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

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2
votes
2answers
242 views

Basic stochastic integral

I am new to this stuff. Can some one explain how I could compute the stochastic integral of the form $\int_0^t W_sds$, where $W_t$ is Brownian process? Thanks!
3
votes
0answers
62 views

Is this a valid method for time-integrating a stochastic process?

I have a stochastic process $X_t$, and I have a function $a(x | t)$ that reflects my beliefs about the value of $X_t$ ($a$ is a density function in its first parameter). I am studying the properties ...
2
votes
1answer
75 views

how to derive this form using product formula

Suppose we have the following SDE $$dS(t) = S(t)(\mu(t)dt + \sigma(t)dW(t))=:S(t)dX(t)$$ where $W$ is a Brownian Motion and the processes $\mu,\sigma$ are well defined, such that the expression ...
1
vote
0answers
42 views

How do you convert an infintesimal generator of a Markov process to a transition function?

Suppose a continuous-time continuous-step Markov stochastic process $X_t$ has infinitesimal generator $\mu(x, t)$, $\sigma(x, t)$ ($\mu$, $\sigma$, and $X_0$ are known). How can we use this ...
3
votes
1answer
282 views

Some basic questions about Stochastic Calculus

I have a transition function for a Markov process $X_t$. I want to find a density function for the stochastic process $Y_t := \int_0^t X_s \,ds$. Some questions about this: Is this the same as the ...
3
votes
1answer
285 views

Funny problem about stochastic integrals and Ito' s lemma

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
1
vote
2answers
368 views

Explicit solution of a SDE

I'd like an explicit formula as a function of $W_t$ (standard brownien motion) and $\lambda >0$ for the solution of the following SDE: $$\mathrm dX_t = \mathrm dW_t - \lambda X_t \,\mathrm dt$$ ...
1
vote
1answer
125 views

Upper bound for the $\sup$ of a martingale defined as a stochastic integral of a general continuous martingale

Consider a probability filtred space $ (\Omega, \mathcal F, \mathcal F_ t, \mathbb P)$ and a continuous $\mathcal F _t$-martingal starting from $0$, $ M = (M_t)_{t \geq 0}$, such that $\left \langle ...
5
votes
2answers
261 views

Brownian Motion Covariance: max instead of min

It is known that $\operatorname{Cov}(B_t,B_s)=\min(t,s)$ where $B$ is Brownian motion. Can one think of an Ito process or integral (preferrably plain Gaussian process) $W$ such that ...
2
votes
2answers
896 views

Expectation of Brownian motion Integral

I want to calculate $\mathbb{E} \left[\left(\int_0^tB_s\text{d}B_s\right)^3\right]$ where $B_t$ is a standard Brownian motion. Using Ito's formula for $f:\mathbb{R}\rightarrow\mathbb{R}$ with ...
1
vote
1answer
305 views

Ito Isometry for conditional expectations

Is Ito's isometry true for conditional expectations too? I mean, is it true that:$$\mathbb{E}\left[\left(\int_0^tX_sdB_s\right)^2\ |\ \mathcal{F}_t^B\right]=\mathbb{E}\left[\int_0^tX^2_sds\ |\ ...
2
votes
2answers
123 views

Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$

Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
0
votes
2answers
421 views

Conditional Expectation of integral of Wiener process

Let $W_t$ be a standard Wiener process. How can we calculate: $$\mathbb{E}\left[\int_0^t|W_r|^2\text{d}r \ |\ \mathcal{F}_s\right]$$ where $(\mathcal{F}_s)_{s\geq0}$ is the natural filtration?
3
votes
1answer
157 views

Applying Ergodic Theorem on fractional Brownian motion

For a fractional Brownian motion $B_H$ consider the sequence for $p>0$ $$Y_{n,p}={1\over n}\sum\limits_{i=1}^n \left|B_H(i)-B_H(i-1)\right|^p.$$ By the Ergodic Theorem it is ...
3
votes
2answers
493 views

Show that this process is a martingale

Let $B_t$ be a Brownian motion and $M_t=\max_{0\leq s\leq t}B_s$. Show that: $$(M_t-B_t)^4-6t(M_t-B_t)^2+3t^2$$ is a martingale for $t\geq0$.
2
votes
1answer
39 views

Quasimartingale is Quasi-Dirichlet process

a paper I read states, that a Quasimartingale (an process $(X_t)_{t\in [0,T] }$ with $\mathbb E[|X_t|]<\infty$ for all $t\in [0,T]$, which suffices $$\sup_\Delta \sum^{n-1}_{j=0} \left\|\mathbb ...
4
votes
1answer
185 views

Show that $M_t$ is a Standard Brownian Motion

Let $M=(M_t)_{t\geq0}$ with $$M_t=\int_0^{\log\sqrt{1+2t}}e^s\text{d}B_s$$ where $(B_t)_{t\geq0}$ is a Standard Brownian Motion. Show that $M$ is also a Standard Brownian Motion and compute ...
2
votes
0answers
68 views

