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

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1answer
289 views

Ito Isometry for conditional expectations

Is Ito's isometry true for conditional expectations too? I mean, is it true that:$$\mathbb{E}\left[\left(\int_0^tX_sdB_s\right)^2\ |\ \mathcal{F}_t^B\right]=\mathbb{E}\left[\int_0^tX^2_sds\ |\ ...
2
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2answers
122 views

Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$

Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
0
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2answers
400 views

Conditional Expectation of integral of Wiener process

Let $W_t$ be a standard Wiener process. How can we calculate: $$\mathbb{E}\left[\int_0^t|W_r|^2\text{d}r \ |\ \mathcal{F}_s\right]$$ where $(\mathcal{F}_s)_{s\geq0}$ is the natural filtration?
3
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1answer
123 views

Applying Ergodic Theorem on fractional Brownian motion

For a fractional Brownian motion $B_H$ consider the sequence for $p>0$ $$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$ By the Ergodic Theorem it is ...
3
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2answers
484 views

Show that this process is a martingale

Let $B_t$ be a Brownian motion and $M_t=\max_{0\leq s\leq t}B_s$. Show that: $$(M_t-B_t)^4-6t(M_t-B_t)^2+3t^2$$ is a martingale for $t\geq0$.
2
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1answer
38 views

Quasimartingale is Quasi-Dirichlet process

a paper I read states, that a Quasimartingale (an process $(X_t)_{t\in [0,T] }$ with $\mathbb E[|X_t|]<\infty$ for all $t\in [0,T]$, which suffices $$\sup_\Delta \sum^{n-1}_{j=0} \left\|\mathbb ...
4
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1answer
182 views

Show that $M_t$ is a Standard Brownian Motion

Let $M=(M_t)_{t\geq0}$ with $$M_t=\int_0^{\log\sqrt{1+2t}}e^s\text{d}B_s$$ where $(B_t)_{t\geq0}$ is a Standard Brownian Motion. Show that $M$ is also a Standard Brownian Motion and compute ...
2
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0answers
68 views

How to calculate the following expectation

I have a problem to find the expectation of the following expression, $$E\left[W_T e^{\int_0^T(W_s)ds}\right].$$ Here, $W_T$ is a Brownian motion. Any suggestions as to how to proceed with it? Many ...
6
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0answers
310 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
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0answers
412 views

Show that this semimartingale is a local martingale

Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a ...
3
votes
1answer
67 views

Checking a solution for a SDE

I want to show that the process $Y(t) = e^t \int_0^t e^{-s}dW(s)$ satisfies the following SDE: $dX(t) = X(t)dt + dW(t), \ \ t\geq 0 , \quad X(0) = 0$ I think the right approach is to use Ito's ...
1
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0answers
78 views

Fractional Brownian motion, selfsimilar

Let $0<H<1$. A real-valued Gaussian process $\left(B_H(t)\right)_{t\geq 0}$ is called fractional Brownian motion (fBm) if $\ \mathbb{E}[B_H(t)]=0$ and ...
2
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2answers
258 views

Martingale representation theorem

Trying to figure out how to solve problems on the 'form': Find a real number $z$ and a square integrable, adapted process $\psi(s,w)$ such that $$G(w) = z + \int \psi(s,w)\,dB_s(w)$$ for som ...
1
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1answer
31 views

a homework question about Levy air

I have a question in my homework: Let $X_t$ and $Y_t$ be two Brownian motions issue de $0$ and define $$S_t=\int_0^tX_s\,dY_s-\int_0^tY_s\,dX_s$$ Show that $$E[e^{i\lambda S_t}]=E[\cos(\lambda ...
3
votes
1answer
191 views

Expectation of stopping time

Let $X_t$be the solution to the SDE: $dX_t=-X_tdt+dB_t$, $X_0=0$ Then $X_t$ is the Ornstein–Uhlenbeck process $X_t=e^{-t}\int_0^te^sdB_s$. I want to calculate $\mathbb{E}[e^\tau X_\tau]$ when ...
1
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2answers
242 views

Ito Process $\Longrightarrow$ continuous semimartingale

I know that the Ito integral is defined in general for continuous semimartingales. But it can also be defined only for Ito processes. My question is if every process $X_t$ satisying a SDE of the form ...
6
votes
1answer
262 views

