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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2
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1answer
176 views

The variance of bilateral filtered random variables

I am glad to have found this great site. There is a problem I am trying to solve for a while. I want to analyze the noise attenuation behavior of the bilateral filter. So given the unnormalized ...
3
votes
1answer
490 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
321 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 ...
-1
votes
1answer
203 views

PDF for the integral of a Stochastic Process

My continuous-time, continuous step Stochastic Process P runs from time $t=0$ to $t=t_f$ and generates a path. I am able to observe its starting and ending position (so $P(0)=a$ and $P(t_f)=b$), but ...
0
votes
0answers
114 views

Why is this a martingale?

In our homework assignment, we are supposed to prove: If $ M $ is a countinuous local martingale and if for each $ T > 0, E[\sup_{t \leq T } |M_t|] < + \infty $ and $ H^T $ is a bounded ...
3
votes
1answer
269 views

Futures pricing and futures price process under the real world measure

This is something that keeps bothering me about the Benchmark approach of Platen, which (very) shortly is as follows: Compare the development of an economic value with a growth optimal portfolio. ...
1
vote
1answer
760 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 ...
1
vote
1answer
119 views

Why is $ N^\tau ( M - M^\tau ) $ a continuous local martingale if $ M $ and $ N $ are?

Working through my stochastic calculus script, I encountered the following identity, for which no proof is given: $ \langle M, N^\tau \rangle = \langle M, N \rangle^\tau $, if $ M, N $ are continuous ...
4
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0answers
150 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 ...
2
votes
2answers
213 views

Is the solution to a driftless SDE with Lipschitz variation a martingale?

If $\sigma$ is Lipschitz, with Lipschitz constant $K$, and $(X_t)_{t\geq 0}$ solves $$dX_t=\sigma(X_t)dB_t,$$ where $B$ is a Brownian motion, then is $X$ a martingale? I'm having difficulty getting ...
3
votes
2answers
114 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 ...
0
votes
1answer
349 views

$d$-Dimensional Brownian Motion Martingales

Let $d > 1$ and let $W_t$ denote a standard $d$-dimensional Brownian motion starting at $x\neq 0$. Let $M_t = \log|W_t|$ for $d = 2$, and $M_t= |W_t|^{2-d}$ for $d > 2$. Show that $M_t$ is a ...
5
votes
3answers
1k 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 ...
1
vote
1answer
325 views

Convergence of quadratic variation of Ito processes

I need to find an example of an Ito process $X=\{X_t:t\in[0,T]\}$ with non-zero Ito integral part and a sequence of Ito processes $\{X_n\}$ such that $X_n$ converges uniformly to $X$, as ...
4
votes
1answer
449 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
2answers
205 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 ...
1
vote
3answers
182 views

Computing Some Integrals via Gauss Integral

$ \displaystyle\int_{-\infty }^\infty e^{-\frac{1}{2} x^2} \; dx $ and $ \displaystyle\int_{-\infty }^\infty x^{2}e^{-\frac{1}{2}x^2} \; dx $ how i compute these integrals via Gauss Integral?
1
vote
1answer
251 views

Differential form of “random walk with reset” based on Wiener process

Assume such a "random walk with reset" X(t) is defined based on Wiener process (GBM) ...
1
vote
1answer
58 views

Hammerstein stochastic integral equation

I'm in trouble with the following integral equation: $$\phi(t)=\rho\int_0^1 t^2 s \phi(s)^2 ds+\nu(t)$$ where $\nu(t)$ is a white gaussian noise with variance $\sigma$ and mean value $\mu$. Is it ...
3
votes
1answer
132 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 ...
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
190 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 $\int _0^1 Y_s dW_s = 0$ ...
4
votes
1answer
174 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
2answers
336 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 ...
2
votes
1answer
232 views

Growth condition for Ito diffusions

Given a one-dimensional SDE $$ \begin{cases} dX_t &= b(t,X_t)dt+\sigma(t,X_t)dB_t, \\ X_0 &= Z \end{cases} $$ for $t\in[0,T]$ where $Z$ is square integrable: $\mathsf E[Z^2]<\infty$ ...
1
vote
1answer
544 views

