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

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13 views

Where does right continuity of the filtration play a role in the poofs of the classical results of stochastic calculus?

Suppose that I define processes in continuous time as mesurable applications from $\Omega$ to the space of continuous functions (with Borel $\sigma$-algebra under the supremum norm), does the right ...
3
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0answers
19 views

Mean value of stochastic process with random variable as the index

Let $\{X_t: t\geq 0\}$ be a stochastic process on the probability space $(\Omega,\mathcal{A},\mathbb{P})$ with values in $\mathbb{R}$ and $T:(\Omega,\mathcal{A},\mathbb{P})\rightarrow\mathbb{R}^{+}_0$ ...
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0answers
46 views

hitting time for a continuous time markov chain

Brad’s relationship with his girl friend Angelina changes between Amorous, Bickering, Confusion, and Depression according to the following transition rates when t is the time in months. They are ...
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0answers
23 views

Black Scholes partial differential equation; Derivation

I have an exam tomorrow and the issue is, my notes just really briefly mentions it. It doesn't even take a full 2 pages to mention the partial differential equation. I haven't even seen it in ...
3
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2answers
57 views

Ito integral for Brownian motion

I know that because $W_t$ is a martingale, $$E\left[\int_{0}^{T} W_t dW_t\right] = 0$$ then what should the value for this equation be: $$E\left[\int_{0}^{T} W_t^{n}dW_t\right]?$$ $n$ is the power of ...
6
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1answer
75 views

Suppose $dX_t = a(X_t) dt + b(X_t) dW_t$ and $Y_s=X_t$ where $s=t^2$. What SDE does $Y_s$ satisfy in the weak sense?

Suppose $dX_t = a(X_t) dt + b(X_t) dW_t$ and $Y_s=X_t$ where $s=t^2$. What SDE does $Y_s$ satisfy in the weak sense? Hint: calculate $E[ dY | \mathcal{F}_s]$ where $dY = Y_{s-ds} - Y_s$. This is ...
1
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1answer
13 views

Ito's formula; when to use one and when to use the other form

I have seen $2$ "forms" of the Ito formula which are essentially, in the end, equivalent. But my question is, having seen quite a few questions on stochastic differential equations, I am wondering ...
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0answers
18 views

Stochastic calculus for continuous time Markov chains

I have absolved a course on stochastic analysis, i.e. integrals with respect to the brownian motion. Now I know that there is a theory of stochastic calculus for diskrete matringales, however I was ...
1
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1answer
35 views

Check solution to the SDE $dX_t = - \mu X_t \, dt+ \sigma \, dW_t$

I get stuck in this problem. I just can't get the hang of how we need to "guess" a function first and almost everything along the process of solving depends on it; It's not entirely logical to me when ...
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0answers
14 views

Problems with finding marginal density from joint density function

For two absolute continuos stochastic variables I have that the joint density function is 8y if 0 I now have to calculate/ show what the marginal density functions are. I got the right answer for y ...
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1answer
29 views

Probability of exponential of brownian motion

$W_t$ is a brownian motion, I have this exponential value: $$v(t)= e^{0.00025 + 0.3W_t}$$ what's the probability that $v(1)<0.5$? By taking natural log on both size, I got $0.00025 + 0.3W_1 < ...
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0answers
13 views

geometric brownian motion start with 0

Say I have an asset that has price 0 today, and $1 at 1 year. Assume it follows GBM, what should the price be? Is the following statement correct? If the asset price follows brownian motion, the ...
1
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1answer
11 views

Find E(X^-1) for stochastic variable

Let $X$ be a stochastic variable with density function: $f(x)=x\exp(-x)$ if $x>0$ and $0$ otherwise. Show that $E(X^{-1} )=1$. I believe I have to integrate but is it simple $x\exp(-x)$ I ...
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0answers
16 views

Markov chain jump

It follows from applying the Markov property that if we start in some point $x \in S$ ($S$ is assumed to be finite here) where $(X_t)_{t \ge 0}$ is a Markov chain that the stopping time ...
1
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2answers
43 views

Compute expectation of the cube of stochastic integral

I want to compute: $$\mathbb{E} \left( \left(\int_0^tudW_u \right)^3 \mid \mathcal{F_s} \right),$$ hence I write $$\int_0^t \text{as} \int_0^s + \int_s^t.$$ Then I need to compute: $$\mathbb{E} ...
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0answers
37 views

Difficult problem on Markov chains

Assume that we have a continuous time Markov chain $(X_t)$ on $\{0,1\}$ and $f(t):=P(X(l)=0 \text{ for all } l\in [0,t]|X(0)=0),$ then I want to show that under the assumption that $f'(0)$ exists, we ...
0
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0answers
29 views

Finding the Levy measure

I am struggling with the derivation of the Lévy-measure of a Gamma-process $X_t$ with law $p_t(x)= \frac{\lambda^{ct}}{\Gamma(ct)}x^{ct-1}e^{-\lambda x}1_{\lbrace x>0 \rbrace }$. The paper I am ...
1
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1answer
22 views