Expected Value of Brownian motion using ito isometry

Find $$ E\ \left[\left(\int_{0}^T e^{s+W_s}dW_s \right)^2\right], $$ where $(W_s)$ is a Brownian motion. I tried to use Ito isometry to solve this question, but still not yet to find the right ...
2
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1answer
360 views

Table of Ito Integrals

Are there any tables with a collection of common Ito Integrals, their equivalent forms, etc. that anyone knows of? Did a search but didn't come up with anything and was wondering if anyone knew of ...
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0answers
126 views

Can we apply Ito's formula?

Suppose that we are given the following processes: $B=(B_t)_{t\geq0}\ $ a standard Brownian motion starting at zero, $I=I_t=\int_0^t|B_s|^2ds,\ S=S_t=\sup_{0\leq s\leq t} B_s$ for $t\geq0$ and a ...
2
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0answers
134 views

Integral representation of fractional Brownian motion

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
2
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1answer
190 views

A question about Itô's representation for $\cos(B_T)$

According to Itô’s representation, any $\xi \in L_2(\Omega, F_T , P)$ has a unique representation: $ \xi = E(\xi) + \int_0^T H_s dBs$ where $(H_s)$ is an adapted process belonging to $L_2$. $B$ is a ...
2
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1answer
107 views

Long Range Dependence, Fractional Brownian Motion

A stationary sequence $(X_n)_{n\in\mathbb{N}}$ exhibits long-range dependence if the autocovariance function $\rho(n):=\mathrm{cov}(X_k,X_{k+n})$ satisfy $$\lim\limits_{n\to\infty}{\rho(n) \over ...
3
votes
0answers
100 views

Stochastic differential equation solution suggestion

Any suggestion on solving the stochastic differential equation \begin{align} dW(t) = d\widetilde{W}(t) + \left(\frac{\kappa - W(t)}{\tau-t} - \frac{1}{\kappa - W(t)}\right)dt \end{align} where ...
6
votes
3answers
723 views

Expected value of average of Brownian motion

For a standard one-dimensional Brownian motion $W(t)$, calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$ Note: I am not able to figure out how to approach this problem. All ...
5
votes
0answers
96 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
4
votes
1answer
190 views

Limiting distribution of Ornstein-Uhlenbeck process

Let $X_t = e^{-\lambda t} \left(X_0 + \int _0^t e^{\lambda u} dW_u\right)$ where $(W_u)_{u \geqslant 0}$ is a Wiener process, $X_0$ random variable of law $\nu$ and independent of $\int _0^t ...
0
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1answer
36 views

Question on weak convergence of random variables

Let $X_n, Y_n, X$ be real random variables such that $X_n \to X$ weakly and $\mathbb{P}_{Y_n} = N(0, 1/n)$ for all positive integers $n$. I am trying to prove that $X_n + Y_n \to X$ weakly as well. ...
0
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1answer
108 views

Weak convergence discrete space

Let $X_n$, $n = 1, 2, 3, \ldots$, and $X$ are random variables with at most countably many integer values. Prove that that $X_n \to X$ weakly if and only if $\lim_{n \to \infty} P (X_n = j) = P(X = ...
0
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1answer
315 views

Show that this continuous local martingale is a martingale

We are given the following SDE: $$dX_t=X_tdt+\sqrt{2}X_tdB_t, \quad X_0=1,$$ and $$F(x,t)=e^{-t}x,\quad t\geq0,\; x\in\mathbb{R}.$$ We are asked to apply Ito's formula to $F(t,X_t)$ for $t\geq0$ ...
17
votes
2answers
816 views

Translations of Kolmogorov Student Olympiads in Probability Theory

I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward. I ...
2
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1answer
1k views

Transition density and distribution: (Ornstein–Uhlenbeck process)

Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE below with $\alpha,\,\beta,\,\gamma$ constants: $$ dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0} $$ ...
2
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1answer
131 views

Black scholes model type

I want to study the following market: $$S_1(t)=S_1(t)(\mu_1dt + \sigma_1dW_1(t))$$ $$S_2(t)=S_2(t)(\mu_2dt+\sigma_2dW_2(t))$$ for $t\in [0,T]$, constants $\mu_i,\sigma_i$, initial values ...
0
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0answers
249 views

