Tagged Questions
0
votes
0answers
14 views
The impact of jump on the returns of portfolio and asset pricing
There exsits jumps in financial market. What will be the impact of jump on the returns of portfolio and asset pricing?
Please explain it both academically and plainly. If you can give some excellent ...
0
votes
1answer
36 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
1
vote
2answers
27 views
Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
3
votes
0answers
30 views
lower bound of expectation of stochastic differential equation
I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
1
vote
1answer
39 views
How is Brownian motion predictable?
Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
1
vote
2answers
53 views
Moment generating function of a stochastic integral
Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then:
$$
\mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
0
votes
1answer
21 views
Proving weak existence of CIR process
Consider the following SDE
$$ X_t = x + \int_0^t \theta (\mu -X_s) ds + \int_0^t\kappa \sqrt{|X_s|} dW_s $$
where W is a brownian motion. I'm trying to show a weak solution exists, does anyone have ...
1
vote
1answer
45 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
0
votes
1answer
36 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
0
votes
1answer
26 views
Brownian motion and convergence in probability of step functions
For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
0
votes
1answer
22 views
Integrating a Poisson Process with respect to itself
I am just learning about Poisson Processes and I feel somewhat comfortable with the basic concepts, but I am a little stuck with the following problem:
Let $N(t)$ be a Poisson process with intensity ...
1
vote
0answers
57 views
$dX_t=1_{X_t\not=0} dW_t$
Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $
how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss
Indeed
Consider the stopping time $$ ...
3
votes
0answers
26 views
Orthogonal projections for minimization problem
I have trouble to understand the existence of a minimization problem in a Hilbert space. Let $(\Omega,\mathcal{F}_T,P)$ be a filtred probability space with filtration $(\mathcal{F}_t),0\le t\le T$. We ...
2
votes
0answers
35 views
Product of predictable process and a characteristic function is integrable
Suppose the time parameter $t\in[0,T]$, $S$ is a Semimartingale and $\theta_t$ a predictable $S$-integrable process such that
$$\int_0^T\theta_u dS_u\ge -a$$
for a $a>0$. Furthermore ...
0
votes
0answers
14 views
Submartingale bounds
Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
0
votes
1answer
36 views
Integral: Is there a closed form?
I wonder whether there is a closed form or way to compute explicitly:
$$\int_0^t e^{\alpha s} dB_s$$
where $\alpha$ is just a real number and the integral is in the Itô sense.
Thank you very much!
0
votes
0answers
19 views
Solve a special non-linear Backward SDE
It is straigtforward to solve a linear Backward SDE. i.e.
$dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.)
How can I solve $dY_t=Z_tdW_t+ ...
1
vote
2answers
39 views
Diffusion process. Distribution vs transition probability.
I need confirmation on the following problem: Take a SDE of the form:
\begin{equation}
dX_t=a(X_t,t)dt+b(X_t,t)dW_t
\end{equation}
where all the conditions, such that the solution $X_t$ is defined ...
1
vote
2answers
35 views
Inequality- Absolute Value general powers
Iam trying to understand the following inequality:$p>0$
Let $$T_{m}:=\sum_{i=1}^{m}\left(\left|\int_{\frac{i}{n}}^{\frac{i-1}{n}}g(s)dW_{s}\right|^{p}-\left|g ...
0
votes
0answers
19 views
Find the distribution of the increments from Langevin equation?
Given a Langevin eq. of a stochastic process:
X[I+1]=X[I]-F(X[I])+W[I]
- where F(X[I]) is a position dependent force, and W[I] is the Wiener process term (i.e. Gaussian, white-noise).
How do I ...
0
votes
0answers
48 views
Drift equation / Girsanov's Theorem
Define $$\frac{dS_t}{S_t} := \mu \, dt + \sigma^B \, dB_t +\sigma^W \, dW_t,$$ $$dS_t^{(0)} := S_t^{(0)}r \, dt,$$ where the constants $\mu, \sigma^B,\sigma^W$ and $r$ all positive and $\mathcal{F}_t$ ...
0
votes
0answers
20 views
SDE(s) satisfied by Radon Nikodym derivatives of martingale measures?
