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

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Does Brownian Motion return to the origin infinitely soon?

Consider a standard unidimensional Brownian Motion $B_t$ (Wiener process). Fact: This process returns to the origin infinite number of times with probability one. Consider a stopping time $\tau = ...
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Girsanov theorem

I work on an exercice and I have to calculate: $$E(W_{t}^2e^{(\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds)})$$ $$\theta$$ is deterministic function I don't know how to ...
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2answers
25 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
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Cumulants in the BN-S stochastic volatility model

Consider a bivariate SDE $$ dX_t =(-\frac{1}{2} \sigma_t -\lambda \kappa(\rho) ) dt + \sigma _t d W_t + \rho dZ_{\lambda t} \\ d\sigma_t^2 = - \lambda \sigma_t ^2 dt + d Z_{\lambda t} $$ with $\rho ...
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1answer
25 views

What is meant by a linear SDE?

I am sure this is a ridiculous question, but I can't seem to find a definition. I know the definition of linear ODE or PDE just by saying that the differential operator should be linear, but how does ...
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14 views

Solving an integral to solve a statistical problem

In solving a statistical problem, I want to know $$\mathbb{P}( (Y_2| Y_1 \geq q_\alpha) \leq q_\alpha)$$ where $Y_1 = X_1 + Z$ and $Y_2 = X_2 + Z$ and $Z \sim N(0,\sigma_z^2), X_i \sim N(0, ...
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15 views

Stochastic Differential Equation for Time Integral of Stochastic Process

Let $X(t)$ denote standard Brownian motion $dX(t) = a X dt + X dW(t)$ with solution $X(t) = e^{a t + W(t)}$. I want to consider the time-integrated process \begin{equation} Y(t) := \int_0^t d\tau~ ...
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1answer
21 views

positiv Martingale process

I would to like to prove that the process: $$e^{\int_{0}^{T}\theta _{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2ds}$$ is a martingale which is positiv and has a mean=1 $$\theta is continuous ...
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24 views

Integral of Brownian Motion with respect to an independent Brownian motion

I have this seemingly simple problem which I haven't been able to solve. I have two standard Brownian motions, $B$ and $W$, on the same probability space and under the same filtration (I am not so ...
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1answer
15 views

Conditions for Expectation of Ito Integral to have Expectation 0

Consider the Ito stochastic process $$X_t = X_0 + \int_{0}^{t} a_s ds + \int_{0}^{t} b_s dW_s$$ What conditions are necessary or sufficient (besides adaptability/measurability) to show that $$ E ...
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30 views

Two-dimensional Brownian motion

Let $B_1$ and $B_2$ be two $\mathbb{R}$-valued Brownian motions with $$\langle B_1,B_2\rangle=\int_0^t\rho_s ds,$$ where $\rho$ is progressively measurable with values in $(-1,1)$. We define ...
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1answer
35 views

Construction of the Itō integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
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1answer
28 views

Measurability of an integral

Let $\{X_t\}_{t\ge 0}$ be an adapted $\mathbb{R}$-valued stochastic process on some filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_t\}_{t\ge 0},\mathbb{P}\}$ such that for each ...
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1answer
21 views

Doubt concerning Stochastic continuity

I know that a stochastic process $X$ is said to be stochastically continuous if $\forall s$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$. But then it is also true that stochastic continuity ...
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1answer
33 views

Ito isometry for bounded Ito integral

Let $(W_t)_{t \in [0, T]}$ be a Brownian motion and $T$ be a finite time. If $\int^T_0 \beta_t d W_t$ is bounded and $\{ \beta_t \}_{t \in [0,T]}$ is locally integrable, I am curious whether the ...
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2answers
25 views

What is this conditional probability?

I have been doing some reading for a project on quantitive finance, and I have been seeing a lot of this kind of conditional probabilities on a "$\mathcal{F}_{t_i}$": $$\mathbb{P} ...
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1answer
17 views

How to use Itō in this very simple case

I want to apply Ito for the following process: \begin{equation*} X_t = tW_t + \int_0^t W_u du, \end{equation*} where $W$ is a Brownian motion. I have no trouble with the part $tW_t$ This can be ...
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1answer
35 views

Independence of linear combinations of Brownian motions

Let $0<s\leq t\leq u\leq v$ and $W_x$ be a Brownian motion. Show that $aW_s+bW_t$ and $\frac{1}{v}W_v-\frac{1}{u}W_u$ are independent for $a,b$ satisfying $as+bt=0$. The question seems easy but ...
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0answers
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What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
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1answer
40 views

Are $X$ and $Y$ necessarily normal if the the sum $Z=X+Y$ is normal?

