The stochastic-integrals tag has no wiki summary.
14
votes
2answers
448 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 ...
11
votes
3answers
283 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 ...
9
votes
2answers
205 views
Area enclosed by 2-dimensional random curve
Consider a 2-dimensional Wiener process $(W_t)_{t \in [0,1]}$. Color every area which is enclosed by the line parametrised by $W_t$ (this means that, when the Wiener process makes a loop and ...
9
votes
0answers
253 views
Probability density function of the integral of a continuous stochastic process
I am interested in whether there is a general method to calculate the pdf of the integral of a stochastic process that is continuous in time.
My specific example: I am studying a stochastic given ...
8
votes
1answer
160 views
Why do people simulate with Brownian motion instead of “Intuitive Brownian Motion”?
I have just recently begun studying Brownian motion and stochastic calculus at the level of an undergraduate or beginning graduate student of applied mathematics. (Textbooks I've looked at are by ...
7
votes
2answers
629 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
1answer
113 views
Very basic doubt about Itô's lemma
While trying obtain the dynamics of $X_t = \exp( \int_t ^T \phi_s ds)$, where $\phi$ is an Ito process following
$$ d\phi_t = \mu dt+ \sigma dW_t \ ,$$
I had some doubt concerning the application of ...
6
votes
1answer
273 views
Expectation of an integral w.r.t. Brownian Motion
I know the following statement:
if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
6
votes
1answer
245 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
131 views
Ito's Lemma and Brownian Motion
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 ...
5
votes
3answers
741 views
On hitting time of Brownian motion and Ito's lemma
I have two possibly related questions. Let $\tau:=\min\{t\geq0:B_t=1\}$, where $B_t$ is a standard Brownian motion.
I am supposed to derive the fact that $\mathbf{E}\tau=\infty$ by applying some ...
5
votes
1answer
816 views
Expected value of the stochastic integral $\int_0^t e^{as} dW_s$
I am trying to calculate a stochastic integral
$\mathbb{E}[\int_0^t e^{as} dW_s]$. I tried breaking it up into a Riemann sum
$\mathbb{E}[\sum e^{as_{t_i}}(W_{t_i}-W_{t_{i-1}})]$, but I get expected ...
5
votes
1answer
154 views
Stochastic Integral which is almost surely zero at fixed time
This is an exercise from Karatzas and Shreve.
Find a $(Y_s)_{s \in [0,1]}$ progressively measurable such that
$ 0 < \int_0^1 Y_s ^2 ds < \infty$ almost surely, and
$\int _0^1 Y_s dW_s = 0$ ...
5
votes
3answers
237 views
Stochastic integral and Stieltjes integral
My question is on the convergence of the Riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. Here ...
5
votes
1answer
46 views
Is this stochastic integral well defined?
Motivation: I want to prove that the existence of a $\sigma$-martingale implies NFLVR (No Free Lunch With Vanishing Risk). This comes from arbitrage theory in mathematical finance and was proved by ...
5
votes
0answers
283 views
Ito's lemma and application
Can someone help me apply Ito's lemma to the function $f(t,x,k)$ where t is the time and x,k dimensions where x and k refer to dynamics
$dX(t)=\mu(t)dt+\sigma(t)dB(t)$
...
4
votes
3answers
1k views
Expectation of geometric brownian motion
I was deriving the solution to the stochastic differential equation $$dX_t = \mu X_tdt + \sigma X_tdB_t$$ where $B_t$ is a brownian motion. After finding $$X_t = x_0\exp((\mu - \frac{\sigma^2}{2})t + ...
4
votes
1answer
632 views
How to derive the Ornstein-Uhlenbeck Stochastic Integral Equation?
I have a question regarding the Ornstein -Uhlenbeck process. We have a simplified version with Stochastic Integral Equation: $X_t=-a\int^t_0 X_s\,ds +B_t$. B is the Brownian motion.
And its analytic ...
4
votes
1answer
66 views
Computation of basic stochastic integral.
I am trying to compute the covariance of a 1 dimensional Ornstein-Uhlenbeck process $dx_t=-\theta x_t dt+ \sigma dW_t$, $\theta>0$ and I am at the stage,
$$\text{Cov }(x_s,x_t)=\sigma^2 ...
