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

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92 views

Representing a stochastic integral as product of a unknown random variable and a standard normal random variable

Consider a probability space $(\Omega,\mathcal F, (\mathcal F_t)_{t\geq0},\mathbb P)$ where $\mathbb F=(\mathcal F_t)_{t\geq0}$ is generated by $B=(B_t)_ { t \geq 0}$ a standard brownian motion ...
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1answer
95 views

Karatzas and Shreve Problem 3.3.38

Let $X$ be a continuous process and $A$ a continuous, increasing process with $X_0 = A_0 = 0$, a.s. Suppose that for every $\theta \in \mathbb{R}$, the process $$Z_t^{\theta} = ...
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1answer
87 views

Exercise 3.3.25 of Karatzas and Shreve

This is the Exercise 3.25 of Karatzas and Shreve on page 163 Whith $W=\{W_t, \mathcal F_t; 0\leq t<\infty\}$ a standard, one-dimensional Brownian motion and $X$ a measurable, adapted process ...
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0answers
51 views

Time series (stochastic process) estimating parameters using characteristic function

I have a time series of assets ${A_1, A_2, ..., A_n}$, which is described by a sophisticated distribution having the following characteristic function: $\phi(u; t;\theta)$, where $\theta$ is a vector ...
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0answers
29 views

New stochastic calculus

I am interested in Kagi and Renko approach and hope I can use it for a random walk process. I searched for it on internet but I couldnt find any basic material to read about it. Can someone please ...
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2answers
61 views

Ito's Isometry for three factors

Ito's Isometry states the following: If $\{W_t\}_{t\ge0}$ is a Brownian motion and $\{\phi_t\}_{t\ge0},\{\psi_t\}_{t\ge0}$ are two non-anticipative piecewise-continous processes with $\mathbb ...
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1answer
121 views

Kolmogorov Backward Equation for Itô diffusion

Let $(X_t)_{t\ge 0}$ be the solution of the SDE $$ X_t = X_0 + \int_0^t \mu(s,X_s) \,ds + \int_0^t \sigma(s,X_s) \,dB_s, \quad t\ge 0 $$ where $\mu(s,x)$ and $\sigma(s,x) $ are Lipschitz continuous ...
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42 views

Is the stochastic integral a Gaussian process [duplicate]

Let $Y_t=\int_0^tW_s^2 d W_s$. It is a martinglae with $\langle Y\rangle_t=t^3$, and then, by computing CF using Ito's lemma, \begin{align*} d e^{i\theta Y_t}&=i\theta e^{i\theta ...
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1answer
98 views

Background for studying and understanding Stochastic differential equations

Assume I have back ground of the following knowledge based on the textbook as : ODE : ODE by Tenenbaum Baby probability : Ross 's baby probability Baby real anlysis : Bartle's introduction to real ...
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2answers
65 views

solving a stochastic differential equation

How to solve $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$ together with the initial condition $X(0) = X_0$.
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44 views

Stochastic control problem

Suppose we have the following stochastic optimal control probelm \begin{equation} V(t,x) = \sup_{u} \mathbb{E}[ g(X_{T}) +\int_{0}^{T}f(t,X_{t},u_{t})dt] + (\mathbb{E}[ ...
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28 views

change of variable in black-scholes equation with dividend

In the black-scholes equation with dividend ${\textstyle{{\partial V} \over {\partial t}}} + (r-q)S{\textstyle{{\partial V} \over {\partial S}}} + {\textstyle{1 \over 2}}{\sigma ...
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2answers
66 views

Martingale Proofs

I havent been able to find an analogous question and our textbook is lacking in good examples, so I could use a little help with this rather straight forward martingale problem: Let X=(Xn) be a ...
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1answer
100 views

What is the probability a random walk hits x before it hits y?

This problem was motivated by my bitcoin trading and recalling some of my math education back in the day. I thought I'd ask people who know this much better than I... Suppose there is a continuous, ...
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0answers
37 views

Clarification on the definition of the îto integral

I have a question regarding the îto integral. In the definition of the integral we basically take the limit in probability of the sum $\Sigma H(t_i)\cdot(B(t_{i+1})-B(t_i))$ for suitable $H$ and a ...
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0answers
35 views

calculation on Ito's Lemma

I have a question on the calculation on Ito's Lemma: ${{Y}_{t}}={{t}^{{{W}_{t}}}}$ solve for $d{{Y}_{t}}$ the following is my solution [\begin{align} & d{{Y}_{t}}=\frac{\partial Y}{\partial ...
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1answer
125 views

proof of Feynman–Kac formula

the article given by wikipedia http://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula#Proof states at some point of the proof that: (line 7) ''the third term is o(dtdu) and can be dropped'' Can ...
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1answer
101 views

