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

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2
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135 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. ...
0
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0answers
147 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 ...
1
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1answer
101 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
48 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 ...
0
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1answer
83 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 ...
1
vote
1answer
68 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
33 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
121 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
55 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, $t \to B_t(\omega)$ ...
1
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1answer
55 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 ...
0
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1answer
45 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 ...
-1
votes
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
votes
1answer
151 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
293 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
83 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
26 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 ...
0
votes
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 ...
0
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2answers
68 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... ...
1
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1answer
36 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
votes
1answer
218 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 ...
2
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2answers
69 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 ...
1
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1answer
75 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
64 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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votes
1answer
47 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
43 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.
1
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1answer
74 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 ...
0
votes
1answer
51 views

Show the following is Local Martingale

$X_t$ bessel square process which satisfies $$\mathop{dx_t}= 2(a+1) \mathop{dt} +2 \sqrt{x_t} \mathop{dB_t}$$ and $u$ is a function which satisfies $x^2 u'' +x u' -u(a^2 + b x^{2p+2})= 0$. How can I ...
1
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1answer
95 views

Representing a stochastic integral as product of a unknown random variable and a standard normal random variable

Consider a probability space $(\Omega,\mathcal F, (\mathcal F_t)_{t\geq0},\mathbb P)$ where $\mathbb F=(\mathcal F_t)_{t\geq0}$ is generated by $B=(B_t)_ { t \geq 0}$ a standard brownian motion ...
4
votes
1answer
114 views

Karatzas and Shreve Problem 3.3.38

Let $X$ be a continuous process and $A$ a continuous, increasing process with $X_0 = A_0 = 0$, a.s. Suppose that for every $\theta \in \mathbb{R}$, the process $$Z_t^{\theta} = ...
1
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1answer
130 views

Exercise 3.3.25 of Karatzas and Shreve

This is the Exercise 3.25 of Karatzas and Shreve on page 163 Whith $W=\{W_t, \mathcal F_t; 0\leq t<\infty\}$ a standard, one-dimensional Brownian motion and $X$ a measurable, adapted process ...
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0answers
63 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
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0answers
34 views

New stochastic calculus

I am interested in Kagi and Renko approach and hope I can use it for a random walk process. I searched for it on internet but I couldnt find any basic material to read about it. Can someone please ...
2
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2answers
67 views

Ito's Isometry for three factors

Ito's Isometry states the following: If $\{W_t\}_{t\ge0}$ is a Brownian motion and $\{\phi_t\}_{t\ge0},\{\psi_t\}_{t\ge0}$ are two non-anticipative piecewise-continous processes with $\mathbb ...
2
votes
1answer
173 views

Kolmogorov Backward Equation for Itô diffusion

Let $(X_t)_{t\ge 0}$ be the solution of the SDE $$ X_t = X_0 + \int_0^t \mu(s,X_s) \,ds + \int_0^t \sigma(s,X_s) \,dB_s, \quad t\ge 0 $$ where $\mu(s,x)$ and $\sigma(s,x) $ are Lipschitz continuous ...
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0answers
46 views

Is the stochastic integral a Gaussian process [duplicate]

Let $Y_t=\int_0^tW_s^2 d W_s$. It is a martinglae with $\langle Y\rangle_t=t^3$, and then, by computing CF using Ito's lemma, \begin{align*} d e^{i\theta Y_t}&=i\theta e^{i\theta ...
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1answer
148 views

Background for studying and understanding Stochastic differential equations

Assume I have back ground of the following knowledge based on the textbook as : ODE : ODE by Tenenbaum Baby probability : Ross 's baby probability Baby real anlysis : Bartle's introduction to real ...
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2answers
83 views

solving a stochastic differential equation

How to solve $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$ together with the initial condition $X(0) = X_0$.
0
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0answers
52 views

Stochastic control problem

Suppose we have the following stochastic optimal control probelm \begin{equation} V(t,x) = \sup_{u} \mathbb{E}[ g(X_{T}) +\int_{0}^{T}f(t,X_{t},u_{t})dt] + (\mathbb{E}[ ...
0
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0answers
33 views

change of variable in black-scholes equation with dividend

In the black-scholes equation with dividend ${\textstyle{{\partial V} \over {\partial t}}} + (r-q)S{\textstyle{{\partial V} \over {\partial S}}} + {\textstyle{1 \over 2}}{\sigma ...
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2answers
103 views

Martingale Proofs

I havent been able to find an analogous question and our textbook is lacking in good examples, so I could use a little help with this rather straight forward martingale problem: Let X=(Xn) be a ...
3
votes
1answer
143 views

What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
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0answers
38 views

Clarification on the definition of the îto integral

I have a question regarding the îto integral. In the definition of the integral we basically take the limit in probability of the sum $\Sigma H(t_i)\cdot(B(t_{i+1})-B(t_i))$ for suitable $H$ and a ...
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0answers
57 views

calculation on Ito's Lemma

I have a question on the calculation on Ito's Lemma: ${{Y}_{t}}={{t}^{{{W}_{t}}}}$ solve for $d{{Y}_{t}}$ the following is my solution [\begin{align} & d{{Y}_{t}}=\frac{\partial Y}{\partial ...
0
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1answer
205 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
152 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. ...
0
votes
2answers
84 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
156 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
votes
0answers
71 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
79 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
vote
1answer
38 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 ...