How to calculate the following expectation

I have a problem to find the expectation of the following expression, $$E\left[W_T e^{\int_0^T(W_s)ds}\right].$$ Here, $W_T$ is a Brownian motion. Any suggestions as to how to proceed with it? Many ...
6
votes
0answers
337 views

Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
1
vote
0answers
420 views

Show that this semimartingale is a local martingale

Let $B_t$ be a standard Wiener motion, $I_t=\int_0^t|B_s|^2\!\text{ds}\ $and $S_t=\max_{0\leq s\leq t}B_s$. Let also $F:\mathbb{R}^2_+\times\mathbb{R}\times\mathbb{R}_+\rightarrow\mathbb{R}$ a ...
3
votes
1answer
69 views

Checking a solution for a SDE

I want to show that the process $Y(t) = e^t \int_0^t e^{-s}dW(s)$ satisfies the following SDE: $dX(t) = X(t)dt + dW(t), \ \ t\geq 0 , \quad X(0) = 0$ I think the right approach is to use Ito's ...
1
vote
0answers
84 views

Fractional Brownian motion, selfsimilar

Let $0<H<1$. A real-valued Gaussian process $\left(B_H(t)\right)_{t\geq 0}$ is called fractional Brownian motion (fBm) if $\ \mathbb{E}[B_H(t)]=0$ and ...
2
votes
2answers
259 views

Martingale representation theorem

Trying to figure out how to solve problems on the 'form': Find a real number $z$ and a square integrable, adapted process $\psi(s,w)$ such that $$G(w) = z + \int \psi(s,w)\,dB_s(w)$$ for som ...
1
vote
1answer
31 views

a homework question about Levy air

I have a question in my homework: Let $X_t$ and $Y_t$ be two Brownian motions issue de $0$ and define $$S_t=\int_0^tX_s\,dY_s-\int_0^tY_s\,dX_s$$ Show that $$E[e^{i\lambda S_t}]=E[\cos(\lambda ...
3
votes
1answer
194 views

Expectation of stopping time

Let $X_t$be the solution to the SDE: $dX_t=-X_tdt+dB_t$, $X_0=0$ Then $X_t$ is the Ornstein–Uhlenbeck process $X_t=e^{-t}\int_0^te^sdB_s$. I want to calculate $\mathbb{E}[e^\tau X_\tau]$ when ...
1
vote
2answers
264 views

Ito Process $\Longrightarrow$ continuous semimartingale

I know that the Ito integral is defined in general for continuous semimartingales. But it can also be defined only for Ito processes. My question is if every process $X_t$ satisying a SDE of the form ...
6
votes
1answer
270 views

Expected Value of Brownian motion using ito isometry

Find $$ E\ \left[\left(\int_{0}^T e^{s+W_s}dW_s \right)^2\right], $$ where $(W_s)$ is a Brownian motion. I tried to use Ito isometry to solve this question, but still not yet to find the right ...
2
votes
1answer
389 views

Table of Ito Integrals

Are there any tables with a collection of common Ito Integrals, their equivalent forms, etc. that anyone knows of? Did a search but didn't come up with anything and was wondering if anyone knew of ...
0
votes
0answers
128 views

Can we apply Ito's formula?

Suppose that we are given the following processes: $B=(B_t)_{t\geq0}\ $ a standard Brownian motion starting at zero, $I=I_t=\int_0^t|B_s|^2ds,\ S=S_t=\sup_{0\leq s\leq t} B_s$ for $t\geq0$ and a ...
2
votes
0answers
145 views

Integral representation of fractional Brownian motion

Let $H\in$ $]0,1[$. A fractional Brownian motion $\left(B_H(t)\right)_{t\geq 0}$ can be represented as $${1\over C(H)}\int_\mathbb{R}\left((t-s)_+^{H-{1\over2}}-(-s)_+^{H-{1\over2}}\right)dB(s)$$ ...
2
votes
1answer
191 views

A question about Itô's representation for $\cos(B_T)$

According to Itô’s representation, any $\xi \in L_2(\Omega, F_T , P)$ has a unique representation: $ \xi = E(\xi) + \int_0^T H_s dBs$ where $(H_s)$ is an adapted process belonging to $L_2$. $B$ is a ...
2
votes
1answer
111 views

Long Range Dependence, Fractional Brownian Motion

A stationary sequence $(X_n)_{n\in\mathbb{N}}$ exhibits long-range dependence if the autocovariance function $\rho(n):=\mathrm{cov}(X_k,X_{k+n})$ satisfy $$\lim\limits_{n\to\infty}{\rho(n) \over ...
3
votes
0answers
102 views

