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

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1answer
53 views

Itô integral with respect to a diffusion

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
3
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1answer
40 views

Why is the solution of a stochastic differential equation wrt the Brownian motion suitable for a model of a disturbed time continuous process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be a Brownian motion on ...
3
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1answer
22 views

Notation in stochastic integrals

There are some notation I don't understand: Given $W_t$, $n$-dimensional Brownian motion, and a smooth function $u:R^n\to R$ my book asserts: $$E^x\left[u(W_0)\right]=u(x)$$ What is the notation ...
4
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1answer
27 views

Itô symmetry for elementary predictable stochastic processes

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
1
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1answer
22 views

Prove that the Itô integral for elementary predictable processes builds a martingale

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathbb F=(\mathcal F)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$ $B=(B_t)_{t\ge 0}$ be an $\mathbb F$-adapted Brownian ...
1
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1answer
59 views

Optional stopping/sampling for right-continuous supermartingales

Let $\mathbb{F}$ be a filtration $(X_t)_{t\ge 0}$ be a right-continuous $\mathbb{F}$-supermartingale $\sigma,\tau$ be bounded $\mathbb{F}$-stopping times with $\sigma\le \tau$ and ...
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1answer
25 views

What is the general Ito formula for a function of two processes

If $f$ i twice differentiable scalar function and $X_t, Y_t$ are Ito processes then Ito lemma holds. But in 90% of sources I can only find the case, when $Y_t=t$ (it is deterministic function). The ...
4
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0answers
40 views

Why is a predictable stochastic process called *predictable*?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I$ be an index set $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be a stochastic ...
1
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1answer
20 views

Fubini's Theorem for Stochastic Integral

Probably a bit trivial, but I was curious about the validity of interchanging the following integrals (where $W_t$ is Brownian Motion): $\mathbb{E}[\int^{t}_{0} W^2_s ds] =? \int^{t}_{0} ...
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1answer
30 views

Proving technique used to show an equivalence to the definition of a Markov process

Let $X=(X_t)_{t\in I}$ be Markov process with values in a Polish space $E$. I want to show, that there exists a stochastic kernel $\kappa:E\times\mathcal{B}(E)^{\otimes I}\to [0,1]$ such that ...
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1answer
23 views

Expected value of time integral of geometric brownian motion

Given that the stochastic process follows, $$ \frac{dS_t}{S_t} = \mu dt + \sigma dW_t $$ How do i calculate the expected value of, $$ \int_0^T S_te^{r(T-t)} dt $$ in terms of T. What I tried ...
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2answers
27 views

If $B$ is a BM and $\mathcal F_t=\sigma(B_s,s\le t)$, then $(B_{s+t}-B_t)_{s\ge 0}$ is independent of $\mathcal F_t^+:=\bigcap_{s>t}\mathcal F_s$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
-1
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1answer
23 views

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Prove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Prove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, $$E[(X_m-X_l)X_k]=0.$$ ...
3
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1answer
57 views

Why can we consider the Brownian motion as being a mapping into the space of continuous functions, even though its paths are only a.s. continuous?

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal{A},\operatorname{P})$. By definition of $B$, for $\operatorname{P}$-almost every $\omega\in\Omega$ ...
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0answers
63 views

Can these random variables be seen as products of indicator functions?

Spin-off from here. The solution given is that $$E[X_{n+1}|X_n] = 1/2\times 2X_n + 1/2\times 0 = X_n$$ How about using indicator functions? I was thinking that $X_n = 2^n 1_{A_1}$, but I guess ...
2
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1answer
40 views

(Elementary) Markov property of the Brownian motion

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
1
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1answer
27 views

Motivation behind the definition of the Itô integral for elementary predictable processes

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space and $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$. A real-valued stochastic process $H=(H_t)_{t\ge 0}$ is called elementary ...
2
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1answer
47 views

Infinitesimal Generator of Ito Diffusion Process

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The following is an excerpt from wikipedia My question is on how to derive this operator? It looks very ...
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1answer
31 views

Prove that the increments of the Brownian motion are normally distributed

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
1
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1answer
38 views

Applying the Multivariate Ito Formula

I want to show that the stochastic process $$ S_t^i = S_0^i \exp\left( \int_0^t \left(\mu_s^i - \frac{1}{2} \sum_{j=1}^m (\sigma_s)^{ij} \right)^2 d s + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j ...
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0answers
27 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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0answers
11 views

Non existence of probabilty measures.

