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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Number of rewards before death

I have a question regarding Poisson events with death. Assume time is continuous $t\in[0,\infty)$. A person may die with intensity $\delta$. Conditional on being alive, he may achieve a reward with ...
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1answer
24 views

The martingale $M_t,\mathcal{F}_t$ is a martingale with respect to the filtration $\mathcal{F}_{t +}$

Let $M_t$ be a right continuous martingale with respect to the filtration $\mathcal{F}_t$. Can we say that $M_t$ is a martingale with respect to the filtration $\mathcal{F}_{t+}$? Attempt: We know ...
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2answers
50 views

$X$ and $Y$ are independent Poisson$(\lambda)$, $\lambda\sim\mathrm{exp}(\theta)$. What is the conditional distribution for $X$ given that $X+Y=n$?

To clarify, the parameter $\lambda$ is a random variable with exponential distribution and parameter $\theta$. Can someone please tell me if I've correctly computed the conditional distribution for ...
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1answer
14 views

Find a family of measures that satisfies the requirements for measurable selection

In chap 12 of Stoock and Varadhan Multidimensional diffusion processes in section 12.2 markov selections page 290 one reads I couldn't find an example that fit the conditions (a)-(d). One would ...
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1answer
19 views

Not independent Increments

For a given Process $K_{t}=\exp(B_{t}+\theta t)$ with $\theta\in\mathbb{R}$ and $B_{t}$ a Wiener process i want to show, that $K_{t}$ does have dependent increments. My idea is: $$ \begin{split} ...
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1answer
50 views

Age distribution when meeting

I have a question regarding Poisson process. I will tell the story in the context of a player-monster game. Consider a player who is born at $t=0$. He will win the game if he can survive until ...
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1answer
37 views

Continuity of a stochastic process

Exercise 3.11 in Oksendal's book "SDEs: an intorduction with applications". Let $W_t$ be a stochastic process satisfying 1) ${W_t}$ is stationary; 2) $t_1\not=t_2\implies W_{t_1}$ and $W_{t_2}$ are ...
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1answer
12 views

Continuous Sigma-martingales and local martingales are equivalent sometimes?

I was reading through this paper, and they mentioned in the beginning-most portion of it that $\sigma$-martingales and local-martingales are equivalent if they are continuous. Why must they be ...
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7 views

Is the function $Z(t,q)$ progressively measurable?

The excercise is taken from Stroock and Varadhan Multidimensional diffusion processes chap 1 page 44 is the following To the first part, I reasoned as follows: Consider the function $F(q_1,q_3) = ...
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20 views

Death process - median time to die out

Question: A pure death process $\{X(t); t \ge 0\}$, where $X(t)$ denotes the number of individuals alive at time $t$, starts with $X(0) = 8$. The lifetime of each of these individuals is exponential ...
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1answer
283 views

What is the difference between calculus and analysis? In stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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94 views

Conditional distribution of mixed process

Let $Y$ be a random variable such that: $$Y \sim \begin{cases} \operatorname{Poiss}(\lambda_0), & x= 0 \\ \operatorname{Poiss}(\lambda_1), & x=1 \\ \end{cases} $$ where ...
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1answer
32 views

A Characterization of the Strong Markov Property

I have a question concerning the strong Markov property: For a strong Markov process $(X_u)_{u\ge 0}$, a real time $t\in \mathbb{R}$ and an optional stopping time $T$ with $t< T$ \begin{align*} ...
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15 views

Modelling commodity price uncertainty with brownian motion - time period impacts

background I have two separate models of a metals resources company. Each model produces a series of accounting and cashflows forecast for different assets, and consolidates these to a overall ...
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17 views

When is the riemann integral of a finite-variation process square-integrable?

Let us say $X_t$ is a continuous finite-variation process and $f(t, x)$ a $C^{1, 1}$ function. We define $$ Y_t = \int^t_0 f(s, X_s)\, \mathrm ds $$ Are there any general results pertaining to ...
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50 views

Difference between the “Hazard Rate” and the “Killing Function” of a diffusion model?

I posted this question on Cross Validated - but I think it applies here too. Also, it increases the chances of the question being answered. Link here If this is not acceptable - administrators ...
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17 views

On the difference between a modification of a stochastic process and two indistinguishable processes. [duplicate]

I am following Brownian motion and Stochastic calculus by Karatzas and Shreve They give the following definitions with an example: I do not understand the difference between definition 1.1 and 1.3 ...
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2answers
435 views

Continuous local martingale of finite variation is constant

Is a continuous local martingale $M$ of finite variation constant? We know that there exists a sequence of stopping times $T_n\nearrow \infty$ a.s. as $n\to\infty$ such that the stopped process ...
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38 views

Let $N$~Pois$(\lambda)$, $X|(N=n)$~Bin$(N,p)$, $Y=N-X$. Show $X$, $Y$ are independent and Poisson with parameters $\lambda p$ and $\lambda (1-p)$.