Check process is a martingale

I have such stochastic process with which I struggle all day, finally I found 2 mistakes, however answer is still unsatisfying. $$X_t = atW_t^2 - \int_0^t(W_s^2+s)ds,$$ I need to check if it is a ...
2
votes
1answer
16 views

Ito's formula but not given $\mu$ and $\sigma$

I have a little question from one of my worksheets(the solution I was given was almost not even a solution, super brief). let $f(t,x)=t\cos(x)$. Use Ito's formula to calculate $df(t,W_t)$. Well, ...
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0answers
31 views

Integration wrt BM

How do I integrate: $\int_{\mathbb{R}} (S_t - K)^+ \phi(t) dt$ where $\phi$ is a normal density and $S_t$ is a geometric brownian motion? I know my answer should be $\Phi(d_1)$, where $\Phi$ is the ...
0
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1answer
38 views

Calculating expectation using martingales

Could anyone help me with this exercise or show me similiar example? Any help appreciated. Using the martingales $M_t^\lambda=\exp(\lambda W_t-\lambda^2t/2)$ and ...
0
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0answers
30 views

Most recent jump probability

Assume I have two independent Poisson processes with respective parameters$$ \sim\text{Poisson}(\alpha_1),\sim \text{Poisson}(\alpha_2)$$ that I observe over a time interval $[0,t].$ What is the ...
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0answers
16 views

What process does this SDE weakly converge to?

So my question is motivated by the following: Note that the ODE $$ dy_t = 2sgn(y_t)\sqrt{|y_t|}$$ $$y_0 = 0$$$$ has no unique solution. However, consider the SDE as follows: $$ dy_t = ...
1
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1answer
37 views

Conditional expectation w.r.t Lebesgue measure

Consider the probability space $(\Omega, \mathcal{F}, \mathbb{P})=((0,1)^{2},\mathcal{B}((0,1)^{2}),\lambda_{2})$, where $\lambda_{2}$ is the Lebesgue measure in $\Omega=(0,1)^{2}$. Then, for ...
2
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0answers
30 views

Versions of Tanaka's SDE

Consider the following versions: $$dX_t=x_0+sgn(X_t)dW_t \tag1$$ $$dX_t=x_0+1_{(0,+\infty)}(X_t)dW_t \tag2$$ $$dX_t=x_0+1_{(-\infty,0]}(X_t)dW_t \tag3$$ SDE (1) is a classical example of SDE with ...
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0answers
21 views

Malliavan Derivative of a Geometric Brownian Motion

I'm trying to understand a proof that requires Malliavan Calculus, but have no experience with the topic. My question revolves around showing that the Malliavan derivative of a geometric brownian ...
2
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0answers
54 views

Markov process and filtration

I would like to restate the question. I'm reading Revuz/Yor's definition of Markov process (P81), they started from transition function, and define the $P_t f(x)$ as usual (let's only consider the ...
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0answers
28 views

Variance of a simple Ito integral

I am trying to apply Ito's lemma to compute variance of the following integral $X(t) = \int_{0}^t W(s)dW(s),$ where $W(t)$ is a Wienner process. Could you please check my calculations? $$E(X(t)) = ...
1
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1answer
43 views

random walk and calculating the probability of paths

Consider a random walk $(X_n)_{n≥0}$ with $p = 0.7$, starting from $X_0 = 3$. Find the probability that $X_{10} = 5$, but $X_n ≥ 1$ for $n = 0, . . . , 10.$. Essentially what I got from the ...
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0answers
23 views

Critical point of contact model

Let $d\geq 2$ and let $\Pi:\mathbb{Z}^d \rightarrow \mathbb{Z}$ be given by $$\Pi(x_1, x_2, ...,x_d)=\sum_{i=1}^d x_i$$ Let $(A_t:t\geq 0)$ denote a contact process on $\mathbb{Z}^d$ with parameter ...
1
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0answers
25 views

differential stochastic equation and solutions

Let E(K):$Z_0=0$ and $dZ_t=K_t(B_t-Z_t)dt+\alpha K_td \beta_t$ with B and $\beta$ two brownian motions, $\alpha>0$ and K a continue function. 1)Show that E(K) has a solution Z with $E(\int_0^1 ...
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0answers
25 views

very simple clarification needed about ito's lemma

I'm looking at this example in this lecture note (page 3, number iv): Let $f(t,x)=t^2+x^2$, and $X_{t}=\mu t + \sigma B_{t}$, where $B_{t}$ is the standard brownian motion. We want to compute ...
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0answers
18 views

Left limit poisson process (stochastic analysis)

Let $N_t$ denote a Poisson process with intensity λ > 0, and let $M_t = N_t − λt$ be the compensated martingale of N . How could I show that $\int_{0}^{t} N_{s-} dN_s =1/2 (N_t^2-N_t)$ Thank you
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33 views

Martingale (stochastic analysis)

