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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8
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3answers
157 views

Stochastic Integrals are confusing me; Please explain how to compute $\int W_sdW_s$ for example

I have been trying hard to understand this topic, but only failing.Reading through my lecture notes and online videos about stochastic integration but I just can't wrap my head around it. The main ...
0
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0answers
18 views

Left limit poisson process (stochastic analysis)

Let $N_t$ denote a Poisson process with intensity λ > 0, and let $M_t = N_t − λt$ be the compensated martingale of N . How could I show that $\int_{0}^{t} N_{s-} dN_s =1/2 (N_t^2-N_t)$ Thank you
2
votes
0answers
33 views

Martingale (stochastic analysis)

Let $N_t$ denote a Poisson process with intensity λ > 0, and let $M_t = N_t − λt$ be the compensated martingale of N . I want to verify that the process Y given by $Y_t = \int_{0}^{t} N_{s-} dM_s$ is ...
0
votes
1answer
13 views

Can I swap conditional expectation and limit

My problem is the following : let $B_t$ be a standard Brownian motion and $H_t$ a progressive measurable process such that $\mathbb{E}\left(\int_0^{+\infty} H_t^2\ dt \right)<+\infty$. Denote ...
0
votes
1answer
17 views

integral of square of Brownian motion

What is expectation of $$\int_0^t B(s)^2ds$$ where $B(s) is standard Brownian motion. Is the integral a well known random variable?
0
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0answers
60 views

Integral inequality of transformed integrand with second order stochastic dominance flavor

Let $f,g : [0,1] \rightarrow [0,1]$ be two functions such that for all $x \in [0,1]$ $\int_0^x f(t) dt \geq \int_0^x g(t) dt$ and $\int_0^1 f(t) dt = \int_0^1 g(t) dt.$ Can I conclude that ...
5
votes
0answers
65 views

Ornstein-Uhlenbeck SDE solution

I'm following this solution of $$dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t \tag1 $$ And the question is whether its solution $$X_t=\theta+e^{-\kappa(t-s)}(X_s-\theta)+\sigma\int_s^t ...
3
votes
1answer
26 views

Application of the Clark-Ocone's Formula to $\mathbb{1}_{S_t > K}$

At page 291 of Nonlinear Option Pricing by Julien Guyon and Pierre Henry-Labordère, the Clark-Ocone's Formula is applied to $\mathbb{1}_{S_t > K}$. I do not get how to get from the second to the ...
4
votes
1answer
54 views

Intuition about Skorohod integral

I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on. In particular ...
2
votes
0answers
16 views

Ito formula for a function of class $C^1$

Can the Ito formula be applied with a $C^1$ function if the second order terms vanish ? For example, let $g(t)$ be a function of class $C^1$ and define $F(x,t)=xg(t)$ which is also of class $C^1$. ...
2
votes
1answer
22 views

Partial differential equations involving Feynman-Kac

I am working on solving the following pde on $[0,T]$; $$f_t(t,x) + 2tf_x(t,x) + \frac{t^4}{2}f_{xx}(t,x) = 0 \qquad f(T,x) = x^2 = h(x)$$ By Feynman-Kac, the solution is given by $$f(t, X_t) = ...
1
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0answers
17 views

Integrating Wishart density

I have several points $\textbf{s} = s_1,...,s_n$ which follow Wishart distribution. In one of my problem, I have to integrate this Wishart pdf over a ball of radius $r$ at origin in $\mathbf{R}^2$ ...
0
votes
0answers
10 views

Ito integral of continuous function is Gaussian process

Let $f\in C[0,T]$. Then $X_t := \int_0^t f(s) dB_s$ is a Gaussian process with independent increments, with zero mean and covariance $\mathbb{E}(X_s X_t) = \int_0^{s\wedge t} f^2(s) \ ds$. This can ...
1
vote
1answer
28 views

Why is the isometry of It\^o integral called so? [duplicate]

For functions $f$ satisfying appropriate (good) conditions, the following property is called to be isometric. $$ ...
1
vote
1answer
42 views

Definition of stochastic integral, square integrable function

Hello I have a question about Stochastic integral. Let $X=(X_{t})_{t \geq0}$ be a Brownian motion started at $0$. I know the following fact: Let $(\varphi(t))_{t\geq0}$ be a progressively measurable ...
0
votes
0answers
19 views

Why is the stochastic integral $\int_0^t \nabla u(B_s)\cdot dB_s $ a local martingale?

