This tag is used for questions about stochastic integrals - especially for calculations . For questions related to more theoretic aspects of stochastic integrals such as its construction. Stochastic-analysis may be a more appropriate tag.

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15
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2answers
1k views

Brownian bridge expression for a Brownian motion

Let $B_t$ be a standard Brownian motion in $\mathbb R$, then the Brownian bridge on $[0,1]$ is defined as $$ Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s} $$ for $0\leq t<1$. Here ...
13
votes
3answers
392 views

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s ...
11
votes
1answer
808 views

Probability density function of the integral of a continuous stochastic process

I am interested in whether there is a general method to calculate the pdf of the integral of a stochastic process that is continuous in time. My specific example: I am studying a stochastic given ...
10
votes
1answer
482 views

Why do people simulate with Brownian motion instead of “Intuitive Brownian Motion”?

I have just recently begun studying Brownian motion and stochastic calculus at the level of an undergraduate or beginning graduate student of applied mathematics. (Textbooks I've looked at are by ...
10
votes
2answers
402 views

Area enclosed by 2-dimensional random curve

Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
9
votes
3answers
5k views

Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
8
votes
1answer
680 views

Calculate $\mathbb{E}(W_t^k)$ for a Brownian motion $(W_t)_{t \geq0}$ using Itô's Lemma

Show by using Ito's Lemma, for $k \geq 2$ the following result hold. $$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$ where $W(t) = N(0,t)$ is standard Brownian motion. I think ...
8
votes
2answers
349 views

Could someone explain rough path theory? More specifically, what is the higher ordered “area process” and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, ...
7
votes
2answers
883 views

Is this local martingale a true martingale?

Using the Ito's formula I have shown that $X_t$ is a local martingale, because $dX_t=\dots dB_t$, where $$X_t = (B_t+t)\exp\left(-B_t-\frac{t}{2}\right),$$ $B_t$ - is a standard Brownian motion I ...
7
votes
1answer
141 views

Trying to integrate a stochastic RV, $\int_0^t sZ_s \, ds$

I'm not taking an official class (actuarial exams), some fellow "students" created a question (forum discussion), considering the integral in title. This is my attempt at a solution with no real ...
6
votes
3answers
3k views

Itō Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itō integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
6
votes
1answer
830 views

Is this a martingale?

Let $W_t$ be a standard Brownian motion with $W_0 = 0$ and let $Z_t$ solve the stochastic differential equation $dZ_t = 2 Z_t W_t \mathrm{d}W_t$. This has solution $$ ...
6
votes
1answer
2k views

Expected value of the stochastic integral $\int_0^t e^{as} dW_s$

I am trying to calculate a stochastic integral $\mathbb{E}[\int_0^t e^{as} dW_s]$. I tried breaking it up into a Riemann sum $\mathbb{E}[\sum e^{as_{t_i}}(W_{t_i}-W_{t_{i-1}})]$, but I get expected ...
6
votes
1answer
202 views

Integral of Wiener Process and Central Limit Theorem

I am trying to solve the following exercise: (1) Given $W$ is a Wiener process, find a constant $M$ such that $\lim\limits_{t\to\infty} \frac{1}{t}\int_{0}^{t}\sin^2W_s ds=M$ (2) Then show ...
6
votes
1answer
232 views

Very basic doubt about Itô's lemma

While trying obtain the dynamics of $X_t = \exp( \int_t ^T \phi_s ds)$, where $\phi$ is an Ito process following $$ d\phi_t = \mu dt+ \sigma dW_t \ ,$$ I had some doubt concerning the application of ...
6
votes
1answer
390 views

Does Itō isometry have different versions?

Itō isometry from Wikipedia: Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to ...
6
votes
1answer
1k views

Expectation of an integral w.r.t. Brownian Motion

I know the following statement: if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
5
votes
3answers
4k views

Expectation of geometric brownian motion

I was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t + ...
5
votes
1answer
365 views

Holder continuity of Ito integral

Let $\sigma(t,\omega)$ be a progressively measurable function and $\mathbb{E}[\int_0^T \sigma_t^2\mathrm dt] < \infty$. Can we say that the Ito process $\int_0^t \sigma_s \mathrm dW_s$ is Hölder ...
5
votes
1answer
190 views

existence/uniqueness of solution and Ito's formula

Given the Ito SDE $$ dX_t=a(X_t,t)dt + b(X_t,t) dB_t $$ where $a(X_t,t)$ and $ b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions. Given a function $f(X_t,t) ∈ C^2$ ...
5
votes
3answers
1k views

On hitting time of Brownian motion and Ito's lemma

I have two possibly related questions. Let $\tau:=\min\{t\geq0:B_t=1\}$, where $B_t$ is a standard Brownian motion. I am supposed to derive the fact that $\mathbf{E}\tau=\infty$ by applying some ...
5
votes
1answer
468 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
5
votes
1answer
1k views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
5
votes
1answer
75 views

Show that $E[X_t^2]<\infty$

Show that $E[X_t^2]<\infty$, where $$ X_t=e^{3W_t-\frac{3t}{2}}-3e^{W_t-\frac{t}{2}}\underbrace{\int_0^te^{2W_s-s}ds}_{A_t},\quad. t\geq0, $$ where $t$ is a fixed number and $W_t$ is Brownian ...
5
votes
1answer
49 views