Given:
Money Market Account: $dR_{t}=R_{t}r_{t}dt, R_{0}>0$
Risky Asset: $dS_{t}=S_{t}(\mu_{t}dt+\sigma_{t}dB_{t}), S_{0}>0$,
where $r, \mu,$ and $\sigma$ are positive processes and $B$ is a ...
0
votes
0answers
23 views
Densities of r.v in stochastic analysis
I have several exercises to solve and there are two which I somehow do not manage to solve...
We consider $W=\{W_t:t\geq0\}$ a standard B.M. issued from zero, for $a\in \mathbb{R}$, ...
2
votes
1answer
22 views
Meyer's Theorem in Williams & Rogers
In Diffusions, Markov Processes and Martingales Volume 2 by Rogers and Williams they state the following theorem due to Meyer:
$\mathbf{Theorem }$ Le $M\in\mathcal{M}^2_0$. Then there exists a ...
0
votes
1answer
20 views
Concepts: time homogenous and independent increments
Can someone give me an illustrative example for a time homogenous process without independent increments and for a process that is not time homogenous, but has independent increments?
0
votes
1answer
43 views
Distribution of stochastic integral w.r.t. to centered Poisson process
Let $N(t)$ be a Poisson process with intensity $\lambda$ and define the centered process as $N_0(t):= N(t)-\lambda t$. A stochastic integral can be properly defined w.r.t. to $N_0(t)$ (but not w.r.t. ...
1
vote
2answers
87 views
Conditional expectation of a functional of an Itô's semimartingale under its equivalent martingale measure
Consider a probability filtered space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfying the habitual conditions and is generated by $1 d $- ...
1
vote
1answer
65 views
Approximation of stochastic integral
Let $f \in C^2_C(\mathbb{R})$ and $$X_t = X_0 + \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds$$ (1-dim.) Itô process where $\sigma,b: [0,\infty) \times \Omega \to \mathbb{R}$ progressively ...
1
vote
0answers
38 views
Rate of increase of maximum process of Brownian Motion
Suppose $M_t=\sup_{0\leq s\leq t}\{B_s\}$, where $\{B_t\}_0^{\infty}$ is a standard Brownian Motion. I would like to know if it is true that $M_t e^{-t}$ converges to 0 almost surely?
Thanks!
1
vote
1answer
66 views
Martingale inequality
Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define
$$
Y^r_t := \int_0^t f(r,s) dW_s
$$
For each fixed $r$, ...
3
votes
0answers
42 views
Spectral process for the Ornstein-Uhlenbeck process
The Ornstein-Uhlenbeck process $X(t)$ is a centered, Gaussian process with covariance function $$B(s,t) = e^{-\vert t-s \vert /2}$$
The spectral measure is abs. cont. w.r.t. the Lebesgue measure ...
2
votes
0answers
20 views
When is the spectral measure absolutely continuous w.r.t. Lebesgue?
According to Bochner's theorem, the covariance function $b(t)$ of a centered, weakly stationary process $X(t)_{t\geq 0}$ can be written as
$$b(t) = \int_{-\infty}^{\infty} e^{i t \lambda} ...
1
vote
1answer
34 views
Time integral over stochastic process depends on distribution only?
Let $X(t),Y(t)$ be two stochastic processes, integrable on $[0,T]$ with $X(t)\stackrel{d}{=}Y(t),\forall t\in [0,T]$.
Does this imply
$$\int_0^t X(s)ds = \int_0^t Y(s)ds, \qquad \forall t \in ...
1
vote
1answer
48 views
Discontinuous Martingales on the interval $[0,T]$
Does there exist a Martingale on continuous time $[0,T]$ such that it is discontinuous for every $t \in [0,T]$?
0
votes
0answers
18 views
spectral representation of discrete time, periodic, weakly stationary sequence
Let $(\xi_n)_{n\geq 1}$ be a sequence such that $\xi_{n+N} = \xi_n$ for some $N$ and all $n$.
What would be the spectral representation of this sequence?
Let $b(t)$ be the covariance function for ...