Any direction on this problem would be much appreciated. So far I know the joint distribution of $X$ and $Y$ is $\begin{align} \mathsf P(X=x, Y=y) & = \mathsf P(X=x, N-X=y) \\ & = \mathsf ...
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10 views

Finding a predictable process $\lambda$ such that $dA_t = d\langle M \rangle_t \lambda_t$ where $Y = M + A$ is an exponential Lévy process

Assume $X$ is a finite-variation Lévy process and let $Y_t = e^{X_t}$. It can be shown that one can decompose $Y_t$ as $Y_t = M_t + A_t$ where $M$ is a local martingale and $A$ is continuous and ...
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46 views

Integral with respect to Brownian motion, Variance

Good day. Imagene we have a martingale $M(t)=\int_0^t f(s)dB(s)$ which satisfies Dambis-Dubins-Schwarz Theorem. At the same time $M(t)^2 - <M>(t)$ is a Martingale starting in $0$ as well. If i ...
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23 views

Expected number of arrival in a poisson process which are not followed by arrival in next time $\delta$ .

Let $a,\lambda,\delta > 0$ . Compute the expected number of arrivals in a Poisson process with intensity function $\alpha(t)=ae^{-\lambda t}$, which are not followed by another arrival with time ...
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1answer
23 views

If $N(t)$ is a Poisson process with parameter $\lambda(t)$ then is $N'(t)=N(t+2)-N(2)$ a poisson process?

If $N(t)$ is a Poisson process with parameter $\lambda(t)$ then is $N'(t)=N(t+2)-N(2)$ a poisson process? I think it should be poisson process as it is like observing a poisson process after time $2$ ...
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2answers
21 views

On the definition of continuous time martingales Stroock Varadhan $\times$ Kallenberg

In the definition of martingales, one finds in Stroock and Varadhan (Multidimensional Diffusion processes - page 20) the strange request that it be right-continuous process. However no such ...
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1answer
33 views

On the proof of lemma 1.2.4 of Stroock and Varadhan A question concerning stopping times

In the book Multidimensional diffusion processes, of Stroock and Varadhan one reads (page 23): This is the proof of $(i)$. Here the authors say Define $f_t$ on $(\{\tau \leq t\}, \mathcal{F}_t ...
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4answers
62 views

Supermartingale vanishing at some stopping time

Let $\left\{X_t\right\}_{t\in[0, T]}$ be a continuous and non-negative supermartingale. We define the stopping time $$\tau_0:=\inf\{t\in[0,T]:X(t)=0\}\wedge T$$ and immediately obtain by continuity ...
5
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2answers
48 views

Suppose $\xi_1, \xi_2,\ldots$ are i.i.d. random variables with mean $\mu$, variance $\sigma^2$. Form the random sum $S_{N} = \xi_{1}+\cdots+\xi_{N}$.

(a) Derive the mean and variance of $S_{N}$ when $N$ has Poisson distribution with parameter $\lambda$. So far, for the mean, I have the following: $E[S_{N}] = E[E[S_{N}\mid N=n]]$ $$ = ...
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1answer
18 views

Do all Stochastic matrix have a stationary probability vector?

I know that a stochastic matrix will have 1 as one of its eigenvalues. But do the stochastic matrices all have a stationary probability vector? Basically, could there be a case where the eigen ...
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1answer
28 views

A credit model. Default time.

In a paper, I find the following situation: Let $(\Omega,\mathcal{G},\mathbb{Q})$ be a probability space. $\mathbb{Q}$ is supposed to be a risk neutral measure. Suppose that ...
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19 views

a conceptual question on markov chain [duplicate]

Suppose $\{X_n,n\ge 0\}$ and $\{Y_n,n\ge0\}$ are two independent discrete-time markov chains (DTMC) with state space $S=\{0,1,2,\ldots\}$. Prove or give a counterexample to: $\{X_n+Y_n,n\ge 0\}$ is ...
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1answer
599 views

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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31 views

Showing that $(X_n)$ obeys the Markov Property. [closed]

Consider a process $(X_n)_{n\geq0}$ where we define $X_0 = 0$ and for $n \geq 1$: $$X_n = X_{n-1} + Z_n$$ where $Z_n$ for $n \geq 1$ are independent random variables on $\{ -1, 1 \}$ with ...
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1answer
88 views

How can I solve this stochastic system of equation?

