Questions on the calculus of stochastic processes, or processes that have a random component.
2
votes
0answers
76 views
Integral representation of fractional Brownian motion
Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
1
vote
0answers
44 views
representation of the solution of SDE
Suppose we have a stochastic process $X:[0,T]\times \Omega\to (0,\infty)$, which is a solution to
$$dX_t=(\lambda X_t-\rho X^n_t)dt +\eta X_tdW_t$$
with $X_0=\xi$ for $n\in \mathbb{N}$, $n>1$, ...
2
votes
1answer
115 views
A question about Itô's representation for $\cos(B_T)$
According to Itô’s representation, any $\xi \in L_2(\Omega, F_T , P)$ has a unique representation:
$ \xi = E(\xi) + \int_0^T H_s dBs$
where $(H_s)$ is an adapted process belonging to $L_2$. $B$ is a ...
2
votes
1answer
70 views
Long Range Dependence, Fractional Brownian Motion
A stationary sequence $(X_n)_{n\in\mathbb{N}}$ exhibits long-range dependence if the autocovariance function $\rho(n):=\mathrm{cov}(X_k,X_{k+n})$ satisfy $$\lim\limits_{n\to\infty}{\rho(n) \over ...
3
votes
0answers
79 views
Stochastic differential equation solution suggestion
Any suggestion on solving the stochastic differential equation
\begin{align}
dW(t) = d\widetilde{W}(t) + \left(\frac{\kappa - W(t)}{\tau-t} - \frac{1}{\kappa - W(t)}\right)dt
\end{align}
where ...
6
votes
3answers
219 views
Expected value of average of Brownian motion
For a standard one-dimensional Brownian motion $W(t)$, calculate:
$$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$
Note: I am not able to figure out how to approach this problem. All ...
3
votes
0answers
57 views
Representation theorem for continuous process of finite variation
There is a martingale representation theorem
If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that
$$
M_t = M_0 + \int_0^t ...
4
votes
1answer
105 views
Limiting distribution of Ornstein-Uhlenbeck process
Let $X_t = e^{-\lambda t} \left(X_0 + \int _0^t e^{\lambda u} dW_u\right)$ where $(W_u)_{u \geqslant 0}$ is a Wiener process, $X_0$ random variable of law $\nu$ and independent of $\int _0^t ...
0
votes
1answer
33 views
Question on weak convergence of random variables
Let $X_n, Y_n, X$ be real random variables such that $X_n \to X$ weakly and $\mathbb{P}_{Y_n} = N(0, 1/n)$ for all positive integers $n$.
I am trying to prove that $X_n + Y_n \to X$ weakly as well.
...
0
votes
1answer
42 views
Weak convergence discrete space
Let $X_n$, $n = 1, 2, 3, \ldots$, and $X$ are random variables with at most
countably many integer values. Prove that that $X_n \to X$ weakly if and only if
$\lim_{n \to \infty} P (X_n = j) = P(X = ...
0
votes
1answer
184 views
Show that this continuous local martingale is a martingale
We are given the following SDE:
$$dX_t=X_tdt+\sqrt{2}X_tdB_t, \quad X_0=1,$$
and
$$F(x,t)=e^{-t}x,\quad t\geq0,\; x\in\mathbb{R}.$$
We are asked to apply Ito's formula to $F(t,X_t)$ for $t\geq0$ ...
16
votes
2answers
475 views
Translations of Kolmogorov Student Olympiads in Probability Theory
I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward.
I ...
2
votes
1answer
326 views
Transition density and distribution: (Ornstein–Uhlenbeck process)
Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE
below with $\alpha,\,\beta,\,\gamma$ constants:
$$
dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0}
$$
...
3
votes
1answer
87 views
Black scholes model type
I want to study the following market:
$$S_1(t)=S_1(t)(\mu_1dt + \sigma_1dW_1(t))$$
$$S_2(t)=S_2(t)(\mu_2dt+\sigma_2dW_2(t))$$
for $t\in [0,T]$, constants $\mu_i,\sigma_i$, initial values ...
0
votes
0answers
117 views
Girsanov Transformation Example
Is this the correct use of Girsanov's transformation where $B_{n}$ is a discrete Brownian motion?
For example computing:
$E[(B_{n}+2n)^{2}]$
Set: $\widetilde{B_{n}}=B_{n}+2n$
And ...
0
votes
0answers
108 views
Integrate Brownian motion with respect to independent Brownian motion
we are wondering what can be said about the distribution of that? We are considering standard Brownian motions and the integral from 0 to 1...
