Questions on the calculus of stochastic processes, or processes that have a random component.

learn more… | top users | synonyms

0
votes
0answers
53 views

Canonical semimartigale truncation function meaning

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: $H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
1
vote
1answer
84 views

Stochastic differential for general semimartingale

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon: "$H = B + H^c + h(x) \ast (\mu − \nu) + (x − h(x)) \ast μ$ where $h = h(x)$ is a truncation ...
2
votes
1answer
80 views

Can an Itō integral be $\infty$?

In other words, can $\int_0^t f(s)dW(s)$ = $\infty$? Thanks!
1
vote
1answer
84 views

Variance of this probability density

I have the function $\rho(x) = \frac{sin^2(x)}{x^2}$ and I want to calculate its variance on $\mathbb{R}$. Does anybody know how to do this? Cause afaik the integral does not converge.
0
votes
2answers
124 views

A mean square derivative

I'm doing an exercise where I have to check some properties about these two stochastical processes: $X(t)=At+B\;\;$ and $\;\;Y(t)=\frac{1}{t}\displaystyle\int_{0}^{t}X(\tau)\;d\tau$, $t>0$. ...
0
votes
1answer
134 views

Riemann integral of a function of the Wiener process

I'm trying to solve this exercise: $\bullet$ Find mean and variance of the next stochastical process, and prove it is a second order stationary process: ...
0
votes
0answers
30 views

A brownian bridge evaluate at a particular random variable

I was wondering of someone could help with the following. I have a random variable given as $\lambda^{*}=\arg \max_{\lambda \in (0,1)} [B(\lambda)-\lambda B(1)]^{2}/\lambda(1-\lambda)$. I am now ...
1
vote
1answer
21 views

Stochastics with induction

prove that for all $n \in \mathbb{N}$: $\sum_{r=0}^n \binom{n}{r}(-1)^{r} = 0$. The base step is easy, i only get lots of problems when i try to mess with the sum boundries.... so far i've tried: ...
3
votes
0answers
44 views

Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
1
vote
1answer
94 views

Markov-chain properties

I have some questions about a Markov-chain $(X_n)$ on a finite state-space $S$ with transition matrix $P$. A function $f:S\rightarrow\mathbb R$ is a columns vector and $Pf$ therefore a matrix ...
1
vote
0answers
90 views

Subtraction of Probability Measures

I have just read that apparently the following two conditions are equivalent: $$ \int f dP \geq \int f dQ \Longleftrightarrow \int f d(P-Q) \geq 0$$ for $f:\mathbb{R}^n\longrightarrow\mathbb{R}$ and ...
0
votes
2answers
72 views

Stochastic differential

Im really new in the stochastic procceses please help me. How can I solve this stochastic differential equation? $$dX = A(t)Xdt$$ $$X(0) = X_0$$ If $A$:[0,$\infty$]$\to$ $R$ is continous and $X$ is ...
1
vote
0answers
49 views

Atypical exponential martingale

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} ...
1
vote
1answer
40 views

Stochastic processes with non-zero higher order variations

I'm under the impression that how non-zero quadratic variation of the Brownian motion results in Itō's lemma or in general, the creation of the Itō's calculus. I'm also aware that stochastic integral ...
0
votes
2answers
137 views

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I'm in the last period of the undergraduate course and I have interest in this field, but my university don't have a specialist in ...
0
votes
1answer
44 views

sup of a submartingale until $t$

My problem: For any submartingale $(X_s)_{s\geq0}$ and for all $t\geq0$ show that $\sup_{s\in[0,t]}\mathbb{E}[|X_s]|]$ is a.s. finite. What I have until now: I know that $\mathbb{E}[X_s]$ is ...
1
vote
1answer
266 views

About stationary and wide-sense stationary processes

I have just started with stochastical calculus, and I need some help with a pair of problems: $\bullet$If $X(t)$ is a mean square differentiable wide-sense stationary stochastical process then the ...
1
vote
1answer
101 views

Basic (continuous) martingale properties

I just learned about martingales in continuous time and solved some basic exercises. But unfortunately there are some seemingly easy and surely basic things I still have problems with. 1) Let ...
2
votes
0answers
155 views