We suppose $X$ solves our SDE $dX_{t}=-X_{t}dt+dW_{t}$ for $t\geq0$ with initial condition $X_{0}=0$ w.r.t to our measure $P$ on $(\Omega,\mathcal{F})$. $W_{t}$ ist standard Wiener. This solution is ...
3
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1answer
39 views

Variation processes and strong solutions of stochastic differential equations

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $\mathbb{F}$ be a filtration on $(\Omega,\mathcal{A})$ $\tau$ be a $\mathbb{F}$-stopping time An $\mathbb{F}$-adapted, ...
1
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1answer
35 views

If $(B_t)_{t\ge 0}$ is a Brownian motion and $\tau$ is a stopping time, then the stopped process $(B_{\min(\tau,t)})_{t\ge 0}$ is integrable

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$. By definition $B_t$ is normally distributed with mean $0$ and variance $t$. Now, let ...
0
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1answer
47 views

Show that $\omega\mapsto\int_a^bX_t(\omega)\;dt$ is measurable, for a real-valued and continuous stochastic process $X$

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $X=(X_t)_{t\ge 0}$ be a real-valued and continuous stochastic process on $(\Omega,\mathcal{A},\operatorname{P})$ $0\le a<b$ I ...
1
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1answer
37 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
1
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1answer
27 views

Infinitesimal Generator of Poisson process

I would like to compute the infinitesimal generator of a Poisson process $N$ with intensity $\lambda$. So I can write: $$\mathbb{E}[\ f(N_{t+s})-f(N_s)\ |\ \mathcal{F_t^0} \ ] = \mathbb{E}[\ ...
3
votes
3answers
36 views

Simulation of interacting Ornstein-Uhlenbeck processes

I would like to simulate the following system of interacting OU processes on $[0,T]$: $$dX_t^1=(X_t^2-X_t^1)\,dt+\sigma_1 \,dW_t^1,\quad X_0^1=x_1$$ $$dX_t^2=(X_t^1-X_t^2)\,dt+\sigma_2 ...
9
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1answer
113 views

What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ...
2
votes
1answer
275 views

Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
3
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1answer
99 views

$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t$ with $x_t$ Ornstein Uhlenbeck process - what to do? [UNRESOLVED]

I consider the following equation: $$\frac{dy_t}{dt} = a \frac{dx_t}{dt} + x_t +y_t, \tag{1}$$ where $a=$ constant and where $x_t$ follows an Ornstein Uhlenbeck process (see here under Alternative ...
1
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1answer
65 views

How do we apply Ito's lemma to a product of functions

In finance an optimal portfolio choice it is common to use some tools of stochastic calculus. Going through a book, I found the following statement, \begin{equation} a_t = \int_s N_t(s) P_t(s) ds ...
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1answer
1k views

Easy proof of Black-Scholes option pricing formula

I use this Book to read the option princing in Black-Scholes model in pages 93-99, The poof of the formula given by $$c(s,t)= N(d_1(s,t)- Ke^{-rT}N(d_2(s,t)))$$ where $$d_{1,2}=\frac{\ln(s/K)+(r\pm ...
1
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1answer
55 views

Prove that the difference of continuous and monotonically increasing functions has continuous variation

Let $G:[0,\infty)\to\mathbb{R}$ be continuous and $$V^1_t(G):=\sup\bigcup_{n\in\mathbb{N}}\left\{\sum_{i=0}^{n-1}\left|G_{t_{i+1}}-G_{t_i}\right|:0=t_0\le\cdots\le t_n=t\right\}$$ be the variation ...
2
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0answers
43 views

The polynomial is dense in $L^2$ with non-lebesgue measure

Assume the function $u\to \mathbb E[e^{iuX}]$ is analytic in a nbhd of $0$ where $X$: $\Omega\to \mathbb R$ is a random variable. Now I want to conclude that the space of polynomial, denoted by ...
2
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0answers
30 views

Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
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0answers
34 views

Definition of Ito Integral

In Kartazas and Shreve, the integral for Bounded Progressively measurable processes is defined first. Then, for Bounded measurable processes ($f(t,\omega)$), the authors say that there exists a ...
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0answers
7 views

How to show the symmetry of a special green's function? (without defining the class of green's functions in general)

Given a two-dimensional simple random walk $ (X_i)_{i\in\mathbb{N}}$ on $ \mathbb{Z}^2 $, a square $ S_N :=\{1,2,\dots, N\} \times \{1,2,\dots, N\} $, and the stopping time $ \tau_{\partial ...
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0answers
15 views

Ito Isometry on Multivariable indicator function

The background of this question is a paper written by Morten O.Ravn and Harald Uhlig, titled "On Adjusting The HODRICK-PRESCOTT Filter For The Frequency of Observations" I will very much appreciate ...
2
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0answers
23 views

What is a stochastic differential equation of the form $dZ = f(Z_{prev}, X_{prev})dt + CdW_t$ called?

At every time step I can approximate the change in $Z$ using the following equation: $$ dZ = f(Z_{prev}, X_{prev})dt + CdW_t, \quad(1)$$ $$dW_t = r\sqrt{dt}$$ where $C$ is some constant, and $r$ is ...
0
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1answer
23 views

Variance of an integral of Brownian Motion

Let $W(u)$ $(u \geq 0)$ be a Brownian motion on a probability space $(\Omega, \mathscr{F}, \mathbb{P})$. Let $I(T) = \int_0^T W(u) du$. One can use integration by parts to show that $I(T) = ...
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0answers
30 views

What math preparation is needed before reading the mathematical method in financial markets?

What math preparation and books are needed before reading the mathematical method in financial markets by Marc Yor if i need to study the whole book? This is one of the advanced finance book
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1answer
59 views

Problem with Ito Isometry

I know that for one-dimensional case, $$ E \left[ \left(\int^T_S f(t,\omega)dB_t \right)^2 \right] = E\left[ \int^T_S f^2(t,\omega) \, dt \right]$$ for adapted, measurable f that satisfies that are in ...
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0answers
27 views

covariance of a function of Wiener processes

Consider two independent Wiener processes, $W_1$ and $W_2$. The covariance of certain functions of Wiener processes is simple, for example ...
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0answers
32 views

Glicko2, odds of winning a match

first of all I have basic knowledge in stochastic, so this is something that's not so easy to understand for me (at least for the moment). I'm trying to figure out how to calculate the odds of ...
0
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1answer
57 views

The quadratic variation of $B \cdot B$, where $B$ is a Brownian motion

Let $B$ be a standard, one-dimensional Brownian motion. Can I show that $[B \cdot B] = B^2 \cdot [B]$, using the "fundamental identity of stochastic integration", namely that $[H \cdot X, Y] = H \cdot ...
1
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2answers
64 views

Most General Theory of Stochastic Integration

I've learnt continuous stochastic integration using the classical books: - Revuz & Yor, - Karatzas & Shreve and - Oksendal. Now I want to learn general stochastic integration, i.e. possibly ...
2
votes
2answers
53 views

Mean Square Error of Monte Carlo

Trying to develop the expression for the Mean Square Error (MSE) of Monte Carlo, I found myself a bit lost when going through a simple proof in the literature. I am working in the context of ...
1
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1answer
24 views

Question regrading to a definition in Stochastic Calculus for Finance 2 by Shreve

I am confused with a definition in Shreve's Stochastic Caclulus for Finance 2 book. In page 129, Theorem 4.2.2, the Ito isometry theorem. It states that The Ito integral defined before satisfies ...
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0answers
69 views

Doubt Concerning Markov Property

Given a Markovian process $(X_t )_{t\geq 0 }$, is the following property accurate? $$\mathbb E \left[ f(X_{t_1}, X_{t_2},X_{t_3}) \mid \mathcal F ^X_{t_2}\right] = \mathbb E \left[ f(X_{t_1}, ...