Questions tagged [stochastic-calculus]
Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.
5,417
questions
1
vote
0
answers
10
views
Existence of a sequence of deterministic measurable kernels (Skorohod Integral and Chaos Expansion)
I'm working on Di Nunno, Øksendal, and Proske's book on Malliavin Calculus and stuck in the problem below. I wrote a possible solution here and discussed it afterward.
Problem. Let $u(t)$, $0 \le t \...
- 11
-1
votes
0
answers
31
views
Exercise with martingales (prove it)
Our professor told us to get familiar with martingales... think I got the theory, however I found this exercise in a book and I'm stuck. Can someone help me with the solution?
Let $x_n$ be a ...
-1
votes
0
answers
26
views
Proof that a Stochastic Integral is a Martingale
Let $(\Omega, \mathcal{F}, \mathbb{P}, \mathbb{F} : = \{ \mathcal{F}_t \} _{0 \leq t \leq T})$ be a filtered probability space. Let $\theta$ be a stochastic process defined as $$ \theta_t = \sum_{i=1}^...
- 191
0
votes
0
answers
27
views
Estimating the mean using antithetic variates [closed]
I'm having trouble understanding variance reduction and in particular this exercise:
We know the value of a stochastic process at the time point $T$. $X(T)=X(0)e^{(r-\sigma^2)T+\sigma W(T)}$ ($W(T)$ ...
1
vote
1
answer
43
views
Stochastic Integral Evaluation
We want to evaluate:
$$
\int_{z=0}^T \int_{s=0}^z dW(s) dW(z)
$$
But I don't really understand how to approach the problem. Is this just zero? Why?
- 37
0
votes
0
answers
16
views
Why is this the expectation of this cross product?
Question:
Show that the average of $B$ i.d. (identically distributed, but not necessarily independent) random variables, each having a variance of $\sigma^2$, with positive pairwise correlation $\rho$ ...
- 652
-3
votes
0
answers
32
views
Markov Chain Question, don't know how to solve it [closed]
enter image description here
Can anyone help me with this problem? Have no clue where to start.
- 1
2
votes
0
answers
49
views
Scaling limit of Ehrenfest chains is Ornstein-Uhlenbeck process?
According to Rick Durrett's book "Stochastic Calculus" (p. 305) rescaled Ehrenfest chains converge to an Ornstein-Uhlenbeck process:
We have two urns containing a total of $2n$ balls. At ...
- 41
3
votes
0
answers
130
views
Ito integral of $\int_0^t B_{s/2} d B_s$
Are the more or less closed formula for Ito integral
$$
\int_0^t B_{s/2} d B_s
$$
where $B_s$ is standard 1-dimensional Brownian motion?
For example, $\int_0^t B_s d B_s = (B_t^2 - t) / 2$, are there ...
- 139
3
votes
2
answers
87
views
About the notation $\mathbb{E}_t [\text{d} s_t]$
My knowledge on stochastic calculus tells me that the notation $\text{d} X_t = \mu(X_t) \text{d} t + \sigma(X_t)\text{d} B_t$ is just an abbreviation of $X_t = X_0 + \int_0^t \mu(X_s) \text{d} s + \...
0
votes
0
answers
14
views
Sufficient condition for invariant sets of stochastic differential equations
Suppose I have an ($n$-dimensional) SDE of the form $dx_t = \mu(x_t) dt + \sigma(x_t)\,dW_t$, and suppose I have some set $S$ defined as the set $x: g(x) = 0$ for some function $g$. Using Itô's lemma ...
- 278
1
vote
1
answer
42
views
How to match this SDE with Ito's formula
I have a SDE of the following form
$\frac{dZ}{dt}=-\lambda Z+\sigma\alpha_t\beta$, where $\sigma$ being the standard deviation and $\alpha_t$ is the stochastic parameter. $\beta$ is deterministic. I ...
- 261
0
votes
0
answers
38
views
Smoothness of the stopping time expectation: $\mathbb{E}[V(\tau)|W_t=x]$?
Let $\bar{x}, \underline{x}: [0,\infty)\rightarrow \mathbb{R}$ be continuous functions, $\bar{x}(t) > \underline{x}(t)$ for all $t \geq 0$ and define $U := \{(t,x) | t \in (0,\infty), x\in [\...
- 2,602
0
votes
0
answers
17
views
Rescaling the time variable in a system of SDEs.