This is from Durrett's book Stochastic calculus: a practical introduction. I don't understand the last sentence in the picture. Could anyone help explain why the first term is a local martingale? ...
1
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2answers
21 views

Fourier transform with the derivative of a function

I have to identify the Fourier transform, defined as $\widehat f(x)=\displaystyle \int_{\mathbb R} e^{-ixy}f(y) dy$ As a task, I have to calculate the the fourier transform of $g(x)= ...
0
votes
1answer
33 views

Ito's formula and Brownian motion

Let $a \in R$,$B=(B^1,B^2)$ a brownian motion. $$X_t=e^{B_t^1}\left(\int_0^te^{-B_s^1}dB_s^2+a\int_0^te^{-B_s^1}ds\right)$$ Show there is a brownian motion $\beta$ such that $$X_t=\int_0^t ...
1
vote
0answers
34 views

Show that $e^{-rt}E\Phi(S_T)=S_0N(d_+)-Ke^{-rT}N_{d_-}$

Show that $e^{-rt}\mathbb E[\Phi(S_T)]=S_0N(d_+)-Ke^{-rT}N_{d_-}$ where $S_t=S_0e^{(r-\sigma ^2/2)t+\sigma W_t}$ for $t\in[0,T]$ , $W_t\sim \mathbb N(0,t)$ and $N$ is the cumulative density function ...
1
vote
1answer
67 views

Uniqueness in law associated to nonlinear SDEs

I do not understand the following when reading a paper on Propagation of Chaos, written by A.S.Sznitman: Consider an $n$- dimensional process $X$ satisfying the following SDE: $$ dX_t = b(t, ...
0
votes
0answers
16 views

Box calculus for sequential differential

A shorthand rule of thumb for Ito calculus is the Box calculus where one assumes that $dtdW^{(i)}=0$ and $dW^{(i)}dW^{(j)}=\delta_{ij}dt$ where $dW^{(i)}$ and $dW^{(j)}$ are increments in two ...
1
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0answers
41 views

Evaluating integral with respect to brownian motion

I am attempting to integrate $$ \int _{0}^{t} \sin(s) dW_s $$ whereas $W_s$ is brownian motion, in some sense a normal random variable with mean 0 and variance $s$. I looked around in stack ...
0
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0answers
38 views

Conditional expectation and Variance

I have the interest rate model: $r(t)= x(t)+y(t)+\phi(t) $ $r(0)=0 $ $dx(t)⁼-ax(t)dt+\sigma dW_1(t) $ $x(0)=0$ $dy(t)⁼-bx(t)dt+\nu dW_2(t) $ $y(0)=0$ $(W_1,W_2) $Brownian (2 dimensions) ...
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0answers
24 views

An Ito integral is normal if the integrand is a deterministic function

Why is an Ito integral normally distributed if the integrand is a deterministic function? This is constantly used in many proofs, and I often take it for granted.
5
votes
1answer
51 views

Can the integral of Brownian motion be expressed as a function of Brownian motion and time?

Let $W_t$ be standard Brownian motion, and define $$ X_t := \int_0^t W_s ~\textrm{d}s. $$ The marginal distributions of $X_t$ are easy to write down (see here), but it doesn't seem possible to express ...
1
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0answers
39 views

Distribution of sum of $n$ i.i.d. symmetric Pareto distributed random variables

Let $X$ be a random variable which follows the symmetric Pareto distribution. For a fix, real parameter set $\alpha > 0$ and $L>0$, its PDF is defined as $$ p_X(x) = \left\{ ...
0
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0answers
24 views

Complete (not heurestic) proof of Ito lemma?

Where can I find a formal and complete proof of Ito lemma. I found a few of them but all are "heurestic" type like on Wikipedia, operating on $dX_t$ notation. Not really proofs. Thank you for any ...
0
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0answers
9 views

Integrating R.V with respect to time

I would like to compute the time integral of a random variable $X(t)$ given by $\int_s^t X(u) e^{k u} du$ where $X(t)$ is a CIR square root process $dX_t = k (\theta - X_ t) dt + \sigma \sqrt{X_t} ...
1
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0answers
28 views

Construction of the Itō integral with (local) martingales as integrators

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$. $\xi_i$ be a real-valued random variable on ...
0
votes
0answers
35 views

Bessel Process and Brownian motion

Let $\beta_s$ be a Bessel process, i.e. the positive solution to the SDE $$\beta_s = B_s + (n-1) \int_0^t \frac{1}{\beta_s} \mathrm{d}s,$$ where $B_s$ is a one-dimensional Brownian motion. Let $U_s$ ...
0
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0answers
9 views

Principal Component Analysis in a stable framework

are you familiar with stable distributions. It is denoted by $S_{\alpha}(\sigma,\beta,\mu)$ where $\alpha$ is the tail index, $\beta$ is the skewness, and $\sigma$ and $\mu$ are the location and scale ...
4
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0answers
73 views

Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
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0answers
26 views

Generalization of the Ito formula

I have a question concerning Ito’s formula for semimartingales with jumps. I am familiar with Ito’s formula in the following setting: Let $X_t=X_0+M_t+A_t$ be an $\mathbb{R}^d$-valued continuous ...
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0answers
25 views

Covariance of two stochastic integrals

Consider the stochastic integral $\int_{0}^{1}J(r)M(r,\lambda) dr$ where $J(r)$ is a demeaned Ornstein-Uhlenbeck process and $M(r,\lambda)=W(r,\lambda)-\lambda W(r,1)$ a Brownian Sheet, independent of ...
2
votes
0answers
38 views

Why isn't this stochastic integral trivial?