Of course, asking the question the other way round is straightforward to answer as via the convolution we find that the sum of two normal distributed variables is again normal. But however, is it ...
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1answer
15 views

What are the conditions for $E[\int_0^tf(W_s,s)dW_s]=0$?

Let $W_t$ be the standard Brownian Motion. I am interested on the conditions on $f(\cdot)$ that guarantee that the expectation of the Ito integral below is zero: ...
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1answer
66 views

An application of Itô's lemma

I found this question in a past exam for a course on Financial Economics. Given the function $f(t,x)$, let $F(t,x)$ be a function such that $∂F/∂x = f$. (a) By writing Itô’s formula in ...
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1answer
18 views

Example of an adapted but not progressively measurable process

I'm looking for an example of a stochastic process $X$ that is $\mathbb{F}$-adapted, but not progressively measurable. One example I found is the following: $(\Omega, \mathfrak{A}) = (\mathbb{R^+}, ...
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1answer
32 views

Stochastic Integral basics

As far as I understand, the stochastic integral is defined so that we can make sense of something like this: \begin{equation*} X_t = x_0 + \int_0^t g(s) ds + \int_0^t f(s) dW(s) \end{equation*} ...
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2answers
38 views

Total Differential / Ito dynamics

I found this process in a scientific paper: $M_t = \int_{0}^t e^{-(t-u)} \frac{dS_u}{S_u}$ where $dS_t = S_t (\phi M_t + (1-\phi)\mu_t) dt + \sigma S_t dW_t$ and I want to compute the ...
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No drift brownian motion and minimization at a given time [closed]

Given two same brownian motion with no drift and different variances: $$(dG_1/G_1)= \sigma_1dW_g $$ $$(dG_2/G_2)= \sigma_2dW_g $$ At a specific given time $ T = \tau $, how can I tell if ...
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1answer
30 views

What is the distribution of $B(t_1)+B(t_2)+…+B(t_n)$ [closed]

$\{ B(t), t\ge 0\}$ is a standard Browian Motion Process. What is the distribution of $B(t_1)+B(t_2)+...+B(t_n)$ ?
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Density of the Absorbed Process

The curiosity arose while reading the Ch.18 of Arbitrage Theory in Continuous Time 3/ed, dedicated to pricing Barrier Options. Definition 18.1 For any $y\in R$, the hitting time of y, $\tau(X,y)$, ...
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1answer
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Slight generalisation of the distribution of Brownian integral

I think I have seen once that if the processes $\sigma$ and $W$, a Brownian motion, are independent then one has that $$ E \left[\exp \left(iu\int_t^T \sigma_s \, dW_s\right) \mid \mathcal{F} \right] ...
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2answers
50 views

Implementing Ornstein–Uhlenbeck in Matlab

I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in ...
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1answer
22 views

How do you count probability from expected value and spread of a discrete variable? [closed]

One garden produces 500kg of fruit on avarage. Another produces 300kg on avarage. (Both yearly.) Their spread is 100kg and 80kg, and their correlation is 0,7. The expected value of both is 800kg. What ...
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why hull white model has normal distribution?

consider hull white model $dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t)$ when we solve the SDE above we have $r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha ...
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1answer
54 views

A question related to reflection principle

Question: $$P(X_1\gt 0, ..., X_n\gt 0, X_n=a-b)=?$$ Its Answer: $= (1,1) \rightarrow (n,a-b) $ that meet neither touch nor cross paths. $=[(1,1) \rightarrow (n,a-b) \ \ \text{all ...
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Clarify a question's answer related to random walk. [closed]

I'm studying Problem5.3 and its solution. However, its solution is not clear for me. Please explanatorily show this answer . I need to learn such type of questions. Please help me. Thank you.
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49 views