I'm trying to change the time variable in the following SDE:
$$
dF(t) = A(t) dB(t)
$$
I'm interested to find the SDE for the process $\overline{F}(t) = F\left(a t\right)$ where $a$ is a non-negative ...
- 115
1
vote
1
answer
68
views
Distribution of $\int_0^t \left(\int_0^s W(u)\,du\right)\,ds$ [duplicate]
I am considering the following Stochastic dynamics:
$$
\mathrm d\mathbf x(t) = \mathbf F\,\mathbf x(t) \mathrm dt + \mathbf L\,\mathrm d\mathbf W(t), \quad \mathbf x(0) = \mathbf x_0
$$
where $\mathbf ...
- 13
0
votes
2
answers
69
views
Use Ito's lemma to find the process's mean and variance
The process is defined as
$$x_t = \exp{(kt+\sigma W_t)}$$
where $k,\sigma\in\mathbb{R}$ and $W_t$ is Brownian motion.
I want to find $Ex_t, Dx_t$.
To find $Ex_t$, I try to solve the equation:
\begin{...
- 25
0
votes
0
answers
40
views
Reference request: showing that solution of an Ito SDE stays bounded with positive probability
Assume that we have a (well-posed) Ito SDE of the form $$\mathrm{d} X_t = b(X_t)\,\mathrm{d} t + \sigma(X_t)\,\mathrm{d}W_t $$ where $b \colon \mathbb{R}^d \to \mathbb{R}^d$, $\sigma \colon \mathbb{R}^...
- 2,622
4
votes
0
answers
73
views
Girsanov from stochastic starting points.
Consider a canonical Wiener space $(\Omega,\mathscr{F},\mathbb{P})$. Consider two SDEs:
$$dX_t = \alpha(X_t,t)dt+dW_t, \ \ \text{Law}(X_0)=\mu$$
and
$$dY_t = \beta(Y_t,t)dt+dW_t, \ \ \text{Law}(Y_0)=\...
- 17
1
vote
0
answers
33
views
Expectation of capped hitting time
Let's suppose we have a random walk $$Z_t=\sum_{i=0}^{t} X_i$$ where each $X_i$ can take values $\pm 1$ with symmetric probability, $P(X=1)=P(X=-1)=\frac{1}{2}$.
It's is known that the hitting time at ...
- 11
1
vote
1
answer
75
views
What is the difference between $dx/x$ and $d(\log x)$?
I am not comfortable with differential forms and do not understand what $d (\log x)$ represents. My understanding is that since
$$
\frac{d}{dx} \log(x) = \frac{1}{x}
$$
then we can write
$$
d \log x = ...
- 2,255
2
votes
1
answer
38
views
Continuous approximation of discrete Markov Chain
I have a tenuous grasp of stochastic processes and am looking for help solving a problem. Apologies for any abuse of notation or naivete. Thanks in advance!
Say we have a random walk on the lattice $\...
- 21
2
votes
1
answer
76
views
Differentiability of a Poisson process
While studying the Poisson process, I had a question about whether this process is differentiable in probability and whether it is differentiable in the mean square sense, i.e. $$E\left|\frac{N_{t+h} -...
- 447
-1
votes
0
answers
16
views
Integral of a process is not a wide-sense stationary process [closed]
Consider $(X_t, t \geq 0)$ which is $L^2$ continuous and stationary in the wide sense process. Suppose it has derivative in $L^2$ and $EX_t \neq 0$, then for no random variable $\xi$ the process $(\xi ...
- 447
0
votes
0
answers
54
views
Confusion about extending the definition of stochastic integral to continuous local martingales
In Jean-François Le Gall's Brownian Motion, Martingales, and Stochastic Calculus, the author defines the stochastic integral for a continuous local martingale $M$ in Chapter 5, which is defined as $H \...
5
votes
0
answers
71
views
Sup-norm of Gaussian process
Let $(G_t)_{t\in T}$ be a centered Gaussian process (with $T = [0,1]$). Can we say anything about the distribution of $$\Vert G\Vert := \sup_{t\in T}\vert G_t\vert?$$
For a multivariate normal (i.e., $...
- 677
0
votes
0
answers
18
views
Why is $X_{t+}$ measurable w.r.t $\mathcal{F}_{t_+}$?
