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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21 views

Applying Ito formula to Ito process

I would like to simplify the expression $\left(\phi(s_{1})\cdot(X_{s_{1}}-X_{s_{2}})+\phi(s_{2})\cdot(X_{s_{2}}-X_{s_{3}})+\ldots+\phi(s_{n-1})\cdot(X_{s_{n-1}}-X_{s_{n}})\right)^{2}$ where $X_{t}$ ...
2
votes
2answers
45 views

Simulating a Stochastic Integral of OU process

The stochastic integral I want to simulate is $$\int_{0}^{1}J_c(s)dJ_c(s)$$ where $J_c(s) = \int_{0}^{s}e^{-c(s-r)}dB(r)$, is an OU process. I simulate the data using Matlab and the sample codes are ...
1
vote
1answer
40 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
2
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0answers
99 views

Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds} $$ where the ...
2
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0answers
70 views

Question on averages of Ito Integral: $E(\int_0^t X_sdB_s \int_0^t X_sds)=?$

Given some probability space, assume $X_t$ is a square integrable continuous process adapted to the filtration $\mathcal{F}_{t}$ generated by the standard Brownian process $B_t$. I denote by $(X.B)_t=\...
2
votes
2answers
512 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 ...
2
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0answers
62 views

How can we prove that $\langle\int_0^t\Phi_s{\rm d}W_s,x\rangle_H=\sum_{n\in\mathbb N}\int_0^t\langle\sqrt{λ_n}\Phi_se_n,x\rangle_H{\rm d}B_s^{(n)}$?

Let$^1$ $U$ and $H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U,H)$ be nonnegative and symmetric operator on $U$ with finite trace $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $...
7
votes
1answer
306 views

Application of the Burkholder Davis Gundy inequality

The proof of the Feynman-Kac formula uses a lemma which I need to prove, but I can not figure it out. The lemma is the folllowing: Let $X$ be a weak solution of $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dW_t$$...
2
votes
1answer
185 views

Conditional expectation brownian motion

Somebody has an idea on how to tackle this quantity $$ \mathbb{E}\left[ \left. \frac{\int_{0}^{T}{{{e}^{\alpha {{W}_{t}}}}}dt}{\int_{0}^{T}{{{e}^{-\alpha {{W}_{t}}}}}dt+\int_{0}^{T}{{{e}^{\alpha {{W}_{...
2
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0answers
77 views

Question about a Bessel process

Are there any explicit path-wise solutions for a 3 dimensional Bessel process? E.g. the Ito SDE: $$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$ where $W_t$ is a standard Wiener process.
1
vote
2answers
54 views

brownian noise and stochastic differential equations

Consider the SDE $$dx=3x(t)dt+dW(t)$$ Where we're dealing with Brownian noise. Now, dW comes from $$dW(t)=\int_0^{dt}ds\ \eta (s)$$ As far as I understood, $\eta$ is the noise distribution (...
0
votes
0answers
18 views

System of stochastic equations

I want to know if this system of SDE: $$dX_{t}=b(X_{t})dt+\sigma( X_{t}) dB_{t}$$ $$dY_{t}=b_{0}(Y_{t})dt+\sigma( Y_{t}) dB_{t}$$...
0
votes
0answers
32 views

System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
0
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0answers
26 views

Quadratic covariation

I am trying to solve small task from stocastic calculus. It can be shown that for stochastic processes X and Y , the quadratic covariation satisfies the polarisation formula - $[X, Y](T) = 1/2 ([X + ...
3
votes
1answer
51 views

Expectation value of stochastic process

For which $k>0$ process $X=(e^{kW_s^2})_{s \ge 0}$ belong to $\mathcal{L}^2_{\infty }(W)$ and for which belong to $\Lambda ^2_{\infty }(W)$. Set one localization sequence $(\tau_n)_{n \ge 0}$ for ...
-1
votes
1answer
18 views