4
votes
1answer
151 views
Simple stochastic integral
Let $(B_1,B_2)$ be a two-dimensional Brownian motion. Let
$$
X_t = \int\limits_0^t B_1(s)\mathrm \; dB_2(s).
$$
Is there a closed form for $X$ or the integral above is all one can get?
4
votes
1answer
184 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 $Y_t$
4
votes
1answer
363 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
2answers
150 views
Stratonovich SDE coefficient selection
Is it possible to find a strictly positive function $\sigma:\mathbb{R}\to\mathbb{R}$, such that a solution $X_t$ to an SDE
$$dX_t=-X_tdt+\sigma(X_t)\circ dB_t,$$
with $X_0$ being arbitrary, is a ...
4
votes
1answer
532 views
Is this a martingale?
Let $W_t$ be a standard Brownian motion with $W_0 = 0$ and let $Z_t$ solve the stochastic differential equation $dZ_t = 2 Z_t W_t \mathrm{d}W_t$. This has solution
$$
...
4
votes
1answer
406 views
Martingale problem
If $X_t$ is an $\mathbb{R}$- valued stochastic process with continuous paths, show that the following two conditions are equivalent:
(i)
for all $f\in C^2(\mathbb{R})$ the process $$f(X_t) − f(X_0) ...
4
votes
1answer
29 views
How to deal with differential in Itô
Suppose I have two Brownian Motion $W$ and $B$ which are connected through Girsanov, i.e. $W_t=B_t-\int_0^t v(u,T)du$. Furthermore I have the following expression
$$\exp{(\int_0^tv(u,T)-v(u,S) ...
4
votes
1answer
155 views
Stochastic integrals and new probability measures
Let $B$ be a standard Brownian motion on $(\Omega, \mathcal{F}, P, ({\mathcal{F}_t})_{t\ge0})$, where the filtration is the one generated by $B$. Fix a time interval $[0,T]$. Define the process $X$ as ...
4
votes
1answer
186 views
How to make this heuristic extension of Itô-Tanaka's formula valid
Here is my story, I have the following function :
$$
g(x)=(1+x)\cdot\exp\left(-\frac{(\log(x+a)+c)^2}{2\sigma^2}\right)1[x\ge y]=f(x)\cdot1[x\ge y]
$$
with $a,c,\sigma$ being "good" reals so that ...
4
votes
1answer
370 views
Brownian hitting time of a _very_ simple linear boundary
I realize that general results on the hitting times of a curve are practically nonexistant, but I am hoping that someone can string together a sequence of tricks to tell me what
$$
\Pr\left( ...
4
votes
0answers
117 views
Calculating $\mathbb{E}[\int_0^T N_{t-} dS_t]$ - an expectation of a simple stochastic integral.
I came across some nasty stochastic integral of which I'd like to calculate the expected value"
$\mathbb{E}[\int_0^T N_{t-} dS_t]$
where $N_t$ is a Poisson process and $S_t$ is, say, a geometric ...
4
votes
2answers
211 views
Distribution of Maximum of Sum of Sum of Gaussians
Let $X_i$ be a sequence of i.i.d. standard normal random variables. Let $Y_i=\sum_{k=1}^iX_k$ and $Z_i=\sum_{k=1}^iY_k$. I am interested in upper and lower bounds for $P(\sup_{1\leq i\leq m}|X_i|\leq ...
4
votes
1answer
177 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
...
4
votes
0answers
124 views
Observable and unobservable parameters of stochastic processes
Consider the following diffusion process
$$
dX_t = \mu\,dt+\sigma(t,X_t)\,dW_t
$$
where $X,W$ are 1-dimensional and. Is it true that given a history $(X_s,s\leq t)$ for each $s< t$ one can find ...
4
votes
0answers
165 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 ...
4
votes
0answers
281 views
Ito's Lemma application
$Z(t) = \int_0^t g(s)\,dW(s)$, where $g$ is an adapted stochastic process.
Find $dZ$ ?
3
votes
2answers
102 views
If $X$ is a martingale, $X(0)=0$; $f$ left continuous, is $\int f X$ dt also a martingale?