Let $N_t$ denote a Poisson process with intensity λ > 0, and let $M_t = N_t − λt$ be the compensated martingale of N . I want to verify that the process Y given by $Y_t = \int_{0}^{t} N_{s-} dM_s$ is ...
0
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0answers
12 views

Relation between $L^2$-equivalence and indistinguishability

Let $\{X_t\}_{t\in[0,T]}$ and $\{Y_t\}_{t\in[0,T]}$ be two stochastic processes defined on some probability space $(\Omega,\mathfrak{A},\mathbb{P})$. Assume that $X$ and $Y$ are $L^2$-equivalent, i.e. ...
1
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1answer
54 views

Show that stochastic process solves certain SDE

1) I want to know the mechanism how to: show that the process $X_t$ solves this SDE 2) know if my friends though, mine though below are correct/incorrect. I have the general linear stochastic ...
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0answers
22 views

Guess quadratic variation of 2 processes, with same/different BM.

I have 2 processes with stochastic parts R, S. $$dR = \mu_1 dt + \sigma_1 dW_{t1}$$ $$dS = \mu_2 dt + \sigma_2 dW_{t2}$$ I am trying to show what precisely quadratic variation $[R,S]$ means for 2 ...
0
votes
2answers
42 views

Solution of an SDE

Can someone guide me on how to solve the SDE \begin{equation} dX_{t} = \gamma(a-\beta X_{t})dt + \delta X_{t}dW_{t} \end{equation} where $a,\beta,\gamma,\delta$ are all positive constants? I tried ...
5
votes
1answer
97 views

Help integrating the transition probability of the Brownian Motion density function.

1. Problem: Given the Brownian Motion with Drift: $$ dx = \mu \, dt+\sigma \, dW $$ It can be shown that the transition density function is the following: $$ p(x, t) = \frac{e^{-\frac{(x-\mu ...
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0answers
25 views

Kullback–Leibler Divergence of between p(x) and self convolved version

Calculate the Kullback–Leibler Divergence between a pdf $f(x)$ and its $n^{th}$ convolution power $g(x)= \underbrace{f * f * f * \cdots * f * f}_n$. I have solved this problem for large $n$ ...
6
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0answers
75 views

Higher math and statistics/probability

So I've heard that certain areas of statistics and probability use manifolds and results from analysis and topology. Given that I lack the background to see where manifolds would become useful in ...
5
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0answers
65 views

Ornstein-Uhlenbeck SDE solution

I'm following this solution of $$dX_t=\kappa(\theta-X_t)\,dt+\sigma\,dW_t \tag1 $$ And the question is whether its solution $$X_t=\theta+e^{-\kappa(t-s)}(X_s-\theta)+\sigma\int_s^t ...
1
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1answer
22 views

Quadratic variation of a Poisson process where time depends on a function F

Let $(N_t)_{t \geq 0}$ be a Poisson process of rate $1$. Then the quadratic variation of $N$ is itself (so $\langle N\rangle_t=N_t$). For a suitable function $f$, what would be the quadratic variation ...
3
votes
1answer
26 views

Application of the Clark-Ocone's Formula to $\mathbb{1}_{S_t > K}$

At page 291 of Nonlinear Option Pricing by Julien Guyon and Pierre Henry-Labordère, the Clark-Ocone's Formula is applied to $\mathbb{1}_{S_t > K}$. I do not get how to get from the second to the ...
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0answers
23 views

Is $\tau(\omega )= \infty \forall \omega \in \Omega$ a stopping time?

My guess is yes since $\{\infty \leq t \}=\phi \in \mathcal{F}_t, \forall t \geq 0$ where $\phi$ is the empty set which is always in the sigma algebra. Am I right?
1
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0answers
16 views

Superposition of two Renewal Processes

Assume we have a renewal process $P_1$ with inter-renewal distribution $p_1(x)$ and rate $\lambda_1 = \lim_{t \to \infty} \frac{N_1(t)}{t}$, where $N_1(t)$ is the total number of renewals of $P_1$ ...
1
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1answer
21 views

Increasing the Rate of a Renewal Process

This problem is a dual question of "Splitting a renewal process". Assume we have a renewal process $P_1$ with inter-renewal distribution $p_1(x)$ and rate $\lambda_1 = \lim_{t \to \infty} ...
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0answers
20 views

Independence of two random variables 2

Assume we put points on the two dimensional $x-y$ plane according to Poisson distribution, consider $r_1$ is the location of the closest point to origin and $r_2$ is the location of the second ...
4
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1answer
54 views

Intuition about Skorohod integral

I'm teaching myself Malliavin calculus and Skorohod integrals and with this kind of math I find myself following the logic through but lacking solid intuition about what is going on. In particular ...
0
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0answers
15 views

Does the law of an Itô process $X_t$ at each time has a density? [duplicate]

Let $X$ be a the strong solution of the SDE on $\mathbb{R}$ $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ $$X_0 = x$$ where $b,\sigma \colon \mathbb{R} \to \mathbb{R}$ are Lipschitz functions. Dose the law ...