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

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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. ...
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1answer
36 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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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
62 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 ...
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0answers
38 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 ...
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1answer
72 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 ...
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2answers
63 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 ...
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33 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 ...
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2answers
110 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
87 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
64 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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40 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 ...
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2answers
50 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
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2answers
51 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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49 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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27 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
160 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) = ...
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1answer
160 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 ...
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1answer
36 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.
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1answer
61 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 ...
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40 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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66 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 ...
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1answer
58 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 ...
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0answers
92 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
94 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
75 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 ...
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1answer
37 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
67 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
48 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
29 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
115 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 ...
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0answers
32 views

How to prove that $P(Z \leq -u \ \text{or} \ Z \geq v) = \frac{4\sigma^2 + (u-v)^2}{(u+v)^2}$ [duplicate]

Let $Z$ be a random variable such that $EZ=0$ and $Var Z = \sigma^2$. Assume that $u,v>0$. How should I prove the following inequality? $$P(Z \leq -u \ \text{or} \ Z \geq v) \leq \frac{4\sigma^2 + ...
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1answer
42 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
37 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
37 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
38 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
100 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
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1answer
185 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
70 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. ...
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0answers
20 views

Weak convergence of discretization scheme with correction

In this article on the Multilevel Monte Carlo method on page 8, http://people.maths.ox.ac.uk/gilesm/files/mcqmc06.pdf, Giles uses a correction term to improve the weak convergence rate of the lookback ...
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1answer
25 views

Meaning of my calculation card game

I have made a calculation and now I do not understand what I did there. It is about the following question: Imagine you have n cards of which there are 2 aces, what is the expectation value to get ...
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2answers
60 views

A stochastic integral computed using Itô's lemma

I need some help with this question: I have to check the following "identity" using Itô's lemma, but I can't see how to do it... ...
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1answer
35 views

About an application of Itô's lemma

I need some help with this exercise. Given the following stochastical differential equation: $dX(t)=\frac{-1}{4}(X(t))^3\;dt+\frac{1}{2}(X(t))^2\;dW(t)$ $X(0)=\frac{1}{2}$ I have to obtain ...
2
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1answer
118 views

The most general version of Ito's lemma

Wiki gives a version of the Ito's lemma for the Ito proccess when we differentiate a function $f(t,X_t)$ of time and some diffusion process. In the general case of multivariate semimartingale ...
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2answers
63 views

Distribution of stochastic integral in small time

Let $W^1$ be a Brownian motion and $\sigma(\cdot)$ be a positive, bounded, continuous function. Define \begin{align*} V_t=\int_0^t\sigma(Y_s)dW_s, \end{align*} where $(Y_t)_{t\geq 0}$ is a ...
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1answer
66 views

Ito Integral surjective?

Let $\Phi\in\mathcal{L}\left(M\right)$ if and only if $\Phi$ is a real predictable process and for every $\left\Vert \Phi\right\Vert_{2,t,M}:=\mathbb{E}\left[\int_{0}^{t}\Phi_{s}^2 d\langle ...
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1answer
61 views

Queueing model - expected outflow

Can anybody please help me how to tackle this question? We have one server. The service time is random with mean 1 minute The arrival rate is constant with 3 customers/minute, but they leave if the ...
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1answer
45 views

$B(t)$ is brownian motion. I want Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,

let $B(t)$ is brownian motion. Find $d(M(t))^2$,where $M(t)=e^{B(t)-\frac{t}{2}}$,
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2answers
41 views

I want Find $d(\frac{X(t)}{Y(t)})$ where $B(t)$ is a brownian motion and $X(t)=tB (t)$ and $Y(t)=e^{B(t)}$.

Let $B(t)$ is a brownian motion and $X(t)=tB (t)$ and $Y(t)=e^{B(t)}$. Find $d(\frac{X(t)}{Y(t)})$ Thanks for help.
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1answer
65 views

meaning of differentiation of stochastic process

Let $X_t,t\in T $ continuous time stochastic process. What is the meaning of $dX_t$ which is differentiation of $X_t$? Does that mean $X_{t+dt}$ and $X_t$ are random variables so $dX_t \approx ...