If $X(t)$ is a martingale, and $X(0) = 0$.
$f(t)$ is a left continuous function,
$$
g(t) = \int_0^t f(s) X(s) ds
$$
is $g(t)$ also a martingale?
I guess it shall be, but don't know how to prove ...
3
votes
3answers
140 views
Which courses before Stochastics?
I would like to know which maths course I need to take before studying stochastics.
Thx for helping,
Stephane
3
votes
3answers
275 views
How to evaluate the following stochastic integral?
How to evaluate the following stochastic integral?
$$\int_0^t M_{s^-}^2 dM_s$$
where $M_t = N_t - \lambda t$ is a compensated Poisson process.
I tried to apply Ito's formula to $M_t^2$ but still ...
3
votes
1answer
159 views
Karhunen-Loève expansion of Poisson process
Let $X_t,t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the Karhunen-Loève expansion of $X$ in interval $[0, T]$. How about the KL expansion of the centered process $X_t−\lambda ...
3
votes
1answer
197 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
453 views
Covariance of Brownian Bridge?
I am confused by this question. We all know that Brownian Bridge can also be expressed as:
$$Y_t=bt+(1−t)\int_a^b \! \frac{1}{1-s} \, \mathrm{d} B_s $$
Where the Brownian motion will end at b at $t ...
3
votes
1answer
183 views
Stochastic integral : $\int_0^T (W(s))^2dW(s)$
How to evaluate this integral $$\int_0^T(W(s))^2 \, dW(s)$$
where $W(s)$ is random variable associated with brownian motion.
I am new to this .Thanks in advance.
3
votes
1answer
192 views
$\mathcal{F_t}$-martingales with Itô's formula?
I need a little help with a problem. I am given some stochastic processes and supposed to show that they are $\mathcal{F_t}-$martingales. The first one is this, and they all look similar:
...
3
votes
1answer
96 views
stochastic analysis problem
Suppose $X$ and $Y$ are Ito processes, $X_t=x+\int^t_0Y_sdB_s$ and $Y_t=y-\int^t_0X_sdB_s,\ t\geq 0$, here $B$ is a standard Brownian motion. I need to prove that ...
3
votes
1answer
134 views
Multidimensional infinitesimal generator of a jump-diffusion
Let $X=\{X_t\}_{t\geq0}$ be an $n$-dimensional Markov process, defined by the SDE
$$dX_t = \mu(t, X_t) \, dt + \sigma(t,X_t) \, dB_t+\beta(t-,X_{t-}) \, dN_t,$$
where $\mu, \sigma$ and $\beta$ are ...
3
votes
1answer
99 views
convergence ito integral
It is easy to calculate the integral $\int_0^T B_t \, dB_t=\frac{1}{2}B_T^2-\frac{1}{2}T$
That means I showed that $\int_0^T S_n \, ...
3
votes
1answer
168 views
Funny problem about stochastic integrals and Ito' s lemma
Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
3
votes
1answer
132 views
Show that $M_t$ is a Standard Brownian Motion
Let $M=(M_t)_{t\geq0}$ with $$M_t=\int_0^{\log\sqrt{1+2t}}e^s\text{d}B_s$$
where $(B_t)_{t\geq0}$ is a Standard Brownian Motion.
Show that $M$ is also a Standard Brownian Motion and compute ...
3
votes
1answer
40 views
$E \left\{ \left( \sum_{i=0}^{n-1} \left[ B_{c_i} \left( B_{t_{i+1}} - B_{t_i}\right)\right] \right)^2 \right\}$, where $c_i \in [t_i, t_{i+1}]$
Let $B$ be a standard Brownian motion and $\{t_i\}_{i=0}^n$ a partition of $[0,t]$.
Define $c_i= (1-c)t_{i+1}+ct_i$, for some $c \in [0,1]$.
Write $B_i$ for $B_{t_i}$ and
$$
S_n=\sum_{i=0}^{n-1} ...
3
votes
1answer
107 views
Existence of solutions to stochastic differential equations by the Banach contraction principle?
I've read a proof for existence of solutions to stochastic differential equation from a book of Ikeda and Watanabe and have a question. Is it possible to prove existence (and uniquness) by means of ...