Quadratic Variation of Sum of Local Martingales

I have a question about calculating covariances of local martingales. Suppose $M_1$ and $M_2$ are local martingales. Put $M = M_1+M_2$. Is there a nice way to calculation $[M]$ in terms of $[M_1]$ and ...
2
votes
1answer
136 views

Covariances of $\int_0^t h(s)\;dB_s$ process

Let $h:[0,\infty) \to \mathbb{R}$ be a measurable, square integrable function on $[0,t]$, for all $t \geq 0$. I want to show that if $H_t = \int_0^t h(s)\;dB_s$, where $(B_t)_{t\geq0}$ is a standard ...
7
votes
2answers
871 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 ...
5
votes
0answers
553 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 $dX(t)=\mu(t)dt+\sigma(t)dB(t)$ ...
3
votes
1answer
191 views

Integral paradox: Deterministic integral interpreted as limiting case of stochastic integral

The value of a stochastic integral, in this case integrating a Wiener process with respect to itself $$\int_0^T W(t)\;dW(t)$$ is dependent on the chosen position of the endpoint of the subintervals. ...
3
votes
1answer
309 views

Stochastic Integral

I've just learned about stochastic integral and only know how to evaluate $\int\limits^{t}_{0} W(s)\mathrm{d}W(s)$. Could anyone give me some instruction on how to evaluate the following integrals? ...
2
votes
1answer
297 views

Deriving SDE(s) and Expectation from Given PDE

We want to solve the PDE $u_t + \left( \frac{x^2 + y^2}{2}\right)u_{xx} + (x-y^2)u_y + ryu = 0 $ where $r$ is some constant and $u(x,y,T) = V(x,y)$ is given. Write an SDE and express $u(x,y,0)$ as the ...
4
votes
1answer
369 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 ...
1
vote
2answers
303 views

Expectation of Stochastic Process Given First Hitting Time Information

Let $V_t$ satisfy the SDE $dV_t = -\gamma V_t dt + \alpha dW_t$. Let $\tau$ be the first hitting time for 0, i.e., $\tau $ = min$(t | V_t = 0)$. Let $s =$ min$(\tau, 5)$. Let $\mathcal{F}_s$ be the ...
4
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 ...
2
votes
0answers
254 views

Show that this is the unique solution of that Stochastic Differential Equation

Reading through a paper, I stumbled across the stochastic differential equation $ dS_t = \sigma S_{t-} dX_t $. The claim there was that $ S_t = S_0 \exp(\alpha N_t - \beta t) $ should be its unique ...
3
votes
1answer
261 views

How shall I prove this Stochastic integral equation?

I want to prove $$ \int_0^T B_t^2 dB_t = \frac{B_T^3}{3} - \int_0^T B_t dt $$ by the definition of Ito integral. I have tried this so far. Given a partition $0=t_0 < t_1 < ... < t_n=T$, I ...
6
votes
1answer
378 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 ...
2
votes
1answer
352 views

First exit time

How to calculate the first exit time of the process $$dx=-M\;dt+dw,$$ where $M$ is a positive constant, and $w$ is a Wiener process? Start from $d>0$, to the boundary $0$. I solved the ...
4
votes
1answer
311 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 $Y_t$
2
votes
1answer
441 views

expected value of product of stochastic processes

Let $X_t=\sigma \int_0^t e^{-a(t-s)} dW_s$, where $\sigma , a $ are constants. How can I find the expected value of the product of $X_t, X_s$ For t>s, $\mathbb{E}[X_t, X_s]$, and $\mathbb{E}[X_t, ...
6
votes
1answer
2k 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 ...
3
votes
1answer
476 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?
4
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0answers
184 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 ...
3
votes
3answers
172 views

Which courses before Stochastics?

I would like to know which maths course I need to take before studying stochastics. Thx for helping, Stephane
4
votes
1answer
658 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( ...
5
votes
3answers
4k 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 + ...
3
votes
3answers
434 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 ...
2
votes
0answers
147 views

Stochastic differential equation

Using stochastic(!) methods find explicit solution to each of the two ($i = 1, 2$) initial value problems $$\partial_t u(t, x) = \frac{1}{2} \beta^2 \partial_x^ 2 u(t, x) + (−\alpha x + \gamma ...
4
votes
0answers
202 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 ...