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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Expectation of squared Ito integral

Let $\omega$ be a standard Brownian motion. How do you compute the expectation involving the square of an Ito integral: $ ...
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9 views

Prove that an operator from $L^2(\Omega;C(s,T;\mathbb R^n ))$ into itself is well defined

I need an help proving the following estimate. First, we fix the notation. Let $L^2(\Omega;C(s,T;\mathbb R^n ))$ be the set of continuous and adapted processes $\{X_t:t\in [s,T]\}$ (valued in ...
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1answer
15 views

Is the integral of Ito processes still an Ito process?

Let $s \in [0,1]$ and define diffusion processes, $$dS(s)_t = \mu(s) dt + \sigma(s) dW_t$$ The question is if the following make sense, $$ \int_0^1 dS(s)_t ds = \int_0^1 \mu(s) ds dt + \int_0^1 ...
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1answer
20 views

Application of martingale representation theorem

I am reading a proof that uses the following fact without proof (a bit strange): Let $W$ be a real Brownian motion generating the right-continuous, completed filtration $\{\mathcal{F}_t \}_{t \geq ...
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1answer
9 views

Proving that a process is a positive martingale

Let $X$ be the strong solution to the SDE $$ dX_t = \tanh X_t \,dt + dW_t, $$ where $W$ is a scalar Brownian motion defined on a probability space $(\Omega, \mathcal{F} ,\mathbb{P})$. (Such solution ...
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18 views

Quality of approximation of an Ito integral

How could I investigate whether $$P(t,T-t)\left[a(T-t-\Delta)-a(T-t)+ (b(T-t-\Delta)-b(T-t))'x(t)+ \frac{1}{2}b(T-t)'\sigma\sigma'b(T-t)+ b(T-t)'(x(t+\Delta)-x(t))\right]$$ is a good or bad ...
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2answers
20 views

System of Stochastic Diff Eq

How can I solve the system of stochastic differential equation $$dX_{1}=X_{2}dt+adW_{1}$$ $$dX_{2}=-X_{1}dt+bdW_{1}$$
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1answer
17 views

How do we apply Ito's lemma to a product of functions

In finance an optimal portfolio choice it is common to use some tools of stochastic calculus. Going through a book, I found the following statement, ...
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17 views

Stochastic Integral martingale if no $dt$ term? [duplicate]

There is a proposition in my book that For a process $M_t$ to be a martingale, it is necessary that its stochastic differential $dM_t$ has no $dt$ term. Why is this exactly? My guess is that it ...
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14 views

Showing that $X_t = \int^{1/[X]_t}_0 f_u dW_u$ is a Brownian motion

Assume we have an Ito process $$ X_t = \int^t_0 f_u d W_u $$ where $f_u$ is a deterministic function of $u$ and $W_u$ is a Brownian motion adapted to $\lbrace \mathcal F_t \rbrace$. I want to show ...
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1answer
44 views

How to compute stochastic integral: $\int_0^t d(B_s^2)$

Here, $B_t$ is Brownian motion at time $t$ What property is used to compute the integreal $\int_0^t d(B_s^2)$? Shouldn't there be some other variable attached with the differential $d$ ?
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16 views

Stochastic integral and usual integral addition

Let's say I have two processes and I would like to say something about their sum. In the case of deterministic functions, $\int f(t)dt + \int g(t)dt = \int f(t)+g(t)dt $, and I can then possibly say ...
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1answer
51 views

Girsanov's theorem and absolutely continuous restrictions

Let $W$ be a Brownian motion on some probability space $(\Omega, \mathcal{F}, P)$. Let $\mathbb{F}^W$ be the filtration generated by $W$ and let $X$ be a process that is progressively measurable ...
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11 views

Reference request for conditional and unconditional covariance of n-times integrated Brownian motion

I'm working through an old Diaconis paper on Bayesian numerical analysis, and am currently calculating the details behind his brief comments on using $n$-times integrated Brownian motion as a function ...
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2answers
82 views

Martingale representation theorem application

Let $X = \exp(W_{T/2}+W_T)$. I try to figure the adapted process $g(s)$ such that according to the MRT we have $$X = \mathbb{E}[X]+\int^T_0 g_s dW_s.$$ I can figure out $X = \exp(2W_{T/2}+W_{T-T/2})$ ...
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1answer
34 views

p.d.f. of a position variable from stochastic velocity p.d.f.