Fokker Plank EQUATION

I would be grateful if you let me know an application of Fokker plank equation in a financial market or introduce a related paper to me. For example, when the price of stocks in our market satisfiy ...
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2answers
47 views

Smallest $n$ where $2 \cdot \sum_{k = 1}^{n} \frac{\left(\frac{n}{100}\right)^k}{k!} \geq e^{\frac{n}{100}}$

I'm having a problem with solving the following relation for $n$: $$2 \cdot \sum_{k = 1}^{n} \dfrac{\left(\frac{n}{100}\right)^k}{k!} \geq e^{\frac{n}{100}}$$ By trial-and-error I was already able ...
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1answer
33 views

Local maximum of brownian motions

Let $B=(B_t)_{t\geq 0}$ be the standard Brownian motion. I want to show that for every $t_0 \geq 0$ $\mathbb{P}$($B$ has a local maximum in $t_0$)=0. I've already shown that for every ...
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1answer
13 views

The joint distribution of running maximum in two dimension

How to calculate the probablity of $P(M_1(t)\leq x,M_2(t)\leq y)$, where $M_1(t)=a_1B_1(t)+b_1B_2(t)$ and $M_2(t)=a_2B_1(t)+b_2B_2(t)$ with $B_1(t)$ and $B_2(t)$ independent Brownian motions?
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Absolute convergence of $\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$

I want to show that if $X \in L^1$, where $X$ is a real-valued random variable, the sum $\sum_{n=1}^{\infty} \mathbb{P}(|X|>n)$ converges absolute. My idea was the following: Since $X \in ...
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Markov Semigroups worked example

I have been reading this excellent paper on Markov semigroups, in which the assertion is made that a markov semigroup $\mathcal{P: L^1 \longrightarrow L^1}$ is defined by $\frac {d\mu}{dm}$ for some ...
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Stochastic integral of local martingales is an extension

I'm trying to prove that the stochastic integral defined for the set of square integrable local martingales is really an extension of ordinary stochastic integral. Define $\mathcal{H}=\{(H_t)_{0\leq ...
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1answer
80 views

Application of the Burkholder Davis Gundy inequality

The proof of the Feynman-Kac formula uses a lemma which I need to proof, but I can not figure it out. The lemma is the folllowing: Let $X$ be a weak solution of ...
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1answer
30 views

Show that $\omega\mapsto\int_a^bX_t(\omega)\;dt$ is measurable, for a real-valued and continuos 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 ...
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39 views

Fundamental theorem of calculus for the Lebesgue integral

Let $\lambda$ be the Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R})$ $f:\mathbb{R}\to\mathbb{R}$ be $\lambda$-integrable What's the easiest way to show $$\frac ...
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1answer
31 views

Closure of the set of elementary predictable stochastic processes

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

Eigenfunctions of an operator using Laguerre Polynomials

I am trying to find the eigenfunctions of the following operator: $$\mathcal{L}f=(-\gamma x+\frac{\mu}{x})f_x+\mu f_{xx}$$ I know that I must somehow use Laguerre polynomials, the solutions to the ...
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1answer
30 views

$\sin(W_T)$ and Ito / Martingale Representation Theorem

I've been solving some exercises which require a function to be represented as an adapted stochastic process such that $$ X = \mathbb{E}[X] + \int_0^T \Theta(s)\,dW(s) $$ For example, $X = W(T)$ ...
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1answer
42 views

An equality involving the Wiener process

The equality below appears as a step in a proof in a chapter titled "Itô Stochastic Calculus" in Brzeźniak and Zatawniak's textbook "Basic Stochastic Processes", Springer 2005 (in a solution to ...
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53 views

Absorbed brownian motion is a Markov process

I have been asked to prove that the Brownian motion absorbed at the origin is a Markov process. Formally, let $B_t^x$ be a Brownian motion originating from $x>0$ and let $\tau^x_0 = \inf\{t>0 : ...
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2answers
21 views

Product rule with stochastic differentials

I am encountering difficulty in seeing how this relationship holds: with $S_T$ being stock price at time $T$, I want to find the sde for $S_t e^{-rt}$ $$dS_t = rS_tdt + \sigma S_t d\hat B_t$$ Where ...