A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Equation Involving Bilateral Laplace Transform

Assume that $f(x,y)$ is a compactly supported, joint probability density function on $\mathbb{R}^2$ and nice enough for the following function to make sense: $$P_t(y):=e^{ty}-\int_{-\infty}^\infty ...
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680 views

Verifying Ito isometry for simple stochastic processes

It is known that stochastic integral must satisfy the isometry property which is $$ \mathbb{E}\left[ \left( \int_0^T X_t~dB_t\right)^2 \right] = \mathbb{E} \left[ \int_0^T X^2_t~dt \right] . $$ I am ...
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Hilbert Spaces and Projections

Suppose that $\{Y_{t}: t \in \mathbb{Z} \}$ is a stationary zero mean time series. Consider the Hilbert space $\mathcal{H}$ generated by the random variables $\{Y_t: t \in \mathbb{Z} \}$ with inner ...
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1answer
334 views

Processus stochastiques et mouvement brownien by Paul Lévy

Does anybody know if there is an English or German translation of the book Processus stochastiques et mouvement brownien by Paul Lévy? If not, can someone recommend a text covering similar contents ...
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Does Brownian motion visit every point uncountably many times?

Let $B_t$ be a one-dimensional standard Brownian motion. Is it true that, almost surely, for every $x \in \mathbb{R}$ the set $\{t : B_t = x\}$ is uncountable? Let $A_x$ be the event that $\{t : ...
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153 views

Existence of a semi-martingale that matches given densities

Let $\{p_t\}_{t \geq 0}$ be a family of densities. Is there any result concerning the existence of a semi-martingale $\{X_t\}_{t \geq 0}$ such that for all $t\geq 0$, the density of $X_t$ is $p_t$ ?
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385 views

How are the pairs of two independent pure-birth processes a Markov process?

A pure-birth Process is a generalization of a homogeneous Poission process. Whereas in the Poisson process the holding times between jumps are iid exponentially distributed random variables with ...
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3k views

Conditional Covariance of Functions of Random Variables

$\newcommand{\Cov}{\operatorname{Cov}} \newcommand{\E}{\mathbb{E}}$ I realize that $\Cov(X,Y) = \E[(X-\mu_X)(Y-\mu_Y)] = \E[\Cov(X,Y|A)] + \Cov(\E[X|A], \E[Y|A])$. But I am not sure how this is ...
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103 views

Compactness of the set of densities of equivalent martingale measures

Consider an incomplete market $(\Omega,\mathcal F,\mathbb P)$ driven by a semimartingale $S=(S_t)_{t\in[0,T]}$. Under the no free lunch under vanishing risk (NFLVR) assumption, the set $\mathcal ...
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152 views

Is sampled square-integrable autocorrelation function for a WSS random process square-summable?

This may be an easy question, but I am just not sure so I am asking the community. Suppose I have a wide-sense (second order) stationary (WSS) continuous random process with autocorrelation function ...
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140 views

Under certain condition, a local martingale is a martingale

It's well known that a local martingale of is a uniformly martingale if and only if it is of class D. I want to show the following: Let $L$ be a continuous local martingale, null at zero such that ...
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123 views

Confusion regarding autoregressive process

I was reading this article related to autoregressive processes of order $1$. According to wiki it is given by $$ x_t = \phi{x_{t-1}} + \epsilon \\ |\phi| < 1 \\ x_t|x_1,\ldots,x_{t-1} \sim ...
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1answer
234 views

Biased alternating random walk on a lattice in 1D

Let's consider a random walk on a fixed lattice with step size 1 in 1 dimensions. In variation to the broadly discussed basic case, with a probability p the next step will be in the opposite direction ...
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1answer
745 views

Probability density function of the integral of a continuous stochastic process

I am interested in whether there is a general method to calculate the pdf of the integral of a stochastic process that is continuous in time. My specific example: I am studying a stochastic given ...
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1k views

Maximum Likelihood Estimation of an Ornstein-Uhlenbeck process

I am wondering whether an analytical expression of the maximum likelihood estimates of an Ornstein-Uhlenbeck process is available. The setup is the following: Consider a one-dimensional ...
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142 views

Lévy measure and compensation

In the Lévy-Itô decomposition it's necessary to compensate small jumps. That's clear. The small jumps are perhaps non-summable. But why are the jumps quared summalbe? In the "ordinary" proofs of the ...
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149 views

Finding an SDE which satisfies $X(t)$

I am attempting the following problem, and was hoping if you guys could provide any feedback on whether my approach is valid. Thank you in advance for your time! The question is as follows: "Let ...
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1answer
175 views

