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

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0
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14 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 ...
-1
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0answers
10 views

Distribution of $M(t) - 2B(t)$, $B(t)$ is Brownian Motion and $M(t)$ is its running maximum.

I'm having trouble with this, though I know its fairly simple. Hoping someone could write it out for me. Thanks. Find the distribution of of $M(t) - 2B(t)$, $B(t)$ is Brownian Motion and $M(t)$ is ...
1
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0answers
27 views

Asymptotic Expansion Method for Pricing American Option

In this Article I faced with Asymptotic Expansion method for pricing American option. the price $P(S,t)$ of this option satisfies the partial differential equation (PDE): $${{P}_{t}}+(r-\delta ...
4
votes
1answer
16 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
18 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 ...
4
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0answers
29 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
24 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 ...
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1answer
14 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} ...
1
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1answer
21 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
24 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
19 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 ...
1
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1answer
26 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 ...
2
votes
1answer
34 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
28 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 ...
2
votes
1answer
41 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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0answers
24 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 ...
0
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0answers
10 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 ...
1
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1answer
29 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 ...
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1answer
46 views
+150

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 ...
3
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1answer
36 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
25 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}[\ ...
1
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1answer
36 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
21 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, $$E[(X_m-X_l)X_k]=0.$$ ...
1
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1answer
35 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 ...
3
votes
3answers
29 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
105 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
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0answers
39 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
29 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 ...
3
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0answers
31 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 ...
1
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0answers
10 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
22 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
votes
1answer
21 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) = ...
0
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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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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 ...
0
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0answers
28 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
52 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 ...
2
votes
2answers
48 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
54 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 ...
1
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1answer
23 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 ...
0
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0answers
17 views

covariance of two ito integrals

Let $X_t=\int_0^t \left(\frac{1-t}{1-u}\right)^k dW_u$. Assume $0\lt s \lt t\lt T$. Is the following the right way to compute the covariation of $X_s$ and $X_t$? $$ \begin{align} \text{Cov}(X_t, X_s) ...
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0answers
57 views

Use of stochastic calculus outside finance?

I have noticed most of the books about stochastic calculus are targeted fo finance and derivatives. Are there any other areas outside finance where stochastic calculus is applicable?
3
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1answer
56 views

Question on applying Ito's formula in this proof

I am reviewing this paper and I'm on page 3 of the document, and I'm having trouble with the proof of uniqueness. First off, the version of Ito's lemma I've learned is: if $X_{t}$ is an Ito process ...
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0answers
19 views

conditional distribution : integral of BM

I have got a question and I have some ideas, but I don't know if I have got the right answer. The question is that Define $W_t=\int^t_0 B_s ds$ ,I have to get the distribution of $W_t$ conditional ...
6
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1answer
39 views

Basic question about the stochastic integral $\int \limits_{0}^{t} X(s) \,dM(s)$

Suppose $(X_{t})_{t \geq 0}$ and $(M_{t})_{t \geq 0 }$ are stochastic processes, where the index is continuous and the probability space is $(\Omega, \Sigma, P)$. We say for each fixed $\omega \in ...
0
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0answers
21 views

Probability that running maximum $M(t) > 2B(t)$, where $B(t)$ is Brownian Motion starting at 0

Looking for where to start with this one. Any hints will be appreciated. Probability that running maximum $M(t) > 2B(t)$, where $B(t)$ is Brownian Motion starting at 0.
0
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0answers
16 views

probabilty of maximum of stochastic process

Given, $$ M_t=exp\left( \int_0^t f(s) dW_s - \frac{1}{2}\int_0^t f(s)^2ds \right) $$ where $W_t$ is a brownian motion. Let $Z_t=W_t-\int_0^tf(s)ds$. How do i show that the above may be used with ...
1
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0answers
66 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}, ...
1
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0answers
12 views

Inverse of Lumped Markov Process

Suppose we have an (ergodic, with dumping factor $\alpha$) Markov chain $P$ on a graph $G$, and we lump the transition matrix (it's a sort of reduction of the matrix using a equivalence relation of ...
1
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1answer
15 views

Expectation of Integral of Brownian Motion

I'm working through some stochastic analysis problems at the moment and I've come across a problem that is a bit tricky (to me) - does anyone know how to calculate this expecation? I'm not sure what ...