process is a martingale

Let $W_t$ be a Wiener process, and let $N_t$ be a Poisson process with intensity $\lambda$. We define a process $Z_t = \lambda Wt^2 − N_t$ Prove that the process $Z_t$ is a martingale
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0answers
41 views

Asymptotic result on quadratic variation of a semi-martingale linear functional estimator

In the same context of this previous question. Consider $$ \mathcal E^{(n)}_t := \sqrt{n}(\widehat\Lambda_n(\phi)_t - \Lambda(\phi)_t )$$ I desire to prove that $$ \left \langle \mathcal ...
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0answers
72 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
2
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0answers
52 views

Is it sensible to always assume that the “usual conditions” always hold?

I've read in several places that it is reasonable to assume that the usual conditions (that the filtered space is complete, and that the filtration is right-continuous) hold since one can always ...
2
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1answer
129 views

Oksendal SDE book mistake?

I am reading through Oksendals SDEs. I think there may be a mistake in question 5.18b and I can not find an errata so I was looking for some confirmation. The problem concerns the following SDE ...
0
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2answers
94 views

Integration of Gaussian process

Let $\textbf{G}(t)$ be a zero-mean tight Gaussian process and $f(t)$ be a deterministic function. What theorem can be used to prove that $\int_0^\tau \textbf{G}(t)df(t)$ is a zero-mean Gaussian ...
0
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0answers
41 views

Expected Value of the minimum stock price where stock price is an exponential brownian process

Hi I am trying to figure out what would be the solution to the following equation: $\tilde{E}[S_{min}]$ where $S_{min}$ is the minimum stock price and the stock price is of the form ...
1
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2answers
287 views

Prove that integral is a Gaussian random variable, compute its mean and variance

I have to prove that $X_t=\int_0^t W_s ds$ is a Gaussian random variable. I need also to compute it's mean and variance. My attempt: Let $W_t$ be a simple adapted process ...
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1answer
98 views

Limit of stochastic integrals?

Let $(W_t)t$ be a Wiener process. I want to find the limit for $\epsilon\to 0$ of $$\frac{W_t^2}{2\epsilon}\chi_{(-\epsilon,\epsilon)}(W_t)-\int_0^t ...
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1answer
90 views

Riemann integral over Itô integral?

let's say I have the Itô integral $I(t) = \int_{0}^{t} f(s)dW_{s} $ How do I then calculate $I_{2}(u) = \int_{0}^{u} I(v)dv = \int_{0}^{u} (\int_{0}^{t} f(s)dW_{s})dv$ ? Is it going to become $0$ ...
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0answers
41 views

How to prove d($\int_t^\infty$$e^{-ru}d\beta_u$)=-$e^{-rt}d\beta_t$?

I found it difficult to state clearly that: d($\int_t^\infty$$e^{-ru}d\beta_u$)=-$e^{-rt}d\beta_t$ , but intuitively it is correct, isn't it? I guess the Gaussian property of the integral may be used ...
3
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2answers
86 views

does continuity of sample paths imply continuity of natural filtration?

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (not necessarily complete) and let $X = (X_t)_{t \in [0, \infty)}$ be a real-valued stochastic process defined on it. In general, is it ...
2
votes
2answers
63 views

Show that process satisfy given equation

I have to show that process (1) $$X_t=e^{-bt}X_0+\int_0^te^{-b(t-s)}\sigma dW_s$$ satisfies the following equation (2) $$dX_t=-bX_tdt+\sigma dW_t$$ My attempt: Multiply both sides of (1) by $e^{bt}$ ...
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0answers
53 views

Which equation does this process satisfy?

1) Which equation does the following process satisfy: $$Y_t:=W_t^{4}$$ Where $W_t$ is Wiener process. 2) Prove that $$\mathbb{E}W_t^{4}=3t^2$$ Using Ito formula for $Y_t$ is a good point to start? ...
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0answers
34 views

Defining the Radon-Nikodym as a solution to an SDE

Can someone please clarify this to me: If I have the Radon-Nikodym $L_t=\frac{dQ}{dP}$, on $\mathcal{F}_t$, then I know that $L_t$ is a non-negative P-martingale. So in many textbooks they say it is ...
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2answers
486 views

Variance of Time-Integrated Ornstein-Uhlenbeck Process

I'm attempting to filter white noise from a deterministic, finite-power signal using a low-pass filter. This filter can be described using an exponentially-decaying response function: $$ h(t) = ...
8
votes
1answer
611 views

Gradient Descent on Non-Convex Function Works But How?

