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

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37 views

Calculate the variance of $\int_{t-m}^t \int_{s-m}^se^{-k(s-u)}dW(u)ds$

I need to calculate the variance of this double stochastic integration: $$ I=\int_{t-m}^t \int_{s-m}^se^{-k(s-u)}dW(u)ds $$ Note that variable $s$ is in the integrand as well as the ...
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16 views

Integrating R.V with respect to time

I would like to compute the time integral of a random variable $X(t)$ given by $\int_s^t X(u) e^{k u} du$ where $X(t)$ is a CIR square root process $dX_t = k (\theta - X_ t) dt + \sigma \sqrt{X_t} ...
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44 views

$X,Y$ independent from sigma algebra $\Sigma$ then also $X+Y$?

Let $X,Y$ be independent from a sigma algebra $\Sigma$. Does this mean that $X+Y$ is independent from $\Sigma$? I just don't know how to show it, but maybe a yes/no answer and a good hint could help ...
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16 views

Find $f_n \to 0$ such that $\phi(f_n) \nrightarrow 0$ (an exercise on boundary conditions)

The claim we are asked to prove in Petr Mandl's boook (An analytical treatment of one-dimensional Markov chains) in exercise 3 of chapter 3 (pg 49) is Condition (41) and theorem 3 are: We ...
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19 views

Is $\int_s^t {B_u}^2 du$ independent of $ F_s$

In order to prove that $(B_t^4)_{t \in \mathbb{R+}}$ is a continuous semi-martingale I need to compute. $E(\int_s^t B_u^2 du |F_s)$ I was first thinking that $\int_s^t B_u^2 du$ independent of $ ...
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19 views

Transition density for reverse stochastic process

Consider an Ito process $$dX_t = a(t,X_t)dt + b(t,X_t)dW_t.$$ Assume that the functions for drift and diffusion, $a$ and $b$ are continuous and differentiable. Also assume that we know the transition ...
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1answer
90 views

Solving StochasticDifferential Equation

Please help me in solving this Stochastic Differential Equation for $Y_t$ $$dY_t = a Y_t dt+ b dX_t \qquad Y(0) = c $$ where $a$ and $b$ are constants. Also find the $\mathbb{E}[Y_t]$ and ...
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28 views

Construction of the Itō integral with (local) martingales as integrators

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$. $\xi_i$ be a real-valued random variable on ...
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0answers
32 views

How to apply Ito's Lemma to this problem?

We have the martingale representation theorem: $G = E[G]+\int_0^t\theta_sdW_s$ Now, given $G=1_A$, where $A=\{exp(W_t)>K\}$, how to find the corresponding $\theta_t$? The hint I received was to ...
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62 views

Expectation of a strongly anticipating Ito like integral

I am using some Ito stochastic differential equations (SDEs) of the form $$dx=A(x)dt+B(x)dW$$ where $dW$ is an increment in a Wiener process and the above are to be interpreted as stochastic ...
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26 views

Integrating random variables correctly

Let $X,Y : \Omega \rightarrow \mathbb{R}$ be two random variables on a probability space $(\Omega , \Sigma, P)$ The expectation value of $f(X,Y)$ is then given by $\int_{\Omega} f(X( \omega) , ...
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0answers
25 views

Proof of the stochastic Fubini's theorem

I am trying to prove the Stochastic Fubini's theorem which is an exercise of An Introduction to Stochastic Calculus Applied to Finance. Let $(W_t)_{t\in[0,T]}$ be a Brownian motion and $H(t,s)$ has ...
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22 views

Assumptions of Exponential Family with possibly a Counterexample

In U.Küchler "Exponential Families of Stochastic Processes" 1997 [p.19-20] one consider a class of probability measures $P:=\{P_{\theta}:\theta \in \Theta\subset \mathbb{R}^{k}\}$ on ...
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0answers
38 views

Bessel Process and Brownian motion

Let $\beta_s$ be a Bessel process, i.e. the positive solution to the SDE $$\beta_s = B_s + (n-1) \int_0^t \frac{1}{\beta_s} \mathrm{d}s,$$ where $B_s$ is a one-dimensional Brownian motion. Let $U_s$ ...
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1answer
25 views

Integrating w.r.t. the pushforward measure

Let $X,Y : \Omega \rightarrow \mathbb{R}$ be independent r.v.'s and $f$ continuous. Then $A \subset \Omega$ $\int_{A} f(X,Y) dP = \int_{(X,Y)(A)} f(z) dP_{X,Y}(z) = \int_{(X,Y)(A)} f(x,y) dP_X(x) ...
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24 views

Stochastic integral and weak integral

Can the stochastic (Skorokhod) integral be seen as a special case of the weak of Pettis integral with the Banach space which win integrate into chosen appropriately?
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9 views

Is the value of the discrete Green's function on a box independent of the position of the points within the box?