Difference of two convex functions

This is an exercise from a probability textbook on Ito's formula, basically Ito's formula extends to functions of this type. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f$ is ...
1
vote
2answers
86 views

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time $$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where ...
1
vote
0answers
80 views

Intensity Function of Stochastic process`

I'm fitting some financial data to a model based on a stochastic process and evaluating the fit of it by looking at the compensator. However, I cannot understand well what does it mean to take the ...
0
votes
0answers
102 views

First hitting time on a element of $\mathcal B ( \mathbb R^d) $ a (right, left) continuous path stochastic process

It's known that, given $\Gamma \in \mathcal B (\mathbb R ^d)$ and $X = > (X_t)_{t\geq 0}$ with right-continuous path, the random time $$T_{\Gamma} = \inf \{ t\geq 0 : X_t (\omega) \in ...
1
vote
0answers
27 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
1
vote
0answers
39 views

is this solution correct about joint distribution?

The question is if $x,y,z$ are independent $x\sim\exp(\lambda), y\sim\exp(\mu), z\sim\exp(\gamma)$ and define $u=\min(x,y), v=\min(y,z)$ what is the probability $p(U>u,V>v)$. Consider the cases ...
0
votes
0answers
58 views

Discrete time equivalent of Constant Elasticity of Variance model

I am looking for a discrete time series model equivalent to the continuous time CEV model (Constant Elasticity of variance). Indeed, I am interested in a (discrete time) mean reversion process ...
0
votes
0answers
20 views

Conditional Empirical Intensity FUnction

I'm trying to compute the empirical conditional intensity function in my data. I have two types of events, event Type1 and event Type2, that can occur in a time interval of, say, 1 day (events don't ...
3
votes
2answers
283 views

Has anyone ever won a field medal for inventing stochastic calculus? [closed]

I somehow wondered today why was Ito never awarded a Fields medal for inventing Ito calculus? I also wonder has anyone ever been awarded a Fields medal for building the foundation of stochastic ...
1
vote
0answers
105 views

Forming a local martingale with continuous increasing process

If $M_t$ is continuous martingale, we know that there exists quadratic variation process which is continuous and increasing. I am interested to know if the converse is also true. To make it precise ...
1
vote
1answer
70 views

limit of sup of a stochastic integral

Let $W$ be a standard, one-dimensional Brownian motion and $0 < T < \infty$. Show that $$\lim_{\beta \to \infty} \sup_{0\leq t \leq T} |e^{-\beta t }\int_0^t e^{\beta s } dW_s| = 0$$ a.s.
5
votes
2answers
201 views

$L^1$ bounded martingale

If $(M_t)_{0\leq t<\infty}$ is continuous martingale and it is $L^1$ bounded, does it imply that quadratic variation $\langle M\rangle_\infty$ is finite a.s. ?
0
votes
1answer
46 views

Gaussian Process / fractional Brownian motion

Let $B_H$ be a fractional Brownian motion, $f\in C^1[0,1]$ and $a>0$. Define the process $$X(t)=f(t)+aB_H(t),\qquad t\in[0,1].$$ I would like to show, that $\mathbb{P}\big(X(0)=X(1)\big)=0$ ...
0
votes
1answer
65 views

Brownian Motion hitting random point

I got a problem that seems to be quite standard and easy, but I have lots of problems with it. I do already know that $T_a:=\inf\{t\geq 0: B_t=a\}$ is a stopping time for any $a\in\mathbb{R}$ fixed, ...
1
vote
1answer
158 views

hitting time of Brownian motion

I'm desperately trying to prove that for a standard BM $B_t$ the stopping time $T:=\inf\{t\geq0: B_t\geq\sqrt{1+t}\}$ is a.s. finite, i.e. $\mathbb{P}[T<\infty]=1$. I actually tried to play around ...
1
vote
2answers
270 views

Linear Stochastic Differential Equation

Please could someone help me with the following proof: Prove that $Y_t = e^{-2t} (Y_0 + 4 \int_0^t e^{2s}d B_s )$ is the solution to the homogeneous linear stochastic differential equation $ dY_t ...
1
vote
1answer
65 views

