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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7 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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0answers
10 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 fi ltration generated by $W$ and let $X$ be a process that is progressively measurable ...
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0answers
9 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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0answers
22 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
24 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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0answers
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 $1$.
2
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1answer
17 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) $E[\int_0^T X_tds]$? ...
-3
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0answers
12 views

Variance of interarrival time of events [on hold]

As shown in the figure, in this problem, there are three types of events where events of each type occur independently. The inter-arrival time distribution between events of the same type is an ...
1
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1answer
21 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 ...
-1
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0answers
18 views

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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0answers
10 views

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 ...
-2
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0answers
21 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
29 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 ...
-3
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0answers
24 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 ...
3
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0answers
27 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 $\{\phi_n\}$ ...
2
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0answers
98 views
+50

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 ...
2
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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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0answers
20 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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0answers
15 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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0answers
25 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, ...
-3
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1answer
40 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} ...
7
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2answers
256 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, ...
3
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1answer
41 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 ...
2
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1answer
32 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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2answers
47 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!
2
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2answers
127 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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0answers
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
19 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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2answers
64 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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0answers
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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0answers
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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0answers
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 ...
1
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1answer
48 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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0answers
24 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 ...
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1answer
35 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, ...
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0answers
24 views

Question on exponential martingale

I was reading the first proof here on exponential martingale, https://fabricebaudoin.wordpress.com/2012/09/27/lecture-23-time-changed-martingales-and-planar-brownian-motion/ It says that "Let $ ...
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0answers
42 views

Covariance of Stochastic Differential Equation

What is the general expression for the covariance $cov \left[ X_s X_t \right]$ of a stochastic process given by \begin{equation} dX_t = f(X_t,t)dt + g(X_t,t) dW_t \end{equation} for some general ...
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1answer
45 views

Integration by parts formula for Wiener integral

Hi I need an help understanding "integration by parts" in Wiener integral. I've defined this integral as in the following: let $T=[0,t]\subset \mathbb R$ we want to define $\int_T f(s) dB_s$ where ...
2
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1answer
87 views

Question related to Kolmogorov equations

Let $d X_t = b(t,X_t)dt + \sigma(t,X_t)dB_t$ be an Ito diffusion. If we choose a continuously twice twice differentiable function $f$ with compact support and define $u(t,x) = E( f(X_t) | X_0 = x)$ ...
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2answers
135 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 ...
2
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0answers
35 views

Is there any standard way of analysing this integral?

I have a compound Poisson process $(X_t)$, with jump distribution $F$, which assigns mass only to $(0,\infty)$. In my working I have an expression of the following form: $$ \mathbb{E} \int_0^{\tau} ...
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1answer
45 views

Most General Theory of Stochastic Integration

I've learnt continuous stochastic integration using the classical books: - Revuz & Yor, - Karatzas & Shreve and - Oksendal. Now I want to learn general stochastic integration, i.e. possibly ...
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0answers
20 views

Pricing of Binary or Digital Options or Feynman-Kac Equation for $\mathbb E f(X_T)$ with diffusion $X$ and discontinuous function $f$.

I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous by means of solving a PDE or ...
2
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0answers
37 views

Quadratic Variation and Semimartingales

It is clear that every (I am particularly interested in continuous) semimartingale has a well defined quadratic variation process. However, what can be said about processes that have a well defined ...
-1
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1answer
31 views

Expectation of B(1) times stochastic integral? [closed]

I need to find the value of this expectation: $$\mathbb{E}\left(B(1) \int_0^1 f(t) dB(t)\right)$$ $B=(B(t))_{0\leq t\leq1}$ is a standard Brownian motion on $[0,1]$ and $f=(f(t))_{0\leq t\leq1}$ is ...