I have a stochastic process, $v(t)$, that represents a velocity, and has a known probability distribution function $f(x,t)$ which is time-varying. I am interested to acquire a probability ...
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1answer
26 views

proving independence of stochastic integrals

Does anyone know how to show that the stochastic integrals \begin{equation} \bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}} \end{equation} are ...
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14 views

Integral of Constant Parameter Martingale

What is the $\int_{1}^{t}W_1W_sdW_s$. This is the question solved by Kuo in his paper an extension of the Ito's Integral (2008) but there limit runs from $0$ instead of $1$.
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1answer
23 views

$n$ times integrated Brownian motion

I have an identity that expresses the $n$ times integrated Brownian motion and I would like to prove that. First, I define what I mean by $n$ times integrated Brownian motion. $$V_1(t) = \int_0^tB_s\, ...
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1answer
28 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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1answer
27 views

Stochastic Integral Question

I'm reading a paper on noise and had a question about the stochastic integral. In the paper, they consider the SDE: $$dX = \lambda Xdt + \epsilon dW$$ which has the solution $$ X(t) = \epsilon ...
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two different Monte Carlo approaches

Assume that the function $f$ is integrable and maps $[0, 1]$ into $[0, 1]$. Consider estimating $\int_0^1 f(x)\,dx$ using two different Monte Carlo approaches. The standard approximation is applied in ...
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Pricing an option on a mean-reverting assets

In an universe we have two assets and a predictor: $\frac {dS_{1,t}}{S_{1,t}}=(\mu_{1,1}+\mu_{1,2}X_t)dt+\sigma_{1,1}dB_{1,t}+\sigma_{1,2}dB_{2,t} $ $\frac ...
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1answer
52 views

How do I solve this SDE (stochastic differential equation)?

I am stuck in trying to solve this equation \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} Here, $b$ is a constant. I am trying to apply my usual methods for ...
2
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1answer
42 views

Stochastic integral where the integrator is zero in probability

We are given a continuous semimartingale $Y$ and a continuous process $B$ of finite variation. Hence, we know that $\langle B \rangle$, the quadratic covariation of $B$, is zero in probability. I now ...
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2answers
31 views

What is the distribution given by $\int^t_0 W_s^2ds$

Define $X_t=\int^t_0 W_s^2ds$, what will be the distribution of $X_t$? My approach is as follow: Let $f(s)=W_s^2s$, by Ito's lemma we have $X_t=W_t^2t-2\int^t_0W_ssdW_s-\frac{t^2}{2}$. Discretize ...
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27 views

The square of an Ito integral is not a martingale

I just had a lecture on martingales and my teacher said something which I thought was interesting but he said wasn't important to the course. I was wondering if you guys could help me on this. We ...
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28 views

Quadratic Variation of Increasing Process?

I am looking through my notes and I came across the following statement: Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, ...
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1answer
51 views

Ito's Integral's definition: Importance of isometry

I'm reading Oksendal's Stochastic Differential Equations (5th edition). He defines the Ito integral of $f$ as the limit $$\lim_{n\to\infty} \int^T_S \phi_n(t,\omega) dB_t(\omega)$$ Where $\{\phi_n\}$ ...
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1answer
144 views

What is the difference between Calculus and Analysis? In Stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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1answer
45 views

If $M_t$ is a martingale, prove $\Bbb E \left[ M_T\int_t^T h_s ds |F_t\right] = \Bbb E \left[ \int_t^T M_s h_s ds |F_t\right]$

If $M_t$ is a martingale, for $0<t<T$, prove $\Bbb E \left[ M_T\int_t^T h_s ds |F_t\right] = \Bbb E \left[ \int_t^T M_s h_s ds |F_t\right]$. I can think of $LHS=M_T \Bbb E \left[ \int_t^T h_s ...
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1answer
65 views

Proving the identity $P( X + Y = a)= \int_{-\infty}^{\infty} P( X + y = a)f_Y(y) \, \text{d}y $

Suppose $\lambda_1, \lambda_2, a \in \mathbb{R}$ and $X,Y$ are random variables. If it is needed, I can assume that $X$ and $Y$ are independent. I want to show, that the identity ...
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21 views

Mean and variance of Gamma distribution

How do I calculate the mean and the variance of a Gamma distribution? I was told to prove the variance was sigma/lambda(^2), I don't know how to find the variance much less the variance.
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18 views