Inequality for Euclidean norm

Let:| | be Euclidean norm on $\mathbb{R}^{n}$ and $b : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n}$ and $\sigma : \mathbb{R}^{n}\longmapsto \mathbb{R}^{n\times m}$ two continuous functions. ...
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2answers
61 views

Variance of sum of two ito integrals

I don't really understand how to solve the following problem: Var(X) where X = $\int_0^2 2t dW(t) + \int_4^6 W(t) dW(t)$ If I use $E [(A+B)^2] = E(A^2) + E(B^2) + 2E(AB)$ I get to the point where I ...
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0answers
121 views

Essential supremum of a conditional expectation

Given the function \begin{equation} P(x,t) := \sup\limits_{t \le \tau \le T} E\left( g(X^{t,x}_{\tau}) \right) \end{equation} where $X^{t,x}$ is the unique solution to the SDE \begin{equation} X_u ...
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45 views

interchange stochastic and deterministic integration

If $f$ is a function in $L^2([0,1]^m)$, W is one-dimensional Brownian motion, $a,b \in [0,1]$, are the following two integrals equal? $$\int_0^1\int_0^{t_{m-1}}\cdots \int_0^{t_2} ...
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1answer
72 views

$\mathbb{E} \int_a^b W^3(t)\,dW(t)=?$

Is it true that $\mathbb{E} \int_a^b W^3(t)\,dW(t)=0$, for $a < b \in \mathbb{R}$ I know that for an adapted process $\Delta(t), t\geq 0$, the integral $\int_0^t \Delta(u)dW(u)$ is a ...
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1answer
36 views

Definition of a random variable in the context of a hypergeometric distribution

We defined a random variable in a probability space $(\Omega, E, P)$ as a map $X: \Omega \rightarrow \mathbb{R}$. Unfortunately, I somehow have the impression that this term "random variable is used ...
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0answers
63 views

Please rate this stochastic processes course. What should my next course be?

I am currently doing a course in Stochastic Processes that uses the book "Adventures in Stochastic Processes" by Sidney I. Resnick. The topics covered in the book are as follows: Discrete Index Sets/ ...
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2answers
63 views

How to show that $\mathbb{E}(\int_0^T t\mathrm \, dW_t) = 0 $?

I just want to know why $\mathbb{E}\left(\int_0^T t \,\mathrm dW_t\right)=0$. I know it's got something to do with the Gaussian distribution but I don't really know what.
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27 views

How small does $dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$ get?

Consider the stochastic differential equation $$dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$$ it is then well known that ($S$ dependence suppressed) the solution is $$S(t)=S(0)\exp(\int_0^t ...
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21 views

DE: $pU^2 = (rx-1)U' + Ur + \frac{1}{2}\sigma^2U''$?

I have been trying to solve the following ordinary differential equation that results from a problem in stochastic control theory. $U$ is a function of $x$. $pU^2 = (rx-1)U' + Ur + ...
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58 views

Joint probability distributions of variables satisfying a certain constraints

Here is my question, given a set of random variables - {x_i}, i=1,2, ...n. And the corresponding pdfs are given by {PDF_i}, i=1,2, ...n. Now if I were it impose a certain set of constraints on ...
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2answers
65 views

Expectation of absolute value of stationary time series

Let $Y_t$ be a stochastic process (time series). We consider stationarity as follows: $Y_t$ is said to stationary if the mean $\mu_t = \mathbb{E}(Y_t)$ is constant (given $\mathbb{E}|Y_t|<\infty$) ...
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44 views

Doob-Meyer Decomposition [closed]

I would like to find the Doob-Meyer Decomposition on the form $X[k] = X_{DM}[k] + E[k]$ for the binomial counting sequence $X[k]$. The binomial counting sequencer is given by $X[k] = ...
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1answer
129 views

Application of Optional Sampling Theorem

Lets assume that Brownian Motion starts from some point $x$ for which $a<x<b$ holds. Let $\tau=\inf\{t:B_t\not\in [a,b]\}$ be a stopping time. Now I want to prove that for $\theta>0$ ,an ...
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1answer
68 views

Changing Brownian motion and law

If I have some variable that depends on Brownian motion, how do I see clearly that replacing that Brownian motion with a different Brownian motion won't affect the law of my variable? To make this ...
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0answers
72 views