Definition question : Two Stopping Times

I have encountered the following question for homework, with our lecturer only requiring us to have a basic idea about stopping times. The question is as follows: Let $X(t)$ be an Ito process and ...
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1answer
246 views

Expectation of a minimum

Why is this: $E(\tau \wedge T) = \int_0^Ttf_\tau(t)dt+T(1-P(\tau\leq T))$, where $f_\tau(t)$ is the pdf of $\tau$. I am thinking its because $E(\tau \wedge T) = \tau P(\tau\leq T)+T(1-P(\tau\leq T))$ ...
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133 views

Simulating first passage times

I have a Brownian motion $X_t = \mu t+\sigma W_t$, where $W_t$ standard Brownian motion. I know that the first passage time $\tau = \min\{t|X_t\leq\alpha\}$, is Inverse Gaussian distributed i.e., ...
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$L^2$-stationary but not $L^2$-continuous process

I have to give a example of $L^2$-stationary (or also weakly stationary) but not $L^2$-continuous process. By definition, for a $X(.)$ $L^2$-stationary process, $EX(t):=m(t)=c$, for all $t\in R$ and ...
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Looking for an example of non-unique weak solution to a SDE but unique strong solution

Everything is in the title, I'm looking for an example of an SDE with a unique strong solution but with multiple weak solutions. Best regards.
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Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
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79 views

Joint distribution of consecutive renewal times

Consider a discrete analog to the Poisson process. Let the sequence $X_i$ be independent geometrically (with parameter $p$) distributed random variables that signify the inter arrival times of events. ...
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169 views

The $\alpha$-Potential-Operator (Definition and resolvent Equation)

during my studies I encountered the following Operator ($X_t$ is the standard Browniang Motion, $\alpha>0$ and $f$ is bounded function ) $U^{\alpha}f(x)=\mathbb{E}^x \int_0^{\infty} e^{-\alpha ...
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268 views

Harmonic oscillator with stochastic forcing

It's well known that the solution of the differential equation: $$\ddot x(t)+\omega^2x(t)=\sin(\psi t)$$ has the form: $$x(t)=C_1 \sin(\omega t)+C_2 \cos(\omega t)-\frac{\sin(\psi ...
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273 views

Can we construct a Hilbert space where the operator $A_u v := -\frac{1}{2} v'' + (vF + v\int_\mathbb{R} Su + u\int_\mathbb{R} Sv )'$ is symmetric?

It seems not to be a easy problem. I'd like to know if one can define a pertinent Hilbert space where the operator $$A_p v := -\frac{1}{2} v^{\prime\prime} + (vF + v\int_\mathbb{R} Sp + ...
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74 views

Applicability of Itô's Lemma for $g\in \mathcal{C}^2((0,1)^2)\cap \mathcal{C}_0([0,1]^2)$

Let the domain be $[0,1]^2$. And let $W^x_t$ be the standard Brownian Motion started in $x\in [0,1]^2$ with absorbption on $\partial [0,1]^2$ and choose some $g\in \mathcal{C}^2((0,1)^2)\cap ...
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264 views

Martingale associated to Markov chain

$X$ is a (continuous time) Markov chain with generator matrix $\Lambda$ and finite state space $G$. I know that for $g\colon G \to R$ $$ M_t = g(X_t) - g(X_0) - \int_0^t (\Lambda g)(X_s)\, ds $$ is a ...
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Are affine SDEs invertible?

If we have an a process $X_t$ with values in $\mathbb{R}^{n \times n}$ which solves a linear Stratonovich SDE $$ dX_t = A_t X_t dt + B_t X_t \circ dW_t $$ then the inverse of $X_t$ exists and solves ...
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1answer
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Sum of Gaussian processes

I would like to prove that the sum of Gaussian processes is also Gaussian, to be precise, $M_t=W_t+W_{t^2}$, where $W_t$ is standard Wiener process. That is kind of obvious, but I am looking for some ...
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3answers
634 views

Compound Poisson process with exponential distribution

Consider the following shock model. The count of shocks within a certain time $t$ is a Poisson process $N(t)$ with parameter $\lambda$, while every shock brings damage $Y_i$ to the subject, which is ...
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1answer
240 views

How to show that these spaces are Banach spaces

I want to show, that the following spaces are Banach spaces: $X_1:=\{M=(M_t)_{0\le t \le T} ;\mbox{ M is an adapted RCLL process }\}$ with the norm $\|M\|_{X_1}:=\|\sup_{0\le t\le T}|M_t|\|_{L^2(P)}$ ...
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1answer
91 views

equivalence of $E[X_\infty]=1$ and $X$ is a u.i. martingale on $[0,\infty]$

Let $(X_t)$ be a strictly positive supermartingale on $[0,\infty)$. Hence $X_t$ covnerge to $X_\infty$ a.s. Now how can I show the following: $E[X_\infty]=1$ is equivalent to $(X_t)$ is a uniformly ...
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1answer
331 views