I currently contemplate over the discrete Green's function on a box and am trying to gain an intuition for its behaviour. Consider the box $B := \{-N,\cdots,-1,0,1,\cdots,N\}^2$ and let $(X_i)_{i \in ...
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25 views

Conditional expectation unique?

Let $A \subset B$ be a subalgebra. This means that $E(X|A)$ is also $B$ measurable. Now, if I can show that $E(X|B)$ is also $A$ measurable, does this imply that the conditional expectations are the ...
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1answer
141 views

Problem 3.24 of “Brownian Motion & Stochastic Processes” by Karatzas and Shreve - Submartingales and stopping times

I'm doing the problem 3.24 of Brownian Motion and Stochastic Processes by Karatzas and Shreve. There is two specific parts troubling me, I need some help to see what to do. Here is the problem: ...
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0answers
59 views

How to calculate analytical solution to fokker planck equation in finite domain?

I am looking for the solution to the following fokker planck equation. $\frac{\partial f(x,t)}{\partial t}= (k_1 x - v)\frac{\partial f(x,t)}{\partial x} + (k_2 x + D) \frac{\partial^2 ...
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34 views

How does the sample space remain constant in filtered

I am trying to understand the concept of filtered event space from the axiomatic probability. From my reference (lecture script by Ramon Handel, Princeton) the filtered probability space looks like ...
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0answers
9 views

Principal Component Analysis in a stable framework

are you familiar with stable distributions. It is denoted by $S_{\alpha}(\sigma,\beta,\mu)$ where $\alpha$ is the tail index, $\beta$ is the skewness, and $\sigma$ and $\mu$ are the location and scale ...
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1answer
65 views

Completeness of Probability Distribution as a Measure.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a complete probability space and $X:\Omega \to \mathbb{R}^n$ be a $\mathcal{F}$-measurable function, i.e., it is a random variable. Then, in the book ...
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93 views

Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
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38 views

Symmetric random walk about $y=2$.

Consider a simple (symmetric) random walk $p=q=\frac{1}{2}$ and $(X_n)_{n\geq 0}$ with $X_0 = 0$. Using the reflection principle, find the probability that $X_{12} = -4$ and $X_1 < 2$, $X_2 < ...
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0answers
16 views

Distribution of a Brownian bridge

I am self studying some stochastic calculus material and come across this question to show that the distribution of $P(W(s)\in dy|W(t)=x)$ with $W(0)=0$ is normal with mean $\frac{s}{t}W(t)$ and ...
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1answer
63 views

Continuous path of stochastic processes

In my probability book a stochastic process is defined as a measurable map $X: \Omega \rightarrow S^T,$ where $S^T$ is equipped with the sigma algebra of cylinder events. Our professor mentioned that ...
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17 views

Ito's Lemma $\ln(S_T / S_t)$

Assume that $dS = \mu d_t + \sigma d_z$, where $dz \sim N(0,\delta t)$ When I apply Ito's lemma to $\ln(S)$ I obtain that $d\ln(S) = [\mu-\frac{1}{2}\sigma^2]dt + \sigma d_z$ Now I thought that ...
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1answer
54 views

Prove that the first hitting time $\tau_x:=\inf\left\{t\ge 0:B_t=x\right\}$ of a Brownian motion is almost surely finite

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion and $$\tau:=\inf\big\{t\ge 0:B_t\in\left\{a,b\right\}\big\}$$ for some $a<0<b$. I want to prove, that $\tau$ is almost surely finite. Let ...
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0answers
11 views

Can we find the boundary condition of a function of diffusion process?

given $dx_t=b(t,x_t)dt+\sigma(t,x_t)dW_t$, if it is in one dimensional case, one can use Feller non-explosion test to see if $x_t$ attains a particular boundary. How about $f(x_t)$, $f$ is any ...
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1answer
56 views

Brownian motion / ito's formula

Little help is needed Can I use geometric Brownian motion here? The question I get: $Let z=(z_t)$ be a one-dimensional standard Brownian motion and define the process $y = ( y_t )$ by $y_t = z_t^2 − ...
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1answer
42 views

Obtaining martingales from Poisson process

All processes here are continuous. Suppose we have a Poisson process $(N_t)_{t\geq 0}$ with parameter $\lambda > 0$ and adapted to the filtration $(\mathcal{F}_t)_{t\geq 0}$. Fix $u\in\mathbb{C}$, ...
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1answer
58 views

Karatzas and Shreve - Problem 3.3.19

I'm struggling here to solve this problem, but with no success. I was able to prove a $\implies$ b, but the next implication is troubling me. In the book, they give a solution, but I think there ...
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1answer
72 views

Do I need topology to study stochastic process?