I encountered a trouble when I read theorem 3.17 in the book named "Brownian Motion, Martingales, and Stochastic Calculus". In this book, it first state that $X_{t+}:= \lim_{s \in D \atop s \...
- 135
0
votes
0
answers
23
views
find and sketch the distribution and density function of the random variable y = g(x). where x ~ N(0,c^2), and g(x) = x^2,
find and sketch the distribution and density function of the random variable y = g(x). If the random variable x ~ N(0,c^2), and g(x) = x^2,
any help on this would be great.
- 13
0
votes
0
answers
48
views
Dynamical equation for covariance term
I have a bunch of covariance expressions for two correlated RVs like $\mathrm{Cov}(X',Y')=E[X'Y']$, where $X'=X-E[X]$, $E[X]$ is the mean of $X$ and $X'$ is the fluctuation. I do not have any given ...
- 261
0
votes
1
answer
79
views
Are the integral and differential definitions of Ito process equivalent? [closed]
I think an ito process $X_t$ can be defined as
$$X_t := X_0 + \int_0^t\sigma_s dB_s + \int_0^t\mu_s ds.$$
(Is this an Ito drift-difussion process?)
(Why use the subscript $s$? Eg. why is it $\sigma_s$ ...
- 2,044
0
votes
0
answers
62
views
About Ito's isometry application
I've been re-reading one passage from the book of Bernt Øksendal for a whole week and still can't understand one moment in proof of that theorem:
Lemma 6.2.7.: $\hat{X_t} = E[X_t] +
\int_{0}^{t}\frac{...
- 153
3
votes
1
answer
61
views
Show that two SDE with similar drifts have similar stationary distribution
Is it true? Suppose
$$ dX_t = \nabla V_1(X_t) + \sqrt{2} dB_t \,,$$
$$ dY_t = \nabla V_2(Y_t) + \sqrt{2} dB_t \,.$$
If $V_1$ and $V_2$ are ``close", would the stationary distributions of $X_t$ ...
- 1,792
0
votes
0
answers
18
views
Extending Schilling's proof of Ito process approximation by simple processes for one-dimensional case to multivariate case
Below is the proof of Lemma 18.5 from Rene Schilling's Brownian motion which states that an Ito process can be approximated uniformly in probability by a simple Ito process.
Now it is stated in the ...
- 5,023
2
votes
0
answers
58
views
Laplace transform of a stochastic process
Let $R := (R_1, R_2)$ be a two dimensional diffusion process defined by the following SDE:
$$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t}\mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \mathrm{d}W_t$$
$$\...
- 21
0
votes
0
answers
59
views
The asymptotic limit of a certain counting measure
Let $P=(p(1),p(2),\ldots,p(i),\ldots)$ be a discrete probability distribution on $\mathbb Z_+$. Thus, $p(i) \ge 0$ for all $i$ and $\sum_i p(i) = 1$. Moreover, assume for simplicity that $p(1) \ge p(2)...
- 9,240
0
votes
0
answers
34
views
Distribution of maximum of random variables taken over an uncountable set
Consider a probability space $(\Omega, \mathcal{F}, P)$ and a one-dimensional Wiener process (Brownian motion) $\{W_t\}_{t\in [0,\infty)}$ adapted to its natural filtration. By Wiener process I mean ...
- 165
0
votes
0
answers
29
views
Long Term Fraction Spent Sitting In Left Seat
In a cafe I frequent, the area where I sit has a left seat and a right seat, and I enjoy switching where I sit. If I'm in the left seat and I feel like switching, there's a 50% chance that I'll move ...
- 775
1
vote
0
answers
40
views
Question about conditional independence of the strong Markov property for the Brownian motion
I have a slight confusion about the definition of independence under a conditional probability measure. So the goal is to show that given $P(T<\infty)>0$, and $B_t^{(T)}=1_{\{T<\infty\}}(B_{T+...
- 5,023
1
vote
0
answers
56
views
(In)equivalence of two definitions of local boundedness of a continuous time stochastic process
I would like to understand whether (and if not, under which conditions) the following two definitions of local boundedness of a stochastic process $ H = (H_t)_{t \geq 0} $ on a filtered probability ...
- 675
2
votes
0
answers
26
views
Conversion of a multivariate Stratanovich SDE into a Ito SDE representing a Brownian motion on a sphere
Sorry, first question here, so I apologize if the formatting is not great.