2
votes
1answer
64 views
Are these two some kinds of generalized Ornstein–Uhlenbeck processes?
An Ornstein–Uhlenbeck process is
$$
d X_t = (\mu - X_t) dt + d W_t
$$
We try to build a model using some generalized Ornstein–Uhlenbeck processes.
The first one is
$$
d X_t = \exp(-|X_t- \mu|) ...
0
votes
0answers
36 views
Intuition: integration of function with respect to stochastic process
Let $X(t),t\in [a,b]$ be a stochastic process with $\mathbb E[X(t)]\equiv 0$ and uncorrelated increments, $f$ a continuously differentiable function.
With the above conditions, the following equality ...
1
vote
1answer
34 views
Joint distribution of Gaussian process and its derivative
Let $X(t)$ be a Gaussian process with zero mean and covariance function $B(t,s) = 1/(1+(t-s)^2)$. Let $X'(t)$ be its $L^2$-derivative. I am looking for the joint distribution of $X(t)$ and $X'(t)$.
...
1
vote
1answer
50 views
Backward martingale property of quadratic variation
Let $\pi_n$ denotes a refining sequence of partitions of a finite closed interval (refining means $\pi_n\subset\pi_{n+1}).$ And we denote $\pi_n B = \sum_{t_i\in \pi_n}(B_{t_{i+1}}-B_{t_i})^2$, where ...
7
votes
1answer
125 views
generating set of predictable sigma algebra
I am solving an exercise in Rogers and Williams and want to ask if my solution is correct. Let me first introduce the notation. The space $b\mathcal{E}$ is the space of processes of the form
...
0
votes
0answers
45 views
Intuitive meaning of Lévy-Khintchine triplet
Let $\varphi$ be the characteristic function of an infinite divisible distribution. It can be expressed in the form $\varphi = e^\psi$ with
$$\psi(\lambda) = i \lambda a - \frac{\sigma^2 ...
5
votes
1answer
46 views
Is this stochastic integral well defined?
Motivation: I want to prove that the existence of a $\sigma$-martingale implies NFLVR (No Free Lunch With Vanishing Risk). This comes from arbitrage theory in mathematical finance and was proved by ...
3
votes
0answers
72 views
locally boundedness of RCLL and LCRL processes
The filrtation in this questions is assumed to fulfill the usual condition. Let $X$ be an adapted RCLL process and we look at $X_-$. It is well known that this process is predictable (hence ...
0
votes
1answer
112 views
Solving Stochastic Differential Equations
Can anyone help me with the following SDE?
Solve the following stochastic differential equation:
$$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$
with $Y_0=0$.
Hint: Try a solution of the form $Z_tH_t$ where $Z_t = ...
1
vote
1answer
49 views
Brownian Motion and the Functional CLT
Suppose we have a time series $(x_t\mid t\in \mathbb{Z})$ for which the partial sum process $X_T$ defined on the unit interval by $$ X_T(\xi)=\omega_T^{-1}\sum_{t=1}^{[T\xi]} ...
2
votes
0answers
43 views
Negative moments of a functional of Wiener process
At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
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 ...
1
vote
1answer
98 views
Distribution of integral with respect to Brownian motion
Let $B$ be a Brownian motion and define the complex sequence $(X(n))_{n\in \mathbb Z}$ as
$$ X(n) := \int^\pi_{-\pi} e^{inx} dB(\pi + x)$$
What is the distribution of $X(n), n\in \mathbb Z$?
0
votes
0answers
58 views
diffusion processes and Ito diffusion processes
If I am correct, a diffusion process is defined as a Markov process with a.s. continuous sample paths.
A Ito diffusion process is defined via a SDE. From Wikipedia:
A (time-homogeneous) Itō ...
0
votes
2answers
40 views
Identity for exponential of Brownian motion using scaling relation
Let $B$ be a Brownian motion and $s\wedge t := \min\{s,t\}$, $s\vee t := \max\{s,t\}$.
I stumbled over the following identity:
$$ \mathbb E[\exp(B(s\wedge t) + B(s\vee t))]
\\=\mathbb ...