So far I dealt with probability from a very intuitive point of view, like guessing frequencies etc. But while studying stochastic process (particularly with application to finance), I came across ...
3
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1answer
19 views

Uniform Integrability and Supermartingal Process

Suppose $\{X_t, \mathcal{F}_t: 0\leq t<\infty\}$ is a right-continuous and nonnegative supermartingal. I want to show that $\lim_{t\to\infty}X_t(\omega)$ exists a.e. and that $\{X_t, \mathcal{F}_t: ...
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0answers
31 views

Correspondence between multi-dimensional Brownian motion and harmonic functions

Let $U\subseteq\mathbb R^d$ be a bounded domain. A continuous function $u:\overline U\to\mathbb R$ is called harmonic $:\Leftrightarrow$ $$u(x)=\frac 1{|\partial B_r(x)|}\int_{\partial B_r(x)} ...
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19 views

Please can you give me an example of a stochastic process that is a Markov process but not a martingale? [duplicate]

Please can you give me an example of a stochastic process that is a Markov process but not a martingale ?
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27 views

Covariance of two stochastic integrals

Consider the stochastic integral $\int_{0}^{1}J(r)M(r,\lambda) dr$ where $J(r)$ is a demeaned Ornstein-Uhlenbeck process and $M(r,\lambda)=W(r,\lambda)-\lambda W(r,1)$ a Brownian Sheet, independent of ...
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1answer
41 views

Modified stochastic processes

I am looking for conditions such that a process $(X_t)_t$ where the $X_t$ are $\text{iid}$ such that there is a process $(Y_t)_t$ satisfying $P(X_t=Y_t)=1$ and $t \mapsto Y_t(\omega) \text{ is ...
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1answer
75 views

Borel $\sigma$-algebra of continuous functions

Let $B(C(T,\mathbb{R}))$ be the Borel sigma algebra of continuous functions mapping from the compact metric space $T$ to $S$ defined by the canonical metric $\|\cdot\|_\infty.$ Now I was wondering ...
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21 views

Condition for covariance of Ornstein-Uhlenbeck processes

I am considering this Ornstein-Uhlenbeck process: $$X_t=e^{-at}X_0+e^{-at}\int_{0}^{t}e^{as}\mathrm{d}W_s$$ in which, $a$ is a constant, and $W_s$ is a standard Brownian motion. Expected value of ...
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58 views

Product of two Ornstein Uhlenbeck processes : conditional distribution

Let $X(t)$ and $Y(t)$ be two independent OU processes (each with some fixed correlation time-scale), and let $S(t) = X(t)Y(t)$. Then is there a expression for the conditional distribution of $X(t)$ ...
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1answer
29 views

Why for a continous local martingale ,on an enlarged probability space, it possibly holds that $M_t=B_{\langle M \rangle _t}$

Why for a continous local martingale $(M_t)_{t\in \mathbb{R}_+}$ ,on an enlarged probability space, it possibly holds that $M_t=B_{\langle M \rangle _t}$ where $(B_u, u \geq 0)$ is Brownian motion . ...
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1answer
27 views

predictable process in continuous times.

If $X_t$ is a stochastic process in $(\Omega,\mathcal F,\mathcal F_t)$ which satisfy usual conditions. Can we get $X_t$ is a predictable process if $X_t$ is adapted to $\mathcal F_{t-}$? since ...
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1answer
89 views

Using Fubini's Theorem in Stochastic Calculus

In basic calculus: 'Fubini's theorem' allows us to switch order of integration in double integrals without changing the bounds provided we are integrating over a rectangle. From here: If the area ...
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2answers
45 views

How to obtain this result using Ito's Lemma?

My book writes that when: $dx = a(x,t)dt + b(x,t)dz$ $x' := x + dx$ Then using Ito's lemma: $E[F[x+ \Delta x, t + \Delta t \mid x)] = F(x,t) + [F_t(x,t) + a F_x + \frac{1}{2} b^2 F_{xx}] ...
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1answer
14 views

Proving linearity of a subspace of $L^2$

I'm reading p.79 of Steele s Stochastic Calculus and Financial Applications. It defines a space of functions $\mathcal{H}^2$ as the space of all measurable adapted functions such that ...
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0answers
40 views

Why isn't this stochastic integral trivial?

I have a stopping time $\tau$ and a stochastic process $f$. Then the following equation is true: \begin{equation} \int^{t\wedge\tau}_{0}f(s)dW(s)=\int^{t}_{0}f(s)\chi_{[0,\tau]}(s)dW(s) ...
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18 views

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$

Let $f \in M^{2}_{\omega} [\alpha, \beta]$, then, $E\{\int_{\alpha}^{\beta}f(t)d\omega (t)|\mathscr{F}_\alpha \}=0$ and $E\{\mid \int_{\alpha}^{\beta}f(t)d\omega (t)\mid^2|\mathscr{F}_\alpha ...
0
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0answers
26 views

Solving a Stochastic PDE with two variables in time

I am trying to work on exercise 5.13 in the book Arbitrage Theory in Continuous time by Thomas Bjork. The equation to solve is; \begin{eqnarray*} \frac{\partial F}{\partial t} (t,x,y) + \frac{1}{2} ...