I have a Stratanovich SDE given by
$$d\theta_{1}=\sin(\theta_{2})\circ dW_{1}-\cos(\theta_{2})\circ dW_{2}$$
$$d\theta_{2}=\...
- 21
0
votes
0
answers
18
views
Expectation of integral of log ito process
Let $dX_t = \mu_t^Xdt + \sigma_t^X dZ_t$. Compute the quantity $E_0[\int_0^{\infty}e^{-\rho t} log X_t dt]$.
Here's what I have so far.
$log X_t = log X_0 + \int_0^{t} dlog X_s ds$. Also, by Ito's ...
- 661
1
vote
0
answers
38
views
What's the Hölder distance between any two distinct brownian motion paths?
We define the $\alpha$-Hölder semi-norm as
$$\| X\|_\alpha = \sup_{s\neq t \in [0,T]} \frac{|X_{s} - X_t|}{|s-t|^\alpha}$$
and the distance function between two rough paths as,
$$\rho_{\alpha}(\mathbf{...
- 421
1
vote
0
answers
33
views
Reverse engineer Ito's Lemma to find $X_t$ for $dX_t = [\nu - \gamma\left(X_t - \nu\,t \right)]dt + \sigma\,dW_t$ [duplicate]
Find an integral expression for $X_t$, where $X$ is an Ornstein-Uhlenbeck-type process
governed by the stochastic differential equation:
$dX_t = [\nu - \gamma\left(X_t - \nu\,t \right)]dt + \sigma\,...
- 481
0
votes
0
answers
17
views
Proof that the quadratic variation of a semi martingale is the quadratic variation of its local martingale part
I am asking this is that I am trying to show that the quadratic variation of a continuous semi martingale is just the quadratic variation of its local martingale part. That is if $X_t = X_0 + M_t+A_t$ ...
- 359
2
votes
0
answers
46
views
Expectation Value of the Product of a Time integral and a Ito Integral
Consider a stochastic process $X_t$
\begin{equation}
dX_t = a(X_t)dt + \sigma dW_t
\end{equation}
where $W_t$ is a Wiener Process.
I know the expectation value of the product of two stochastic ...
- 21
0
votes
1
answer
76
views
How to compute the second derivative of a function of a stochastic process?
Background
Suppose we have a state $\mathbf{x}(t) \in \mathbb{R}^{d \times 1}$ evolving according to
\begin{equation}
\mathrm{d}\mathbf{x}(t) = \mathbf{F}(\mathbf{x}(t)) \mathrm{d}t + \mathbf{G}(...
- 119
0
votes
0
answers
24
views
Why is the quadratic covariation of two independent brownian motions zero?
In my notes there is a remark saying that if $B$ and $B'$ are independent Brownian motions, then $\langle B,B'\rangle = 0$.
I know the following properties of quadratic covariation:
$\langle M, N \...
- 359
1
vote
1
answer
65
views
Differentiating Stochastic Integrals (Ito Integrals)
If I wanted to differentiate a stochastic integral, is this logic correct? $$X_t = \int_0^t B^2_s dB_s\\ dX_t = d \int_0^t B^2_s dB_s \\ dX_t = B^2_tdB_t - B^2_0dB_0 \\
dX_t = B^2_tdB_t \quad (B_0 = ...
5
votes
2
answers
132
views
SDE of conditional expectation
Assume I have an SDE of the form
$$
dX_t = a(X_t, t) dt + b(X_t, t) dW_t, X_0 = x_0.
$$
Let now with $T>t$ denote $f(X_t)$ the conditional expectation of $X_T$,
$$
f(X_t) = E[X_T|X_t]
$$
My ...
- 129
0
votes
2
answers
80
views
j-th order derivative of function $f(x) = 1 - \sqrt{1 - x}$ [closed]
Can you please help me with this part of book (Stochastic Calculus for Finance I: The Binomial Asset Pricing Model; Shreve; p.140):
Define $f(x) = 1 - \sqrt{1 - x}$ so that
$$f^{’}(x)=\frac{1}{2}(1-x)^...
0
votes
1
answer
34
views
No arbitrage for one period Binomial model option pricing - understanding the return function
I am looking at lecture notes in mathematical finance. The prince $C$ of a European call option with strike $K$ and expiry time $T$ in a one period binomial model is derived via no arbitrage principle....
- 1,540