Vasicek equation [closed]

Vasicek interest rate stochastic differential equation is $$dR(t)=(\alpha-\beta R(t))dt+\sigma dW(t)$$ where $\alpha , \beta$ are positive constants. I need to use Ito-Doeblin formula to compute $...
1
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0answers
44 views

Stochastic calculus with normal distribution

For $l=1,2......$ prove that $E[W^{2l+1} (t)]=0$ I am trying to find the ways of solving the task from Stochastic calculus, but it seems to be very difficult to start with. I am really appreciate ...
0
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0answers
14 views

Interpretation of the stochastic integral of simple functions

A simple function $h$ (or random staircase function) is defined as $$ h(t) = \sum_0^{n} \xi_i I_{(t_i,t_{i+1}]}(t) $$ where $0<t_1< \ldots < t_{n+1}<T$ and the $\xi_i$ are random ...
0
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1answer
36 views

Using of Ito formula with martingales

We have exam test - $\alpha,\beta \in \mathbb{R}$ and $N(t)=e^{\beta t}cos(\alpha W(t)).$ It is necessary to calculate $\mathbb{E}[cos(\alpha W(t))]$. I know that $\beta$ can be chosen so that $N$ ...
3
votes
2answers
63 views

What's the variance of the following stochastic integral?

The stochastic integral is defined as $$u_t = \int_{t-1}^t e^{-\kappa(t-s)}\int_0^s e^{-c(s-r)} \, dW(r) \, ds.$$ where $W(t)$ is a standard Brownian motion, $\kappa$ and $c$ are both positive. I ...
0
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1answer
26 views

Ito integral for simple stochastic process

I need for $l=1,2......$ prove that $E[W^{2l} (t)]=$ $\frac{(2l)!}{2^l l!}$ and $E[W^{2l+1} (t)]=0$ I know that Ito integral for simple stochastic process satisfies $E[I^2 (t)]=E\int_0^t\Delta^2(u)...
5
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 ...
1
vote
1answer
41 views

Girsanov theorem calculations help

I need help to understand a couple of calculations in this Girsanov theorem related SDE problem. I have five questions as stated below. Let $X_t$ solve the Ornstein-Uhlenbeck equation $$dX_t = X_t\, ...
3
votes
1answer
44 views

Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
0
votes
1answer
25 views

Question about Ito integration in SDE in Stochastic optimal control

Here is my question statement. I cannot understand the last equality. Let $U=[-1,1]$. \begin{equation} \mathcal{U}[0, T] = \left\{ u:[0,T] \rightarrow U \mid u \text{ is } \{\mathcal{F}_t\}_{t\geq0}\...
2
votes
0answers
29 views

The integral is the area under the curve. Is there a similar notion for stochastic integrals?

As discussed in the answers to this question, the integral is defined to be the (net signed) area under the curve. The definition in terms of Riemann sums is precisely designed to accomplish this. ...
2
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0answers
52 views

A Simple Stochastic Integral Asymptotics

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ $(\mu,\sigma)$ obeys the linear growth condition $...
5
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0answers
50 views

How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
1
vote
1answer
84 views

A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that $$\infty&...
3
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0answers
33 views

Find (a,b) such that aX+bY is a Brownian motion

Let $$\begin{cases} dX_t = \mathrm{sin}(X_t+Y_t) dW_t \\ dY_t = \mathrm{cos}(X_t+Y_t) dV_t \\ X_0=Y_0=0 \end{cases}$$ Where $(W,V)$ is a two-dimensional Brownian motion and $(X,Y)$ be a strong ...
0
votes
1answer
30 views

Stochastic integral of $\mathrm{sin}(X_t+Y_t)\mathrm{cos}(X_t+Y_t)$ [closed]

As part of an exercise I am doing from stochastic analysis I need to compute the following integral: $$E(\int_0^x \mathrm{sin}(X_t+Y_t) \mathrm{cos}(X_t+Y_t)dt)$$ where $X_0=0$ and $Y_0=0$. Presumably ...
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0answers
51 views

what does this integral stand for?