Moments of Multivariate Normal Distributions

I have two questions. Suppose we have two multivariate normal distributions $X \sim N(\mu,\Sigma)$ and $Y\sim N(c\mu,\Sigma)$ where $0<c<1$ is a constant, $\mu$ is a vector and $\Sigma$ is a ...
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26 views

$\int X dt$ integral of random variable

Define $$\int X dt$$ where X is a continuous uniform random variable that can take on any value (0,1). Also, $\int X dt \not = X \int dt$. In other words, $X$ takes on a different, but still random, ...
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1answer
41 views

Variance and expectation of the stochastic intergal [closed]

Compute the unconditional expected value and variance, and describe, as far as possible, the distribution of the random variable $Y_{t} = \int^{t}_{0} W_{s} ds $ with the hint below $\int^{t}_{0} ...
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2answers
265 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, ...
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1answer
42 views

Computation of a stochastic integral with respect to a local martingale

I am trying to compute the stochastic integral $$\int_{(0,t]}\mathbb{1}_{[a,b)}(s)dM_s$$ where $0 < a < b< \infty$ are constant and $M$ is a continuous local $L^2$-martingale. I am guessing ...
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1answer
33 views

Is this Brownian Integral identity correct?

$$\int_0^1 B_t dt=\lim_{\omega \to\infty}{1 \over {\omega}}{\int_0^{\omega}{Y_0+}X_t dt}$$ Where $B_t$ is simple brownian motion, and $X_t$ is a discrete random variable that can be 1 or -1 with ...
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49 views

Limit of time integral of brownian motion

Can someone help explain the following, $$ \lim \limits_{t \to 0} \frac{1}{t} \int_0^t W_u\, du=\lim \limits_{t \to 0} \frac{W_0t}{t}=W_0=0\,? $$ Thanks!
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2answers
132 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
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25 views

How to show that stochastic exponent is integrable?

I need to prove that if $u: [0,T]\rightarrow \mathbb{R}$ is a deterministic square integrable function then stochastic exponential process defined : $M_{t} = exp(-\int_0^t \! u(s) \, \mathrm{d}W_{s} ...
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2answers
44 views

Distribution of a random measure is determined by the characteristic function

I ham trying to understand a proof from a book I am reading. It says the proof follows directly from the prior theorem and I just can't see that. Let $X$ be a random measure on a locally finite, ...
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1answer
21 views

Integral of square of Brownian motion with respect to Brownian Motion

While trying to compute $\int_0^TB_t^2\ dB_t$, $B$ being the standard Brownian motion, I got stuck at showing the following. $$\sum_{i=0}^{n-1}B_{t_i}(B_{t_{i+1}}-B_{t_i})^2 \rightarrow \int_0^TB_t\ ...
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66 views

Integral of time with respect to Brownian motion

I am trying to compute $\int_0^T t\ dB_t$ where $B$ is the standard Brownian motion. To this end I define the sequence of simple predictable functions $$ f_n = ...
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34 views

What is the solution to these SDP?

I am in trouble with my homework, the quesetion is to solve a pair of stochastic differential equation. $dX_t^1 = X_t^2dt + \alpha dB_t^1$ $dX_t^2 = -X_t^1dt + \beta dB_t^2$ $\alpha \ and \ \beta$ ...
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30 views

Is the variance of an Ito process strictly increasing?

Consider the Ito equation: $dX_t = f(t, X_t) dt + G(t, X_t) dW_t$ where $f:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^n$, $G:\mathbb{R}\times\mathbb{R}^n\to\mathbb{R}^{n\times m}$, $X_t \in ...
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22 views

Stratonovich SDE and generator in divergence form

Let $a:\mathbb{R}^d\rightarrow\mathcal{S}_{d\times d}$ be a smooth map that takes its values in the space of $d\times d$ symmetric matrices and suppose there exists a $d\times d$ matrix valued ...
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1answer
50 views

Prove that the quadratic covariation is a bilinear form

If we take $X,Y,Z$ to be square integrable martingales starting at zero, we want to show that for any $\alpha\in\mathbb{R}$ we have $\langle X + Y , Z \rangle = \langle X,Z\rangle + \langle Y, Z ...
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25 views

Preservation of the Markov Property for SDEs

Let $X$ be a continuous Markov process on $\mathbb{R}^d$ that is also a semimartingale. Let $V=(V_1,...,V_d)$ be a collection of suitably nice vector fields on $\mathbb{R}^d$ such that there exists a ...