I have a stopping time $\tau$ and a stochastic process $f$. Then the following equation is true: \begin{equation} \int^{t\wedge\tau}_{0}f(s)dW(s)=\int^{t}_{0}f(s)\chi_{[0,\tau]}(s)dW(s) ...
0
votes
0answers
21 views

How can I formally arrive at solution of “deterministic SDE”

Let $dX_t=\mu X_t dt+\sigma X_t dW_t$. We know that this is a shorthand for integral equation: $X_t=X_0+\int_0^t\mu X_s ds + \int_0^t\sigma X_s dW_s$ Now: what if our equation looks like this ...
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0answers
18 views

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$ and $E\{\mid \int_{\alpha}^{\beta}f(t)d\omega (t)\mid^2|\mathscr{F}_\alpha ...
0
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0answers
24 views

Solving a Stochastic PDE with two variables in time

I am trying to work on exercise 5.13 in the book Arbitrage Theory in Continuous time by Thomas Bjork. The equation to solve is; \begin{eqnarray*} \frac{\partial F}{\partial t} (t,x,y) + \frac{1}{2} ...
1
vote
1answer
26 views

Prove that: $E[\int^{\tau}_{0} f(t)d\omega(t)]=0$ and $E\mid \int^{\tau}_{0} f(t)d\omega(t)\mid^2=E[\int^{\tau}_{0} f^2(t)dt]$.

Suppose $f \in L^{2}_{\omega} [0, \infty]$, and $\tau$ is a stopping time such that $E[\int^{\tau}_{0} f^2(t)dt]<\infty$. Prove that: $E[\int^{\tau}_{0} f(t)d\omega(t)]=0$ and $E\mid ...
1
vote
1answer
33 views

Why writing $[X,Y]_t$ as $dX_t dY_t$ is so called “abuse of notation”

Why writing $d[X,Y]_t$ as $dX_t dY_t$ or $[B]_t$ as $\int_0^tdt$ is so called "abuse of notation"? Is it because $[B]_t \rightarrow \int_0^tdt$ a.s. but they are not equal?
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0answers
13 views

Moments of the integrated Bessel process

I am trying to compute the moments of the integrated and the integrated-inverse Bessel process. For simplicity, if $X_t$ is a BES$(d)$ assuming $d>2$, I am trying then to compute $$\mathbb E ...
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0answers
11 views

Which point is the correct one when integrating w.r.t a discrete martingale?

first post here so I will try to explain as well as possible. In Durrett's book of Brownian Motion and Martingales, he uses the following example: $$ X_t = \begin{cases} 0, t<T \\ \xi, t\geq T ...
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0answers
21 views

What does it mean by totality of borel cylinder set?

I understand what cylinder set is, but what does it mean by totality of cylinder set? I encounter this term in stochastic book quite often but I do not get the idea quite well. Does the totality mean ...
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vote
2answers
41 views

Why is a stochastic integral w.r.t a martingale always a local martingale?

In my course on stochastic calculus, the professor mentioned that stochastic integral w.r.t a martingale always a local martingale? How can I rigorously show this? I know that when integrating wrt to ...
9
votes
2answers
302 views

Itô's formula: Differential form

I've started a course on financial mathematics and I'm currently being introduced to stochastical analysis, spesifically Itô's formula. From the book: It is sometimes useful to use the following ...
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0answers
26 views

Strong Markov property of Ito Diffusion - why must the stopping time be a.s. finite ? (Oksendal 6th edition p117 )

I am reading the proof of the Strong Markov property for Ito diffusions In Oksendal 6th edition p117 Theorem 7.2.4, and I do not understand where the fact that the stopping time has to be almost ...
0
votes
2answers
30 views

Ito's Lemma Simple Application

I need help applying Ito's Lemma to show a given result. $B_t$ is standard Brownian motion $dS_t = 0.4 S_tdt + 0.5 S_tdB_t$ I need to find $dlog(S_t)$ I am told it is $(0.4-1/8)dt + 0.5 dB_t$ ...
0
votes
1answer
33 views

what's the usage of purely discontinuous martingale in stochastic integral?

Recently I'm reading Jacod's Limit Theorems for Stochastic Process ,chapter 1 and I'm confused with the general stochastic integral for semimartingales. $H$ is locally bounded predictable process. ...
4
votes
1answer
72 views

Motivation behind Ito integral

Today my professor introduced the Ito integral as a way to make sense of $$\int \sigma(u) \cdot "noise"du$$ where noise is modeled as Brownian motion. He then said: With Riemann integrals you ...
0
votes
1answer
23 views

Verifying the identity $E\left( \int^t_0 X_s ds \right)^2 = \int^t_0 \int^t_0 E(X_s X_u)\,ds\, du$

I am doing the following exercise: The thing I am struggling with is the identity given in the hint: $$ E\left( \int^t_0 X_s ds \right)^2 = \int^t_0 \int^t_0 E(X_s X_u)\,ds\, du $$ I am unable to ...