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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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 ...
-1
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1answer
62 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
34 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
32 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
54 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 $\{\phi_n\}$ ...
4
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1answer
155 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
46 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
29 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
22 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
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
45 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
295 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
47 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
35 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 ...
0
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2answers
52 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
155 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
27 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
47 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
26 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\ ...
4
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2answers
72 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
32 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
34 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
55 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
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 ...
5
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1answer
42 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
26 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
54 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 ...
1
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1answer
51 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
89 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 $u(t,x) = E( f(X_t) | X_0 = x)$ ...
4
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2answers
143 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
36 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
47 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
23 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
44 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 ...
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1answer
32 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 ...
3
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0answers
160 views

Integration of independent Brownian motions

I am wondering if the following integral of stochastic Brownian motions has an analytical solution? $$ \int_{0}^{t}e^{\nu \tilde{V}_{\tau} - \frac{1}{2}\nu^{2}\tau}d\tilde{W}_{\tau} $$ where ...
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1answer
53 views

Differential of stochastic term

Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it ...
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2answers
42 views

Predictable Processes in Brownian Setting

Maybe it's a silly question. I've been reading Protter's book on stochastic integration. And all the integrands are required to be predictable. But from what I can recall, in the traditional ...
2
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1answer
30 views

Bayes formula on a general $\sigma$-algebra

I want to prove the following Bayes formula: Let $\Omega$ be a sample space, $\mathbb A$ a $\sigma$-algebra over $\Omega$ and $\mathbb B$ $\sigma$-algebra of $\mathbb A$. For $A\in\mathbb A$ and ...
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0answers
4 views

Error from bias and noise in a linear operator

There's a result $S$ that depends linearly on some forcing $F$: $S=\int dt' G(t-t')F(t')$ Let's say I need to predict $S$, but can't measure $F$ exactly. I have both bias and noise in my ...
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40 views

martingale representation property

Suppose there are two independent Brownian motions $B_1$ and $B_2$ and the natural filtration of $B_1$ is denoted by $F^1_t$. Define ${g}:=\sigma\{B_2(t),t\in[0,T]\}$ and filtration $F_t$ is defined ...
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1answer
44 views

How to solve this question with Itô lemma?

Let $$M(t) = \int_{0}^t Y (u)dB(u).$$ where $$E \left[ \int_{0}^t Y^2(u)du\right] < \infty.$$ Use Itô’s rule to find the differential $dQ$ of the process $$Q(t) = M^2(t) − \int_{0}^t Y^2(u)du$$ ...
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1answer
69 views

Prove identity in law for stochastic process driven by Brownian Motion

Let $B = (B_t)_{t\geq 0}$ be a standard brownian motion started at $0$. Consider the two following stochastic equations: \begin{equation} \begin{split} dX_t &=& (13 + 2X_t)\,dt + (6 + ...
0
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1answer
33 views

Convergence properties of the Ito integral

I am currently going through the proof of the existence of a solution of the SDE \begin{align} dX_t = bdt + \sigma dB_t \end{align} where $B_t$ is a Brownian motion wrt a filtration ...
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1answer
38 views

The Itō Integral

In stochastic calculus and specifically for mathematical finance Ito's lemma is used for time varying processes I need to know intuitively why the Ito Integral is stochastic?
2
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1answer
57 views

Show that a process is gaussian

I need an help with the following exercise. I would like to know if what I've done is correct. Let $(X_t)_{t\geq 0}$ be the process define as $$X_t=e^{\lambda t} X_0-\sigma \Big(\lambda \int_0^t ...
0
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1answer
16 views

Conditional expectations one more time

Please someone verifies my results: 1) $E \Big( \int_0^3W_t^2dt|F_1\Big)=$(editing in progress) 2) $E \Big( \int_0^2 (tW_t+t^2)dt|F_1\Big)=E \Big( \int_0^2 tW_tdt|F_1\Big)+E \Big( \int_0^2 ...