Advanced urn problem

Imagine there are two urns — urn A and urn B. Urn A contains 3 blue balls and 7 red balls. Urn B contains 7 blue balls and 3 red balls. Balls are now randomly drawn from one of these urns where the ...
2
votes
1answer
61 views

Ito integral show $\int_0^t Z X_u dM_u = Z\int_0^tX_u dM_u$

Fix a continuous local martingale $M$ starting at $0$. Suppose $X \in \mathscr{P}^*(M)$, i.e. $X$ is progressively measurable and $\int_0^tX_u^2 d\langle M \rangle_u<\infty$ a.s. Then suppose $Z$ ...
1
vote
1answer
89 views

Properties of concave,two-parameter function

I already showed that the function $\psi(\mu,\sigma)=\mathbb{E}U(X)$ is concave in $(\mu,\sigma)$, where $X$ is normally distributed with mean $\mu$ and variance $\sigma^2$. $U$ is a nice concave ...
4
votes
1answer
188 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
2
votes
5answers
335 views

Why do people write stochastic differential equations in differential form?

I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ...
0
votes
1answer
94 views

2 dimensional Brownian motion but not 3 dimensional Brownian motion

Let $W_t = (W_t^{(1)},W_t^{(2)},W_t^{(3)})$ be 3 dimensional Brownian motion. Let $X=sgn(W_1^{(1)})sgn(W_1^{(2)})sgn(W_1^{(3)})$. Define a 3 dimensional process $M_t$ as follows : $M_t^{(1)} = ...
1
vote
2answers
136 views

Average distance to perimeter of a polygon?

Trying to calculate heat transfer which is a function of distance of each molecule to the closest wall for various container shapes. For example, a rectangular prism versus a cylinder. So I think ...
2
votes
1answer
220 views

Conditional Independence and Mutual information

I have a question concerning conditional independence. According to wikipedia (yes, maybe not the best source) two random variables are conditionally independent given a third if $$p(x,y|z) = ...
2
votes
1answer
149 views

Why can I exchange the order of integration in a multiple Ito stochastic integral?

Stochastic Processes for Physicists by Jacobs says that we can exchange the order of a multiple Ito stochastic integral, giving the example: I don't see how this works either for a regular integral ...
0
votes
1answer
28 views

find the soultion $Y(t)$ of the SDE $dY(t) = \left ( \theta - \gamma Y(t) \right )dt + \sigma dw(t)$

find the soultion $Y(t)$ of the SDE $$dY(t) = \left ( \theta - \gamma Y(t) \right )dt + \sigma dw(t)$$ as a function of the inital conditon $Y(0) = y_0$ where $\theta$, $\gamma$ and $\sigma$ are ...
1
vote
2answers
79 views

Name of the formula transforming general SDE to linear

For SDE's of the general form $$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t \tag{1}$$ @saz taught me that there is a formula to transform it into a linear SDE, quoting from René L. Schilling/Lothar ...
0
votes
1answer
169 views

Why does the function in Dynkin's formula need to have compact support? [closed]

I'm reading Oksendal's SDE book and I don't quite understand why Lemma7.3.2 and Theorem 7.4.1 requires compact support condition.
1
vote
0answers
64 views

Characterization of hitting time's law. (Proof check)

Under the same assumptions of this early question, consider also a the random time $T_a := \inf\{ t > 0: B_t \geq a\}$ which is a stopping time. Since $M^\lambda$ is a continuous martingale, Doob's ...
2
votes
0answers
130 views

Checking proof that a given process is a martingale

I am interested in justify the well known result about the process $M^\lambda _t =\exp\left(\lambda B_t - \frac{\lambda^2}{2} t\right)$ being $\mathcal F_t$-martingale in the filtered probability ...
0
votes
1answer
104 views

Question regarding Ito integral

I have a question regarding Ito integral, in particular, when I am trying to prove the normality of Ito integral, I encountered the following differential equation I need to solve: $$dX_{t} = ...
0
votes
1answer
235 views

Joint Distribution of two correlated ito integral

I have a question regarding finding the joint distribution of two process$$dX_{t}=a_{t}dB_{t}$$$$dY_{t}=b_{t}dW_{t}$$where $B_{t}$ and $W_{t}$ are two Brownian motions with correlated increments, in ...