Questions on the calculus of stochastic processes, or processes that have a random component.
0
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0answers
9 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
0
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1answer
16 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 ...
2
votes
0answers
20 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
32 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
43 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:
$$
\mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
0
votes
0answers
8 views
Proving weak existence of CIR process
Consider the following SDE
$$ X_t = x + \int_0^t \theta (\mu -X_s) ds + \int_0^t\kappa \sqrt{|X_s|} dW_s $$
where W is a brownian motion. I'm trying to show a weak solution exists, does anyone have ...
0
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0answers
20 views
For which p>0 does $S_t=W_t+t^p$ admit an equivalent martingale measure?
Let W be a brownian motion and p>0.
For which p does $S_t=W_t+t^p$ admit an equivalent martingale measure?
I recently saw at my lectures that
NFLVR cond:
There does not exist a sequence $\{H_n\}_{n ...
1
vote
1answer
41 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
0
votes
1answer
35 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
0
votes
1answer
25 views
Brownian motion and convergence in probability of step functions
For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
0
votes
1answer
22 views
Integrating a Poisson Process with respect to itself
I am just learning about Poisson Processes and I feel somewhat comfortable with the basic concepts, but I am a little stuck with the following problem:
Let $N(t)$ be a Poisson process with intensity ...
1
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0answers
39 views
$dX_t=1_{X_t\not=0} dW_t$
Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $
how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss
Indeed
Consider the stopping time $$ ...
2
votes
0answers
24 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
34 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
13 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 $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
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0answers
9 views
A (notational) question on composition of maps involved in time change
This comes from a paper:
we have
$X_t = x+\int^t_0 a(X_s)Y_sdB_s$
If we let $M_s = \int^t_0 Y_sdB_s$. But time change $X$ by the inverse of $\langle M\rangle$, we have
$G_t=x+\int^t_0 a(G_s)dW_s$
...
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
18 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
38 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 ...
0
votes
0answers
22 views
Help solving the (degenerate) SDE: $X_t =\int_0^t |X_s|^\alpha ds$
In a homework exercise I am, as an example of non-uniqueness of SDE's with drift only Hölder continuous of index in (0,1) , asked to show that
both the zero process and $X_t=C\cdot t^p$ where ...
4
votes
0answers
51 views
What are the prerequisites for stochastic calculus?
I am not a math student, and only kind of picking up something whenever I need it. After emerged in the field of machine learning, probability, measure theory and functional analysis seem to be quite ...
1
vote
2answers
34 views
Inequality- Absolute Value general powers
Iam trying to understand the following inequality:$p>0$
Let $$T_{m}:=\sum_{i=1}^{m}\left(\left|\int_{\frac{i}{n}}^{\frac{i-1}{n}}g(s)dW_{s}\right|^{p}-\left|g ...
0
votes
0answers
19 views
Find the distribution of the increments from Langevin equation?
Given a Langevin eq. of a stochastic process:
X[I+1]=X[I]-F(X[I])+W[I]
- where F(X[I]) is a position dependent force, and W[I] is the Wiener process term (i.e. Gaussian, white-noise).
How do I ...
0
votes
0answers
44 views
Drift equation / Girsanov's Theorem
Define $$\frac{dS_t}{S_t} := \mu \, dt + \sigma^B \, dB_t +\sigma^W \, dW_t,$$ $$dS_t^{(0)} := S_t^{(0)}r \, dt,$$ where the constants $\mu, \sigma^B,\sigma^W$ and $r$ all positive and $\mathcal{F}_t$ ...
0
votes
0answers
19 views
SDE(s) satisfied by Radon Nikodym derivatives of martingale measures?
Given:
Money Market Account: $dR_{t}=R_{t}r_{t}dt, R_{0}>0$
Risky Asset: $dS_{t}=S_{t}(\mu_{t}dt+\sigma_{t}dB_{t}), S_{0}>0$,
where $r, \mu,$ and $\sigma$ are positive processes and $B$ is a ...
0
votes
0answers
23 views
Densities of r.v in stochastic analysis
I have several exercises to solve and there are two which I somehow do not manage to solve...
We consider $W=\{W_t:t\geq0\}$ a standard B.M. issued from zero, for $a\in \mathbb{R}$, ...
1
vote
1answer
75 views
american put option
For a perpetual american put option $v(s)$, satisfies the following problem:
$$\frac12\sigma^2S^2\frac{\mathrm d^2V}{\mathrm dS^2}+(r-D)S\frac{\mathrm dV}{\mathrm dS} - rV = 0\quad\text{for ...
2
votes
1answer
21 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
63 views
Will I have learned the prerequisites for self learning stochastic calculus and monte carlo method?
I'm an undergraduate econ major, and my main focus is in actuarial sciences, which as you may or may not know it's pretty mathematical. Some of the topics I will have to learn at some point on my own ...
0
votes
1answer
40 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
85 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
64 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 ...
0
votes
0answers
5 views
Does the domain of the generator of an Ito diffusion contain $C^2$ or just $C_c^2$?
Does the domain of the generator of an Ito diffusion contain $C^2$ or just $C_c^2$?
From Wikipedia
(For the generator $A$) One can show that $C_c^2$, i.e. any compactly-supported $C^2$ (twice ...
1
vote
0answers
35 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
41 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
0answers
50 views
Quadratic variation process of $G$–Brownian motion
I would like to prove the inequality
$$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$
where $\langle B ...
0
votes
0answers
24 views
Is $Z_t$ defined by $d Z_t = a(t) Z_t dW_t$ necessarily martingale?
The process $Z_t$ is defined by $d Z_t = a(t) Z_t dW_t$.
Some claims that since $Z_t$ can be represented as the Ito integral of $a(t) Z_t$, $Z_t$ is Martingale.
But I think $Z_t$ has to be ...
1
vote
1answer
46 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
44 views
Probability computation $P(X_n/\log(n))$
Let $X_1, X_2, ...$ denotes a sequence of i.i.d. random variables such that $X_1$ ~ $exp(1)$ and c>0.
What is $P( X_n/\log(n) > c$ for infinitely many $n$'s) ?
Can I simply say that $P(X_n > c ...
1
vote
0answers
20 views
Can Ito's formula apply to $f(t, B_t)$ if $f(t,x)$ itself is random?
Can Ito's formula/lemma apply to $f(t, B_t)$ if $f(t,x)$ itself is random?
I asked this, because in Ito's formula, $f$ is assumed to be a deterministic function?
For example, define $f$ as
$$
f(t, ...
1
vote
1answer
80 views
How to compute $\int_0^t s d B_s$ and $\int_0^t B_s ds$?
Consider the Itō integral $X_t := \int_0^t s \,dB_s$.
Here is my attempt. Let $f(t,x) = tx$. By Itō's formula,
$$ d f(t, B_t) = B_t dt + t dB_t $$
so
$$ t B_t = \int_0^t B_s\, ds + \int_0^t s \,dB_s. ...
1
vote
1answer
25 views
Questions about existence and uniqueness theorem for stochastic differential equations in Oksendal's SDE book
In Oksendal's SDE book, Theorem 5.2.1. (Existence and uniqueness theorem for stochastic differential equations) assumes $Z$ is a random variable which is independent of the sigma algebra $\mathcal ...
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
61 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
$$
d X_t = \exp(-|X_t- \mu|) ...
5
votes
1answer
124 views
Ito's Lemma and Brownian Motion
Show by using Ito's Lemma, for $k \geq 2$ the following result hold.
$$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$
where $W(t) = N(0,t)$ is standard Brownian motion.
I think ...
0
votes
0answers
33 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 ...
