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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4
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1answer
196 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 ...
2
votes
1answer
220 views

A question related to Novikov's condition

The well-known 'Novikov condition' says: Let $ L = (L_t)_{t \geq 0} $ be a continuous local martingale null at 0 and $ Z = \exp(L - \frac{1}{2} \langle L \rangle) $ its stochastic exponential. If ...
1
vote
0answers
191 views

Constructing Ito integral for adapted process

I am trying to construct Ito integral for adapted process. However, I am stuck at some point. Let $X^n(t)$ be a sequence of simple processes convergent in probability to the process $X(t)$. Then the ...
2
votes
1answer
568 views

Verifying Ito isometry for simple stochastic processes

It is known that stochastic integral must satisfy the isometry property which is $$ \mathbb{E}\left[ \left( \int_0^T X_t~dB_t\right)^2 \right] = \mathbb{E} \left[ \int_0^T X^2_t~dt \right] . $$ I am ...
2
votes
1answer
74 views

Covariation Paradox??

we can see that $\left\langle \int_0^t \! W_s \, \mathrm{d} s ,W_t \right\rangle_t = 0$ However if I am to use the expression $$\int_0^t \! W_s \, \mathrm{d} s= t W_t - \int_0^t \! s\, \mathrm{d} ...
2
votes
1answer
155 views

One correlated Stochastic Integral

If $${\rm Cov}[dW_t,dB_t]=\rho dt$$ then what is $$\mathbb{E} \left[\int_0^t\sigma_{1s}dW_s \int_0^t\sigma_{2s}dB_s\right]$$ where $\sigma_{1s}$ and $\sigma_{2s}$ are two deterministic functions ...
11
votes
1answer
672 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 ...
4
votes
1answer
981 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 ...
1
vote
0answers
73 views

Applicability of Itô's Lemma for $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}_0([0,1]^2)$

Let the domain be $[0,1]^2$. And let $W^x_t$ be the standard Brownian Motion started in $x\in [0,1]^2$ with absorbption on $\partial [0,1]^2$ and choose some $g\in \mathcal{C}^2((0,1)^2)\cap ...
2
votes
0answers
70 views

Are affine SDEs invertible?

If we have an a process $X_t$ with values in $\mathbb{R}^{n \times n}$ which solves a linear Stratonovich SDE $$ dX_t = A_t X_t dt + B_t X_t \circ dW_t $$ then the inverse of $X_t$ exists and solves ...
2
votes
0answers
450 views

Why is it true that the continuous local martingale with quadratic variation “t” is a square integrable continuous martingale?

I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $M_t$ be a continuous local martingale. On page 157, it wrote that "because $\langle M\rangle_t = t$, we have $M \in ...
0
votes
1answer
124 views

properties about stochastic integral

I have a question about stochastic / Lebesgue Stieltjes integrals. I'm following Revuz / Yor. The space $H^2$ is the space of all $L^2$ bounded continuous martingales. If $M\in H^2$ then they call ...
2
votes
1answer
174 views

Stochastic differential equation: Itô's formula?

I came across a problem with SDE and need your help once again: $$dX_t=tX_t \, dt+\exp \left(\frac{t^2}{2}\right)$$ and I'm supposed to solve this, in the way $X_t=f(t,W_t)$. So I use Itô's formula: ...
2
votes
1answer
160 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
409 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
292 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
190 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
108 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
251 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. ...
0
votes
1answer
602 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
114 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
votes
0answers
146 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
184 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
113 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
334 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
308 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
398 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
193 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
177 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
241 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
57 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
123 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
922 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
184 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
170 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
310 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
205 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
426 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
134 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
820 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
513 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
180 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. ...
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votes
0answers
706 views

Square root of a Wiener process

Ito integral is generally defined through the sums $$S_n=\sum_{i=1}^nG(\tau_i)(W(t_i)-W(t_{i-1}))$$ then the limit $\lim_{n\rightarrow\infty}S_n$ must exist in the rms sense. This definition can be ...
2
votes
1answer
277 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? ...
0
votes
0answers
205 views

Exercises for “Limit Theorems for stochastic processes”

I am reading the book of Jacod and Shiryaev: Limit Theorems for Stochastic Processes. But there are no exercises in this book. Does anyone know a good source with exercises?
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0answers
364 views

Min and Max of Geometric Brownian motion

I am trying to derive the distribution of $M_X(t) = \max\limits_{0\leq s\leq t}X(s)$ and $m_X(t) = \min\limits_{0\leq s\leq t}X(s)$, where $dX(t)=\mu X(t) dt+\sigma X(t)dB(t)$ and $B(t)$ is standard ...
2
votes
1answer
262 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
336 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
300 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 ...