For Netflix Prize competition on recommendations one method used a stochastic gradient descent, popularized by Simon Funk who used it to solve an SVD approximately. The math is better explained here ...
0
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1answer
39 views

Solution to stochastic differential eqn [closed]

How do you solve this stochastic differential equation? Not sure how to start on this. Need some guidance.
4
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1answer
82 views

Laws and Moments of two dimensional brownian motions

I am a bit rusty on this. So let us consider the following two dimensional standard Brownian motion issued from zero defined on the probability space $(\Omega, \mathcal{F},\mathbb{P})$ (note that, in ...
4
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0answers
53 views

2-D exponential functional brownian motion

I'm looking for the distribution of $X = \int_0^T e^{-W_t} dt \int_0^T e^{W_t}dt$ and $Y = \frac{\int_0^T e^{-W_t} dt}{ \int_0^T e^{W_t}dt}$ (where $W_t$ is a standard brownian motion) On most ...
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0answers
95 views

Solution of the problem 1.2.2 from “Brownian Motion and Stochastic Calculus” of Karatzas & Shreve

Does anybody have the solution of that problem, please? I don't understand the relation between random variables $X$ and $T$. Regards Edit : Thank you for the comments. Let me first apologize for ...
1
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1answer
74 views

Solution of two (first) SDEs.

I'm about to study SDE's for the first time and I'm kinda having troubles "guessing"/"finding" solutions. Also I don't really know how and when analogies to simple ODEs are allowed (e.g. to get a ...
2
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0answers
161 views

When does almost sure convergence of stochastic integral imply $L^2$ convergence?

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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0answers
195 views

Proving the martingale property of stochastic exponentials of pure jump processes

I am playing with different versions of compound-Poisson like processes with regime-switching features. Then I take stochastic exponentials of these to define a change of measure process. However, how ...
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1answer
105 views

Quadratic covariation of Itô processes

I haven't found any similar question in the forum, so I trust some of you will find this thought-provoking (at the very least). Perhaps you can help me. Let's consider first the two following ...
0
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1answer
53 views

Expectation of stopping times

Let B = (Bt)t¸0 be a standard Brownian motion started at zero, let $X_t$ be a non negative stochastic process solving: $dX_t=1/X_tdt+dB_t$ Compute $E[\sigma]$ when $\sigma=\inf \{ t\ge 0 : X_t= 1 ...
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1answer
91 views

expectations of Brownian motions

Let $B_t$ be a standard Brownian motion started at zero, and let $M_t$ be a stochastic process defined by $M_t=3\int_0^{t^{1/9}} s^4dB_s$ Compute $E\left[1+\int_0^t(1+M_s)^4 dM_s\right]$. Compute ...
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1answer
77 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
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0answers
34 views

Stochastic Increments

Can anybody help me generate the increments $\Delta$$W_n$ in mathematica. I Know $W_{i+1}=w_i+Z_{i+1}\sqrt{\Delta t}$ where the $Z_i$ are independent and standard normal. But I cant make any code to ...
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2answers
123 views

Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
1
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1answer
59 views

What is wrong with my example where the Itô Integral and Riemann-Stieltjes Integral don't coincide?

I have an interesting question concerning those two integrals. Considering a Brownian motion $(B_t)_{t \geq 0}$ with start in $x$. We can choose an $\omega \in \Omega$ such that, $t \to B_t(\omega)$ ...
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1answer
56 views

Stochastic differential equation for $Y(t)=\sqrt{X(t)}$

Assume that $X(t)$ solves the stochastic differential equation $$dX(t)=\sigma(t)dW(t)+\mu(t)dt$$ with $\mu(x)=bx+c$ and $\sigma^2(x)=4x.$ Assume that $X(t)\ge 0$. Find the stochastic differential ...
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1answer
46 views

How to show $Y(t)=\ln(\frac{X(t)}{1-X(t)})$ has a constant diffusion coefficient.

A PROCESS $X(t)$ on $(0,1)$ has a stochastice differential with coefficient $\sigma(x)=x(1-x)$,Assuming $0<X(t)<1$ , show that the PROCESS defined by $Y(t)=\ln(\frac{X(t)}{1-X(t)})$ has a ...
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2answers
51 views

Let $X(t)=(1-t)\int_{0}^{t}\frac{dB(s)}{1-s}$ I want find $dX(t)$ [closed]

Let $X(t)=(1-t)\int_{0}^{t}\frac{dB(s)}{1-s}$, where $0\le t < 1$.Find $dX(t)$. thanks for help.
2
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1answer
160 views

How to solve $\mathrm dX(t)=B(t)X(t)\mathrm dt+B(t)X(t)\mathrm dB(t)$ with condition $X(0)=1$?

I want to solve the stochastic differential equation $$\mathrm dX(t)=B(t)X(t)\mathrm dt+B(t)X(t)\mathrm dB(t)$$ with condition $X(0)=1$.
3
votes
1answer
376 views

Expectation of stochastic integrals related to Brownian Motion

I'm trying to solve a problem that's now doing my head in a bit. I'll share with you the question and let's see if somebody can shed some light into the matter: Let B be a standard Brownian Motion ...
0
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0answers
87 views

Jump diffusion process with sum of Poisson processes a martingale?

Hi Mathematics community, assume you have dynamics of a jump diffusion process consisting of a Brownian motion and a sum of compensated (not necessarily independent) Poisson processes, i.e. ...