Stochastic differential equation solution suggestion

Any suggestion on solving the stochastic differential equation \begin{align} dW(t) = d\widetilde{W}(t) + \left(\frac{\kappa - W(t)}{\tau-t} - \frac{1}{\kappa - W(t)}\right)dt \end{align} where ...
6
votes
3answers
801 views

Expected value of average of Brownian motion

For a standard one-dimensional Brownian motion $W(t)$, calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$ Note: I am not able to figure out how to approach this problem. All ...
5
votes
0answers
98 views

Representation theorem for continuous process of finite variation

There is a martingale representation theorem If $M$ is a continuous $L^2$-martingale, there is a Brownian motion $B$ and a cadlag adapted function $\sigma$ such that $$ M_t = M_0 + \int_0^t ...
4
votes
1answer
195 views

Limiting distribution of Ornstein-Uhlenbeck process

Let $X_t = e^{-\lambda t} \left(X_0 + \int _0^t e^{\lambda u} dW_u\right)$ where $(W_u)_{u \geqslant 0}$ is a Wiener process, $X_0$ random variable of law $\nu$ and independent of $\int _0^t ...
0
votes
1answer
38 views

Question on weak convergence of random variables

Let $X_n, Y_n, X$ be real random variables such that $X_n \to X$ weakly and $\mathbb{P}_{Y_n} = N(0, 1/n)$ for all positive integers $n$. I am trying to prove that $X_n + Y_n \to X$ weakly as well. ...
0
votes
1answer
132 views

Weak convergence discrete space

Let $X_n$, $n = 1, 2, 3, \ldots$, and $X$ are random variables with at most countably many integer values. Prove that that $X_n \to X$ weakly if and only if $\lim_{n \to \infty} P (X_n = j) = P(X = ...
0
votes
1answer
331 views

Show that this continuous local martingale is a martingale

We are given the following SDE: $$dX_t=X_tdt+\sqrt{2}X_tdB_t, \quad X_0=1,$$ and $$F(x,t)=e^{-t}x,\quad t\geq0,\; x\in\mathbb{R}.$$ We are asked to apply Ito's formula to $F(t,X_t)$ for $t\geq0$ ...
17
votes
2answers
856 views

Translations of Kolmogorov Student Olympiads in Probability Theory

I am deeply interested in Kolmogorov's probability contest whose tests could be found in English for the five first years but there is no English translation to its problems from round 6 onward. I ...
2
votes
1answer
1k views

Transition density and distribution: (Ornstein–Uhlenbeck process)

Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE below with $\alpha,\,\beta,\,\gamma$ constants: $$ dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0} $$ ...
2
votes
1answer
132 views

Black scholes model type

I want to study the following market: $$S_1(t)=S_1(t)(\mu_1dt + \sigma_1dW_1(t))$$ $$S_2(t)=S_2(t)(\mu_2dt+\sigma_2dW_2(t))$$ for $t\in [0,T]$, constants $\mu_i,\sigma_i$, initial values ...
0
votes
0answers
260 views

Girsanov Transformation Example

Is this the correct use of Girsanov's transformation where $B_{n}$ is a discrete Brownian motion? For example computing: $E[(B_{n}+2n)^{2}]$ Set: $\widetilde{B_{n}}=B_{n}+2n$ And ...
1
vote
1answer
210 views

Integrate Brownian motion with respect to independent Brownian motion

we are wondering what can be said about the distribution of that? We are considering standard Brownian motions and the integral from 0 to 1... More precisely: What can be said about the distribution ...
1
vote
2answers
425 views

Sum of independent random variables almost surely constant

I am trying to solve the following problem: Let $(\Omega, \mathbb{A}, \mathbb{P})$ be a probability space and $X_1, X_2, \ldots, X_n$ independent real random variables. Prove that the sum $X_1 + X_2 ...
2
votes
1answer
387 views

Expectation of exponential martingale and indicator function.

Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$. I want to evaluate $$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
2
votes
1answer
82 views

Readings necessary to understand Ito Integrals?

I searched for this question but couldn't find a direct answer. Basically I want to understand (and possibly compute some simple instances of) the Ito integral. I am coming from a physics background ...
2
votes
0answers
189 views

Explaining Ito formula to an analyst

From the point of view of analysis, what is Ito formula? Is it an integral by substitution, or, a radon-nikodym derivative? Define the probability space $$ \left(C\left(\Bbb ...
0
votes
1answer
36 views

Is it true that $X(t)^a > K \iff X(t) > K^\frac1a$

Let $a \in \mathbb{N}$, $K \in \mathbb{R^+}$ and $X(t)$ be a geometric Brownian Motion. Is the following true? $$X(t)^a > K \iff X(t) > K^\frac1a$$ The context of the above is that I want to ...
2
votes
1answer
118 views

Bounded variation and continuous local part when using Ito's Formula

When we apply Ito's Formula to a continuous semimartingale, which is the bounded variation part and which is the continuous local martingale part? Is there a general rule or does it depend on the ...