Girsanov Transformation Example

Is this the correct use of Girsanov's transformation where $B_{n}$ is a discrete Brownian motion? For example computing: $E[(B_{n}+2n)^{2}]$ Set: $\widetilde{B_{n}}=B_{n}+2n$ And ...
1
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1answer
207 views

Integrate Brownian motion with respect to independent Brownian motion

we are wondering what can be said about the distribution of that? We are considering standard Brownian motions and the integral from 0 to 1... More precisely: What can be said about the distribution ...
1
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2answers
388 views

Sum of independent random variables almost surely constant

I am trying to solve the following problem: Let $(\Omega, \mathbb{A}, \mathbb{P})$ be a probability space and $X_1, X_2, \ldots, X_n$ independent real random variables. Prove that the sum $X_1 + X_2 ...
2
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1answer
376 views

Expectation of exponential martingale and indicator function.

Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$. I want to evaluate $$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
2
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1answer
79 views

Readings necessary to understand Ito Integrals?

I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background ...
2
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0answers
180 views

Explaining Ito formula to an analyst

From the point of view of analysis, what is Ito formula? Is it an integral by substitution, or, a radon-nikodym derivative? Define the probability space $$ \left(C\left(\Bbb ...
0
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1answer
36 views

Is it true that $X(t)^a > K \iff X(t) > K^\frac1a$

Let $a \in \mathbb{N}$, $K \in \mathbb{R^+}$ and $X(t)$ be a geometric Brownian Motion. Is the following true? $$X(t)^a > K \iff X(t) > K^\frac1a$$ The context of the above is that I want to ...
2
votes
1answer
117 views

Bounded variation and continuous local part when using Ito's Formula

When we apply Ito's Formula to a continuous semimartingale, which is the bounded variation part and which is the continuous local martingale part? Is there a general rule or does it depend on the ...
4
votes
1answer
490 views

covariance of integral of Brownian

What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$. Any ideas? Thanks you very ...
3
votes
1answer
176 views

Optional sampling exercise

I came across the following exercise in Stochastic Calculus: Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process: $M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
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0answers
167 views

Independent Exponentially Distributed Random Variables - Athletes Problem??

Q) At a javalin competition two athletes (1 & 2) are competing against each other. Each has one attempt to throw the javalin. Assume the acheived distance of a throw ($L$1 & $L2$) [note these ...
0
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1answer
190 views

$dt$ terms have zero quadratic variation

Why does $ds$ integral have zero quadratic variation? Even if I have a integral of the form $$\int X_s ds$$ where $X$ is a stochastic process? I know that a continuous process of finite variation ...
1
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0answers
193 views

Integral with respect to Wiener process.

Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process. My First Question What is ...
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0answers
57 views

Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation

Let $B$ be a standard Brownian motion, and, $$ X_t=e^{\int_0^t f(B_s)ds}, $$ for some function $f$. What are the condition on $f$ for $X_t$ to be of finite variation? Let $Y_t=\int_0^t f(B_s)ds$, if ...
2
votes
1answer
143 views

Quadratic variation of $X_t=\int_0^t B_s \, ds$

Let $B$ be a standard brownian motion and $$ X_t=\int_0^t B_s \, ds. $$ What is the quadratic variation $[X]_t$ of $X$? I see $dX_t$ as an sde with drift term $B_t$.
2
votes
0answers
152 views

Multivariate Stochastic Process

I am dealing with a multivariate Ornstein Uhlenbeck style SDE. Specifically $dx_{t,j}=\kappa_{j}(x_{t,j-1}-x_{t,j})dt+\sigma dW_{t,j} $ here j=1,2,...,6 , $x_{t,0}=\theta$ , ...
1
vote
0answers
657 views

Stochastic integral: Interchanging the order of expectation and integration

Let $B$ be a standard Brownian motion and $$ X_t=\int_0^t f_s ds+\int_0^t g_s dB_s, $$ where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$. Is it true that $$ ...
0
votes
0answers
76 views

Local martingale iff each component is a local martingale?

This is probably an easy question: A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a ...