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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12
votes
3answers
6k views

Integral of Brownian motion is Gaussian?

Let $(W_t)$ be a standard Brownian motion, so that $W_t \sim N(0,t)$. I'm trying to show that the random variable defined by $Z_t = \int_0^t W_s \ ds$ is a Gaussian random variable, but have not ...
13
votes
3answers
415 views

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s ...
0
votes
1answer
323 views

how to do such stochastic integration $dS = a S^b dt + c S dW$?

How to do stochastic integration $dS = a S^b dt + c S dW$, where $a$, $b$ and $c$ are constant, $b > 0$, and $W$ is the Wiener process. I know how to do integration for $dS = aS dt + cS dW$, or ...
8
votes
1answer
764 views

Calculate $\mathbb{E}(W_t^k)$ for a Brownian motion $(W_t)_{t \geq0}$ using Itô's Lemma

Show by using Ito's Lemma, for $k \geq 2$ the following result hold. $$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$ where $W(t) = N(0,t)$ is standard Brownian motion. I think ...
6
votes
3answers
4k views

Itō Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itō integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
4
votes
1answer
255 views

“Continuity” of stochastic integral wrt Brownian motion

I'd like to prove a nice property of a stochastic integral with respect to Brownian motion. Let $(H_t)_{t\geq0}$ be a progressive and bounded process that is continuous at $0$ and $B$ a standard ...
2
votes
1answer
1k views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
0
votes
2answers
41 views

How to show stochastic differential equation is given by an equation

I I tried using substitution and I got an extra integral at the end and do not know how to proceed. Can anyone help me to break this down?
3
votes
1answer
534 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
1
vote
0answers
97 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
6
votes
0answers
229 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
1
vote
1answer
957 views

Scalar product of Gaussian process

Assume that $n(t)$ is a White Gaussian Noise (WGN) process with $E[n(t)]=0$, $E[n(t)^2]=\sigma^2$ and $x(t)$ a deterministic function defined in $[0,T]$. How can I compute from first principles the ...
5
votes
1answer
1k views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
4
votes
0answers
206 views

stochastic differential equation

Xt is a weak solution to the SDE with dXt = ( −αXt + γ )dt + β dBt , ∀t ≥ 0 X0 = x0. α, β , and γ constants, and Bt is a brownina motion. need to find the PDE for the transition density of X at ...
1
vote
1answer
50 views

positive martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}\,dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2\,ds}$$ is a martingale which is positive and has a mean=1, where $\theta_s$ is ...
1
vote
1answer
173 views

Evaluating Stratonovich integral from definition

$\bf 3.9.$ Suppose $f\in\mathcal V(0,T)$ and that $t\to f(t,\omega)$ is continuous for a.a. $\omega$. Then we have shown that $$\int\limits_0^T f(t,\omega)dB_t(\omega)=\lim_{\Delta ...
0
votes
0answers
65 views

Convergence in $L^2$ of the stochastic integral $\int\limits^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s$

Let $e\in \mathbb{R}^+$ and $B_t$ 1-dimensional Brownian motion. Consider $$X_t=\int^{t}_{0}\frac{B_s}{e}1_{B_s\in(-e,e)}dB_s.$$ How to show that $X_t \to 0$ in $L^2$ as $e\to0$? Obviously the ...
0
votes
0answers
203 views

Expected value of correlated stochastic integrals

I do not understand the following result: Suppose $dz_\chi$ and $ dz_\xi$ are correlated increments of standard Brownian motion with $dz_\chi dz_\xi=\rho dt$ you have the following expectation ...
0
votes
3answers
145 views

Show that $E(Y\mid X=x)$ is a linear function in $x$

Let $Y$ and $X$ be bivariate normal distributed with expectationvector $\mu=(\mu_Y,\mu_X)^T$ and covariance matrix $\Sigma=\begin{pmatrix}\sigma_Y^2 & p_{XY}\\p_{XY} & ...
2
votes
2answers
1k views

Matlab Code to simulate trajectories of Ito process.

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
2
votes
1answer
124 views

Limit of a stochastic integral

Let $W_t$ be a one-dimensional Brownian motion and I would like to prove $$\lim_{\beta\rightarrow+\infty}\sup_{0\leq t\leq T}\left|e^{-\beta t} \int_0^te^{\beta s}\mathrm dW_s\right|=0$$ This is an ...
2
votes
2answers
196 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
1
vote
1answer
165 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
0
votes
1answer
137 views

Prove directly from the definition of the Ito's integral

I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question. Prove directly from the ...
-1
votes
1answer
64 views

What is the integral of a family of diffusion processes? [closed]

Let $S$ be an infinite subset of $[0,1]$. For all $s \in S$, let W_s(t) be a standard Wiener process. Definite P(s)_t = \mu(P,s,t) dt + \sigma(P,s,t) dW^s_t Can we characterize? $$F_t= \int_S P(s)_t ...
15
votes
2answers
1k views

Brownian bridge expression for a Brownian motion

Let $B_t$ be a standard Brownian motion in $\mathbb R$, then the Brownian bridge on $[0,1]$ is defined as $$ Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s} $$ for $0\leq t<1$. Here ...
5
votes
1answer
555 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
4
votes
1answer
325 views

Solution to the stochastic differential equation

Let $X_o=x$, $dX_t=\frac{1}{X_t}dt+X_tdW_t$, $W_t$ is a brownian motion i am thinking of trying $Y_t=\frac{X_t^2}{2}$ and apply ito's lemma on $Y_t$
3
votes
6answers
674 views

Why do people write stochastic differential equations in differential form?