Solutions to a stochastic differential equation

$$dX_t = -\frac{1}{2}e^{-2X_t}\ \ dt+e^{-X_t}dB_t, X_0=x_0$$ Hint: solve this equation using the substitution $X_t=u(B_t)$, show that the solution blows up at a finite random time.
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Why is it true that the continuous local martingale with quadratic variation “t” is a square integrable continuous martingale?

I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $M_t$ be a continuous local martingale. On page 157, it wrote that "because $\langle M\rangle_t = t$, we have $M \in ...
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1answer
2k views

Questions and Solutions in Brownian Motion and Stochastic Calculus?

I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
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240 views

Conditional Expectation.

Are the following two the same: $E[V(X_{t_{k+1}})|g(X_{t_{k+1}}),X_{t_k}]$ and $E[E[V(X_{t_{k+1}})|g(X_{t_{k+1}})]|X_{t_k}]$ Where $X$ is Markov chain $X_{t_k} \in \mathcal{R}^n$ $V: ...
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210 views

Determining a square integrable martingale

I'm preparing for an exam in my course Martingales & Stochastic Integrals. Currently I'm having a look at some old exams, and there's a question on one of them that I'm not able to figure out. The ...
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237 views

stochastic exponential

I was able to show the following: $X$ a semimartingale, $X_0=0$ then the SDE $$ dZ_t=Z_tdX_t$$ with $ Z_0=1$ has the unique solution $Z_t:=\exp{(X_t-\frac{1}{2}\langle X\rangle_t)}$. I was able ...
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Ratio Distribution: Poisson Random Variables

Suppose two Poisson processes. For example, during the time interval, $\Delta t_{1} = t_{1} - t_{o} = 50\mu s$ , $x$ photons are incident on a detector with rate $\lambda_{1} = 10$x$10^4 s^{-1}$. At ...
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440 views

Application of Monotone class Theorem in the proof of Kunita-Watanabe Inequality

The Kunita-Watanabe Inequality says: Let $X,Y$ be two continuous locale martingales and $H,G$ two product-measurable functions on $(0,\infty)\times \Omega$, then $$ \int_0^t|G_s||H_s|d|\langle ...
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1answer
3k views

How can I spot positive recurrence?

Can someone please explain to me the intuition behind Positive recurrence. What does it mean and why is it different to normal recurrence?
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Stochasticity of Fermi problems

The great physicist Enrico Fermi was known for his ability to make good guesses with little or bad data by multiplying series of estimates. 1 I've seen this described as corresponding to a stochastic ...
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1answer
50 views

Generating functions and tumour cells

I have got G(s) = p+ $\ rs^2$ a p.g.f for a family size. Let K be the total number of tumour cells produced from a single original tumour cell Let R(s) = P[K=0] + sP[K=1]+.... be the p.g.f of this ...
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1answer
139 views

2D simple symmetric random walks

I have a theorem which says that 2D symmetric random walks are recurrent. I understand this, the way my lecturer shows it is as follows; $\ p_{(0,0),(0,0)}^{(2n)} = (p_{0,0}^{(2n)})^2 = ...
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96 views

Question about a proof in probability theory

If $ M$ is a continuous local martingale, then it exists a sequence of partitions of $[0,\infty)$ with $|\Pi_n| \to 0 $ ($|\Pi|$ denotes the mesh side) such that $$ P(\lim_{n\to \infty} Q^{\Pi_n}_t = ...
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2answers
492 views

Mean Square Equality of Random Processes

It is given in Papoulis that: Two random processes X(t) and Y(t) are equal in the MS sense iff \begin{equation} E{|\mathbf X(t)-\mathbf Y(t)|^2} = 0 \end{equation} for every t. It follows that ...
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1answer
348 views

How to calculate the volatility of a compensated poisson process?

Poisson process $N(t)$ with density $\lambda$, could generate a compensated Poisson Process $$M(t) = N(t) - \lambda t,$$ $M(t)$ is a martingale with mean of $0$. Now, how could I calculate the ...
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1answer
189 views

a question on Stochastic Calculus

I encounterred a question on Stochastic Calculus as following, but I don't understand the meaning of $\mathcal{N}$ here, can any expert explain me a little bit? Thank you very much in advance! ...