$(B_1(t),B_2(t))$ is a 2-dimensional standard Brownian motion. $\alpha , \beta$ are constant. The system of equations is: $$dX_1(t)=X_2(t)dt+\alpha dB_1(t)\\dX_2(t)=-X_1(t)dt+\alpha dB_2(t)$$ I tried ...
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23 views

How to obtain accumulated counts of past events by time $t$?

Given $f: [0, \infty) \to \{0,1\}$, $f(t)$ represents whether there is an event occurring at time $t$. How can we obtain $g: [0,\infty) \to \mathbb{N}_0$ so that $g(t)$ represents the number of ...
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22 views

What is the interpretation of $\nu(dy - x)$ where $\nu$ is a Lévy measure?

In a paper I am reading, it is seemingly suggested that, if $\nu(dx)$ is a Lévy measure, then the following holds for a function $f(x)$ which is smooth (and satsifies some integrability conditions): ...
2
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2answers
163 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
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19 views

Compute distribution in Hidden Markov models

Let $Z_1, Z_2, ..., Z_n$ be the latent variables, and $X_1, X_2, ... X_n$ be the observed ones in a hidden markov models. Let's assume that the parameters of the hidden Markov models are known: the ...
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0answers
17 views

Density of intersection of sets with boundary condition

I would like to prove that $$E:=\bigcap_{n\geq 1} \left\{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f\left(\frac{k}{n}\right)g_n (k)\right\}$$ is a dense subset of: $$F:=\left\{f\in C^2 (\mathbb{R}) : ...
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1answer
38 views

hitting times and stopping times

stopping times are always hitting times, but not the other way around. As an example of this, Last exit times are not stopping times as they depend on future information. the last exit time of $A$: ...
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1answer
21 views

Weak convergence for composition of cadlag stochastic processes

Let $(X^n_t)_{t \geq 0}$ be a sequence of cadlag stochastic processes, that is $X^n$ is a random element in the Skorokhod space $D([0, \infty), \mathbb{R}$) for each $n \in \mathbb{N}$. Also for each ...
2
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1answer
32 views

Probability that a stochastic process is below a special random level

Given a stochastic process $x(t)$ over time $t \in [0,T]$, and a given (deterministic) $\tau$, where $0<\tau<T$, define a random variable $x^{*}$ as $$ x^{*} \triangleq \inf\bigg\{y: ...
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1answer
20 views

Predictability of $\int^t_0 f(X_s)\,\mathrm ds$ where $X$ is a Lévy process

Let $X_t$ be a Lévy process and $f(x)$ some smooth function. Under what conditions is $$ Y_t = \int^t_0 f(X_s)\,\mathrm ds$$ predictable? Not sure how to investigate this. It is clearly adapted, so ...
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14 views

Lognormal approximation of the sum of successive values of a lognormal process

I would like to use a lognormal process to approximate the successive values of another lognromal process. Let $X_t$ be a lognormal process. I would like to approximate $$ Y_t := \sum_{t=0}^T X_t $$ ...
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1answer
17 views

Find $P(\eta_t=m)$, $m=0,1,2,\dots,$

Let $\epsilon_t$, $t=1,2,\dots$ independent random variables with $P(\epsilon_t=1)=p$ and $P(\epsilon_t=-1)=1-p$. If $\eta_0=0,\eta_t=\eta_{t-1}+\epsilon_t$ , $t=1,2,\dots$ where $\eta_t$ is ...
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1answer
28 views

Is there a way to estimate moments of strong solution to SDE

Suppose the SDE $$\mathsf dX_t =b(t,X_t)\mathsf dt + \sigma(t,X_t)\mathsf dW_t,\; X_0 = x$$ where $t\in[0,T]$ has a strong solution. I know in general we can't find an explicit formula for the ...
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1answer
41 views

Understanding Quadratic Variation

I think part of the trouble a lot of people (or at least me personally) have with making the jump from calculus to stochastic calculus is the notion of quadratic variation. It doesn't have as much ...
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13 views

Stopping of quadratic covariaton

I am given two local martingales $M$ and $N$ and a stopping time $\tau$. We work on a finite time interval $[0,T]$. I want to prove $$\langle M,N\rangle^{\tau}=\langle M^\tau,N\rangle$$ using the ...
2
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0answers
48 views

one dimensional SDE with zero drift

I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, ...
2
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1answer
171 views

Density of subspace with nonlocal/Wentzell boundary condition

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by ...
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1answer
17 views

Expected value of distance between independent Brownian motions

Suppose $\{W^{(1)}_t, t\geq 0\}$ and {$W^{(2)}_t, t\geq 0\}$ are two independent Brownian motions. If I recall correctly, the distance between the two at a given time has the following property: ...