More precisely: What can be said about the distribution ...
1
vote
2answers
162 views
Sum of independent random variables almost surely constant
I am trying to solve the following problem:
Let $(\Omega, \mathbb{A}, \mathbb{P})$ be a probability space and $X_1, X_2, \ldots, X_n$ independent real random variables.
Prove that the sum $X_1 + X_2 ...
2
votes
1answer
232 views
Expectation of exponential martingale and indicator function.
Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$.
I want to evaluate
$$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
2
votes
1answer
65 views
Readings necessary to understand Ito Integrals?
I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background ...
1
vote
0answers
126 views
Explaining Ito formula to an analyst
From the point of view of analysis, what is Ito formula?
Is it an integral by substitution, or, a radon-nikodym derivative?
Define the probability space
$$
\left(C\left(\Bbb ...
0
votes
1answer
30 views
Is it true that $X(t)^a > K \iff X(t) > K^\frac1a$
Let $a \in \mathbb{N}$, $K \in \mathbb{R^+}$ and $X(t)$ be a geometric Brownian Motion. Is the following true?
$$X(t)^a > K \iff X(t) > K^\frac1a$$
The context of the above is that I want to ...
2
votes
1answer
79 views
Bounded variation and continuous local part when using Ito's Formula
When we apply Ito's Formula to a continuous semimartingale, which is the bounded variation part and which is the continuous local martingale part? Is there a general rule or does it depend on the ...
4
votes
1answer
98 views
covariance of integral of Brownian
What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$. Any ideas?
Thanks you very ...
3
votes
1answer
120 views
Optional sampling exercise
I came across the following exercise in Stochastic Calculus:
Let $B=(B_t)_{t\geq0}$ be a standard Brownian motion. Let also $M$ be the following process:
$M_t=B^4_t-6t(B^2_t-\dfrac{t}{3})$ for ...
1
vote
0answers
94 views
Independent Exponentially Distributed Random Variables - Athletes Problem??
Q) At a javalin competition two athletes (1 & 2) are competing against each other. Each has one attempt to throw the javalin. Assume the acheived distance of a throw ($L$1 & $L2$) [note these ...
0
votes
1answer
72 views
$dt$ terms have zero quadratic variation
Why does $ds$ integral have zero quadratic variation? Even if I have a integral of the form
$$\int X_s ds$$
where $X$ is a stochastic process? I know that a continuous process of finite variation ...
1
vote
0answers
119 views
Integral with respect to Wiener process.
Suppose that $\sigma(t,T)$ is a deterministic process, where $t$ varies and $T$ is a constant. We also have that $t \in [0,T]$. Also $W(t)$ is a Wiener process.
My First Question
What is ...
0
votes
0answers
24 views
relation between foward curve and interest rates
I am working through the book "term structure models a graduate course" by damir filipovic. Suppose the interest rate is given by the following sde
$$ dr(t)=(b(t)+\beta r(t))dt +\sigma dW^*(t)$$
...
0
votes
0answers
41 views
Condition on $f$ for $e^{\int_0^t f(B_s)ds}$ to be of finite variation
Let $B$ be a standard Brownian motion, and,
$$
X_t=e^{\int_0^t f(B_s)ds},
$$
for some function $f$.
What are the condition on $f$ for $X_t$ to be of finite variation?
Let $Y_t=\int_0^t f(B_s)ds$, if ...
2
votes
1answer
107 views
Quadratic variation of $X_t=\int_0^t B_s \, ds$
Let $B$ be a standard brownian motion and
$$
X_t=\int_0^t B_s \, ds.
$$
What is the quadratic variation $[X]_t$ of $X$?
I see $dX_t$ as an sde with drift term $B_t$.
2
votes
0answers
116 views
Multivariate Stochastic Process
I am dealing with a multivariate Ornstein Uhlenbeck style SDE. Specifically
$dx_{t,j}=\kappa_{j}(x_{t,j-1}-x_{t,j})dt+\sigma dW_{t,j} $
here j=1,2,...,6 , $x_{t,0}=\theta$ , ...
1
vote
0answers
126 views
Stochastic integral: Interchanging the order of expectation and integration
Let $B$ be a standard Brownian motion and
$$
X_t=\int_0^t f_s ds+\int_0^t g_s dB_s,
$$
where, $|f|$ and $|g|$ are both bounded, almost surely, by some positive constant $M$.
Is it true that
$$
...
0
votes
0answers
64 views
Local martingale iff each component is a local martingale?