I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ...
1
vote
2answers
496 views

What is an alternative book to oksendal's stochastic differential equation: An introduction?

What is an alternative book to oksendal's stochastic differential equation: An introduction? But also An alternative that is over 300 pages and at the same level? Some professor refer that book as a ...
6
votes
1answer
418 views

Does Itō isometry have different versions?

Itō isometry from Wikipedia: Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to ...
5
votes
1answer
656 views

what's the difference between RDE and SDE?

what's the difference between random differential equation and stochastic differential equation? does stochastic differential equations include random differential equation?
3
votes
1answer
159 views

Proving that $T_t := S_t -\left| x \right| -\frac {n-1}{2} \int _0 ^t \frac {1}{S_u}~du$ is a brownian motion

Consider $B=(B_t)_{t\geq 0}$ $\mathcal F_t$ - brownian motion in $\mathbb R ^n, \ (n\geq 2)$ starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. ...
2
votes
2answers
374 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
votes
1answer
159 views

Show that $dX_t=1_{X_t\not=0} dW_t$ does not have a pathwise unique solution.

Given the SDE : $$dX_t=1_{X_t\not=0} dW_t \qquad \text{with} \quad X_{0}=\xi $$ how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss Indeed Consider the ...
1
vote
1answer
673 views

Covariance of two geometric Brownian motions

Assume we have two geometric Brownian motions $$ dX_t = \mu X_t dt + \sigma X_t dW^1_t, \qquad \qquad dY_t = \mu Y_t dt + \sigma Y_t dW^2_t $$ where the Wiener processes are correlated with $E[dW^1_t ...
7
votes
2answers
911 views

Is this local martingale a true martingale?

Using the Ito's formula I have shown that $X_t$ is a local martingale, because $dX_t=\dots dB_t$, where $$X_t = (B_t+t)\exp\left(-B_t-\frac{t}{2}\right),$$ $B_t$ - is a standard Brownian motion I ...
6
votes
2answers
1k views

Why isn't the Ito integral just the Riemann-Stieltjes integral?

Why isn't the Ito integral just the Riemann-Stieltjes integral? What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral: $$\int_0^Tf(t)\;dB(t).$$ ...
4
votes
1answer
74 views

How can a random variable have random variance?

This seems counter-intuitive to me since variance is a difference of expectations and afaik, unconditional expectation is a real number. Apparently, $X_t$ where $dX_t = Y_t dW_t$, where $Y_t$ is an ...
4
votes
2answers
785 views

ito vs Stratonovich

I need to sum up the advantages of ito and stratonovich. I often heard, that the Stratonovich integral lacks the important property of the Itō integral, which does not "look into the future". Can you ...
4
votes
1answer
743 views

Brownian hitting time of a _very_ simple linear boundary

I realize that general results on the hitting times of a curve are practically nonexistant, but I am hoping that someone can string together a sequence of tricks to tell me what $$ \Pr\left( ...
2
votes
2answers
257 views

Conditioning on a random variable

The number of storms in the upcoming rainy season is Poisson distributed but with a parameter value that is uniformly distributed between (0,5). That is Λ is uniformly distributed over (0,5), and ...
2
votes
1answer
113 views

Ito Integral surjective?

Let $\Phi\in\mathcal{L}\left(M\right)$ if and only if $\Phi$ is a real predictable process and for every $\left\Vert \Phi\right\Vert_{2,t,M}:=\mathbb{E}\left[\int_{0}^{t}\Phi_{s}^2 d\langle ...
2
votes
1answer
188 views

generalized derivative of Wiener process

Defined a standard Wiener process $W = (W_t , \mathcal F_t)_{t≥0}$ and a deterministic, continuously differentiable function $f : [0, ∞) → \mathbb R$. Prove that ...
1
vote
2answers
160 views

Martingality Theorem: Solving expectation of a stochastic integral

I am trying to prove that: $$ \Bbb E\left[\int_s^t\sigma e^{-k(t-u)}\sqrt{V_u}dW_u\right] =0$$ Where: $$ dV_t=k~(\theta-V_t)~dt+\sigma\sqrt{V_t}dW_t $$ I have attempted to use Ito's formula on the ...
1
vote
1answer
170 views

Expectation of Ito integral

The expectation of an Itô stochastic integral is zero $$ E[\int_0^t X(s)dB(s)\,]=0 $$ if $$ \int_0^t E[X^2(s)]ds\,<\infty $$ It is sometimes possible to check this condition directly if the ...
1
vote
2answers
496 views

Moment generating function of a stochastic integral

Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then: $$ \mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
0
votes
1answer
63 views

Positivity of a stochastic process

I want to simulate the paths of a stochastic process $$ dS_t = r S_t dt + \sigma S_t dW_t$$ Using the Forward Euler method, we can write: $$ S_{n+1} = (1 + r \Delta t_n + \sigma \Delta W_{n}) S_n $$ ...
4
votes
2answers
164 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 ...
3
votes
1answer
388 views

Expectation of Ito integral, part 2, and Fubini theorem

I previously asked a question (Expectation of Ito integral). I have additional questions on the same subject. Let's say that we have an Ito process such as $$ X(t)=X(0) + \int_0^t a ds + \int_0^t b ...