Compute the distribution of $\int_0^1 B_t dt$

I need an help with the following: let $(B_t)_t$ a Brownian motion. Compute the distribution of $X:=\int_0^1 B_t dt$. Integrating by parts we have that: $$\int_0^1 B_t dt=B_1-\int_0^1 t dB_t.$$ Now, ...
5
votes
1answer
190 views

Stochastic Integral which is almost surely zero at fixed time

This is an exercise from Karatzas and Shreve. Find a $(Y_s)_{s \in [0,1]}$ progressively measurable such that $ 0 < \int_0^1 Y_s ^2 ds < \infty$ almost surely, and $\int _0^1 Y_s dW_s = 0$ ...
5
votes
3answers
314 views

Stochastic integral and Stieltjes integral

My question is on the convergence of the Riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. Here ...
5
votes
1answer
85 views

Reading list to master Numerical Analysis' research literature

As of lately I have been going through many research papers in my current job, and even though I have a Mathematics background at Masters level in Mathematical Finance, I sometimes struggle to follow ...
5
votes
1answer
36 views

Density of stochastic integral

I am working on finding the PDF of $X_t^2$, where $X_t = \int_0^t A(u) \,dW_u$, a Wiener integral, i.e., $W_t$ is Brownian motion and $A(t)$ is a deterministic function. Here, would like to ask that ...
5
votes
1answer
191 views

Brownian motion, reproducing kernel Hilbert space, and the Laplace operator

Consider the standard Brownian motion on $[0,1]$: $$ dB_t, \; B_0 = 0, $$ defined on the probability space $(\Omega, P)$. It covariance function is $K(s,t) = \min \{s , t\}$ on $[0,1] \times ...
5
votes
1answer
117 views

Expectation of a stochastic integral

Let $M$ be a right-continuous local martingale, $s,t$ two times (stopping times, if you like). Under what conditions does the following hold: $$E\left(\int_s^t X \, dM\mid\mathcal{F}_s\right)\le 0$$ ...
5
votes
2answers
884 views

Why isn't the Ito integral just the Riemann-Stieltjes integral?

Why isn't the Ito integral just the Riemann-Stieltjes integral? What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral: $$\int_0^Tf(t)\;dB(t).$$ ...
5
votes
1answer
112 views

Calculation of Radon–Nikodym derivative

Suppose the function $X \colon \mathbb{R} \longrightarrow \mathbb{R} \colon x \longmapsto X(x) : = x^2$. I want to calculate the Radon–Nikodym derivative $\frac{\text{d}\lambda_X}{\text{d}\lambda}$, ...
5
votes
0answers
188 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
5
votes
0answers
117 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 ...
5
votes
0answers
563 views

Ito's lemma and application

Can someone help me apply Ito's lemma to the function $f(t,x,k)$ where t is the time and x,k dimensions where x and k refer to dynamics $dX(t)=\mu(t)dt+\sigma(t)dB(t)$ ...
5
votes
2answers
408 views

Stochastic integral inequality

Let $W_t$ be a Brownian motion with $m$ independent components on $(\Omega,F,P)$. Let $G(\omega,t)=[g_{ij}(\omega,t)]_{1\leq i\leq n,1\leq j\leq m}$ in $V^{n\times m}[S,T]$ such that ...
4
votes
1answer
98 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
4
votes
1answer
168 views

Computation of basic stochastic integral.

I am trying to compute the covariance of a 1 dimensional Ornstein-Uhlenbeck process $dx_t=-\theta x_t dt+ \sigma dW_t$, $\theta>0$ and I am at the stage, $$\text{Cov }(x_s,x_t)=\sigma^2 ...
4
votes
1answer
2k views

Covariance of Brownian Bridge?

I am confused by this question. We all know that Brownian Bridge can also be expressed as: $$Y_t=bt+(1−t)\int_a^b \! \frac{1}{1-s} \, \mathrm{d} B_s $$ Where the Brownian motion will end at b at $t ...
4
votes
1answer
2k views

How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?

I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion. And its analytic ...
4
votes
1answer
234 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
4
votes
1answer
534 views

what's the difference between RDE and SDE?

what's the difference between random differential equation and stochastic differential equation? does stochastic differential equations include random differential equation?
4
votes
2answers
155 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...
4
votes
1answer
343 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
4
votes
1answer
183 views

Simple stochastic integral

Let $(B_1,B_2)$ be a two-dimensional Brownian motion. Let $$ X_t = \int\limits_0^t B_1(s)\mathrm \; dB_2(s). $$ Is there a closed form for $X$ or the integral above is all one can get?
4
votes
1answer
315 views

Solution to the stochastic differential equation

Let $X_o=x$, $dX_t=\frac{1}{X_t}dt+X_tdW_t$, $W_t$ is a brownian motion i am thinking of trying $Y_t=\frac{X_t^2}{2}$ and apply ito's lemma on $Y_t$
4
votes
1answer
325 views

Funny problem about stochastic integrals and Ito' s lemma

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
4
votes
2answers
213 views

Stratonovich SDE coefficient selection

Is it possible to find a strictly positive function $\sigma:\mathbb{R}\to\mathbb{R}$, such that a solution $X_t$ to an SDE $$dX_t=-X_tdt+\sigma(X_t)\circ dB_t,$$ with $X_0$ being arbitrary, is a ...
4
votes
1answer
61 views

Expectation of an Itô integral

I'm interested in computing the following expectation: $$\mathbb{E}\left[W_T\cdot\int_0^T f(s)\mathrm{d}W_s\right].$$ Here $\{W_t\}_{t\ge 0}$ is a standard $\mathbb{R}$-valued Brownian motion and ...