This is probably an easy question:
A local martingale is an adapted, cadlag process for which there is an increasing sequence of stopping times (going to $\infty$) such that the stopped process is a ...
0
votes
1answer
44 views
Condition for existence of a stochastic differential equation
With $B$ a standard Brownian motion, write
$$
dX_t=f_tdt+g_tdB_t.
$$
What are the conditions on $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ for $X_t$ to exists?
I think ...
1
vote
1answer
172 views
Integration of Wiener process: $\int_{t_1}^{t_2} dB(s)$
We all know that $\int_0^t dB(s) = B(t)$, where $B(t)$ is a standard Brownian Motion.
However, is the following identity true? Also, why or why not?
$\boxed{ \displaystyle \ \ \int_{t_1}^{t_2} ...
2
votes
1answer
213 views
Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$
Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$
Provided Question
The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
1
vote
1answer
28 views
Integrating $d(e^{-ut}X(t))$, where $X(t)$ is stochastic.
Given that $\sigma e^{-ut}dB(t) = d(e^{-ut}X(t))$, where $X(t)$ is a stochastic process and $B(t)$ is a Wiener process, we have that:
$$
\int_0^t d(e^{-ut}X(s)) = X(0) + \sigma \int_0^t e^{-us}dB(s)
...
1
vote
1answer
54 views
Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion
Original Question:
Derive an SDE for $B^2(t)$, where $B(t)$ is standard Brownian Motion.
Attempt at an answer:
Apply Ito's calculus over $f(t,b):= B^2(t)$.
$$df(t,b) = \frac{\partial ...
1
vote
1answer
116 views
Min of two stopping times is also a stopping time.
Preface: I'm having trouble with the correct solution.
The Original Question: Given that $\mathscr{F}_t$ is a filtration that satisfies all the usual conditions, and given ...
3
votes
1answer
218 views
Itô Integral has expectation zero
I have a question about the following property, which I didn't know so far:
Why does the Itô integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
1
vote
1answer
39 views
Rephrasing a Stochastic Process as a Stochastic Differential Equation
I have a continuous-time stochastic process $X$, described as follows:
(1) If the process is at $x_0$ at time $t_0$, then the function $f(t_f, x_f \, | \, t_0, x_0)$ is a PDF in the parameter $x_f$ ...
2
votes
4answers
257 views
Where to begin in approaching Stochastic Calculus?
I have experience in Abstract algebra (up to Galois theory), Real Analysis(baby Rudin except for the measure integral) and probability theory up to Brownian motion(non-rigorous treatment). Is there a ...
2
votes
2answers
82 views
Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$
Is there a way to solve
$$
\partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0?
$$
This appeared as a condition for
$$
X_t=u(t,B_t)e^{\int_0^tv(B_s)ds}
$$
to be a martingale.
With $B$ a ...
0
votes
1answer
299 views
Applying Ito formula to the Brownian bridge
Let $B$ be a standard Brownian motion and
$$
W_t=(1-t)\int_0^t \frac{1}{1-s}dB_s
$$
be a Brownian bridge.
Calculate $dW_t$.
To apply Ito formula define
$$
f(t,B_t)=(1-t) \int_0^t\frac{1}{1-s}dB_s
$$
...
0
votes
1answer
89 views
superlinear and convex function
Assume $X = \mbox{random variable} X>0$, $\mathbb{E}X<\infty \implies \exists \phi:\mathbb R_+\to\mathbb R_+$, superlinar, convex with $\mathbb{E}\phi(X)<\infty$
superlinear means ...
1
vote
0answers
144 views
Analogue of Leibniz Rule for Stochastic Integrals
Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w}$$, where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ ...
1
vote
1answer
55 views
Why all previsible processes are also optional?
My doubt concerns a step on demonstration of the inclusion of the set of previsible processes in the set of optional processes.
The idea of the demonstration consists in:
Given a filtered ...
1
vote
0answers
57 views
general semimartingale theory
Last semester I attended a course about stochastic calculus. There we constructed the stochastic integral with respect to continuous semimartingales. We restrict ourselves to the continuous case. ...
0
votes
1answer
138 views
Levy's theorem? [duplicate]
Possible Duplicate:
how to show convergence in probability imply convergence a.s. in this case?
Good evenig! I stumbled upon this theorem:
For independent random variables ...
1
vote
1answer
17 views
Economics simplification of stochastic transition of capital
I'm taking an macro-econ paper and I can't seem to work out the following simplification. Basically somehow by combining equation 4.14 and ...