Expectation of $n$-dimensional Inverse Bessel Process

I think the main problem for me is to calculate the integral of $$\int_{0}^{\infty}\frac{e^{-\frac{r^2}{2t}}}{\sqrt{x^2+r^2}}r^{n-1}dr,n\geq2$$ For n=2, change of variable $y=\sqrt{x^2+r^2}$ would ...
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1answer
85 views

Expectation value quantum mechanics momentum operator

What is the random variable that belongs to the expectation value of momentum in quantum mechanics. Or in general: Is there any way we can define the expectation values that occur in quantum mechanics ...
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3answers
130 views

Distribution of stochastic integral

Assume that $\mathrm{d}S = \sigma \, \mathrm{d}W$ with initial level $S(0)$ and where $\mathrm{d}W$ is usual Brownian motion. Now $A(T) = \frac{1}{T} \int_0^T S(t) \, \mathrm{d}t$ . What is the ...
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1answer
153 views

How do I find the stochastic differential equation given the solution?

Homework Exercise. I've been given the question, find the stochastic differential equations satisfied by the following processes and determine which are martingales? B_t is a standard Brownian ...
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1answer
30 views

Probability Space and proof of existence for my specific problem involving stochastic differential equations

I have a question regarding the probability space for my problem. This deals with radiation therapy. If X(t) and Y(t) represent the number of two types of cancer cells. X(t) and Y(t) satisfy the ...
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1answer
44 views

Cadlag process integration

Let $A,B$ be non-decreasing cadlag processes such that $A_0 = B_0 = 0$ and limits $A_\infty = \lim_{t \to \infty} A_t$ and $B_\infty = \lim_{t \to \infty} B_t$ are finite. I am trying to prove that ...
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2answers
68 views
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1answer
124 views

determine Fisher information of $N(0,\sigma^{2})$ over $\sigma^{2}$

So far i've got that $I(\sigma^{2}) = E_{\sigma^{2}}[\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(X)]^{2}$. And i got that $\frac{\delta}{\delta \sigma^{2}}\log_{\sigma^{2}}(x) = ...
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1answer
24 views

How to show $F(x_1,\ldots,x_n) = \int_0^{x_1}\cdots\int_0^{x_n}\prod p(t_k,t_{k-1},y_{k-1},y_k)dy_1\cdots dy_n$?

Suppose $B_t$ a Brownian motion, and I fix $0=t_0<t_1<\cdots<t_n$. I want to show that $$P(B_{t_1} \leq x_1, \ldots, B_{t_n} \leq x_n) = F(x_1,\ldots,x_n) = ...
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1answer
150 views

Ornstein - Uhlenbeck Process

I'm considering the Ornstein - Uhlenbeck process $$ V_t = e^{\lambda t} v_o + \int_0^t e^{-\lambda (t-s)} dB_s $$ with $ \lambda > 0$, $v_0 \in \mathbb{R}$, and $B$ a brownian motion. I want to ...
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1answer
42 views

covariation of martingales

For a bivariate brownian motion $(B_t,W_t)_t$ we have that the covariation is given by $\langle W,B\rangle _t = t\rho$ where $\rho$ is the (constant) correlation between $W$ and $B$. Does this hold ...
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1answer
101 views

$L^1$ convergence of a sequence of stochastic integrals and convergence of their quadratic variations

On a filtered probability space $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$ containing a Brownian motion $W_t$. Let $\sigma^n_t$ be a sequence of square intergable adapted processes and consider: ...
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1answer
61 views

Weird equality of expectations involving stochastic integral

First of all, sry for the title. I just couldn't figure out any better description for this weird problem: Let $X$ be a bounded real r.v. and $(A_t)_{t\geq0}$ an increasing bounded process (hence ...
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1answer
32 views

Special adapted process

I want to construct a process $(Z_n)$ adpated to a filtration $\mathcal{F}_n$ such that $$E[Z_{n+1}\mid Z_n]=Z_n, E[Z_{n+1}\mid\mathcal{F}_n]\not=Z_n$$ I start by taking the three values $1,2,3$; ...
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1answer
72 views

Simple random walk two properties

First suppose that $X_1,...,X_t$ are IID random variables and $P(X_1=1)=p, P(X_1=-1)=1-p=q$ for $p\in (0,1)$, then $S_t=X_1+...+X_t, S_0=0$ is the simple random walk. I have two questions: (1) Why ...
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0answers
46 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 ...
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1answer
73 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
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1answer
79 views

Can an Itō integral be $\infty$?

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