i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ...
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0answers
7 views

stochastic calculus, stopping time, ito integral vector brownian motion

I'm referring to chapter 4, question 7 in Harrison's book 'Brownian Motion and Stochastic Flow Systems.' Problem In the setting of (9) let $f_{n}(x)=E_{x}[\int_{0}^{T}X_{t}^{n}dt]$. Use Ito's ...
1
vote
1answer
37 views

Ito formula when g(t,x) is an integral

Suppose we have a stochastic process which is written as an Ito process. $$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$. If $Y_t$ is defined as a stochastic process as a function of $X_t$, then we can find $dY_t$ ...
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0answers
19 views

Indistinguishable Processes under local Lipschitz Condition

Let $a,b, \rho, \sigma$ be locally Lipschitz functions on $\mathbb{R}^d$, G an open subset of $\mathbb{R}^d$ and assume that on $G$ we have the equalities $a=b$ and $\rho=\sigma$. Let $\xi \in G$ and ...
1
vote
1answer
115 views

Stochastic Integral with respect to Compensated Poisson Process

Proposition: Let $N_t$ be an $\mathcal{F}$-Poisson process and $M_t=N_t-\lambda t$ its compensated process. Then for any $\mathcal{F}$-predictable bounded process $H_t$, the stochastic integral $$(H\...
0
votes
1answer
45 views

Martingale and local martingales

I have to show that $e^{B_t^1}\cos(B_t^2)$ is a martingale ($B=(B^1,B^2)$ is a two-dimensional Brownian Motion). I used Ito's formula and got $e^{B_t^1}\cos(B_t^2)=1+\int_0^t e^{B_s^1}\cos(B_s^2)dB_s^...
2
votes
1answer
52 views

Why Are Semimartingales the Largest Possible Class of Stochastic Integrators?

I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ...
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vote
2answers
54 views

How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
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0answers
26 views

Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv \int\...
6
votes
2answers
485 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 $$\limsup_{\...
3
votes
2answers
82 views

Integral of Wiener Squared process

I don't have a background of stochastic calculus. It is known fact that definite integral of standard Wiener process from $0$ to $t$ results in another Gaussian process with slice distribution that ...
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0answers
31 views

Lebesgue-Stieltjes integral and related topics

The theory of stochastic integration relies on the concept of the Lebesgue-Stieltjes integral. However, it is hard to find a textbook that handles this concept in detail. Take, for instance, Chung ...
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26 views

Integrating over random boundary

What are some correct stochastic integral notions or theories which make formal sense of the problem of "integrating a function over the boundary of random domain"?
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13 views

Solving the following SDE with a constant

Given is the stochastic differential equation: $\frac{dX(t)}{X(t)}=\mu+\sigma \theta dt+ \sigma dW(t)$, where $W(t)$ is the standard Wiener process and $X(0)=x_0$ I try to solve this by the Itos ...
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0answers
43 views

Some Kind of Generalized Brownian Bridge

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not ...
5
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0answers
47 views

Can Stochastic Integration be Further Generalized?

Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ...
3
votes
0answers
480 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf E\bigg[\exp\Big(\...
0
votes
1answer
15 views

Question about law of substitution in stochastic integral

I need to compute some Integrals for my stochastic course. And i have the following problem: $$ \frac{\lambda^n}{\Gamma(n)} \int_0^{\infty} \exp(-\frac{\lambda}{y}) \frac{1}{y^n} dy = \star$$ so i ...
1
vote
1answer
65 views

Covariance of stochastic integral

I have a big problem with such a task: Calculate $\text{Cov} \, (X_t,X_r)$ where $X_t=\int_0^ts^3W_s \, dW_s$, $t \ge 0$. I've tried to do this in this way: setting up $t \le r$ $$\text{Cov} \, (...