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

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17
votes
2answers
816 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 ...
11
votes
1answer
405 views

Motivation of Feynman-Kac formula and its relation to Kolmogorov backward/forward equations?

Kolmogorov backward/forward equations are pdes, derived for the semigroups constructed from the Markov transition kernels. Feynman-Kac formula is also a pde corresponding to a stochastic process ...
11
votes
3answers
353 views

Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s ...
10
votes
1answer
377 views

Why do people simulate with Brownian motion instead of “Intuitive Brownian Motion”?

I have just recently begun studying Brownian motion and stochastic calculus at the level of an undergraduate or beginning graduate student of applied mathematics. (Textbooks I've looked at are by ...
9
votes
0answers
155 views

Removing deterministic discontinuities from semi-martingales

Let $X:=(X_t)_{0 \le t \le T}$ be a solution of the SDE $$ X_t = X_0 + \int_0^t \sigma(s,X_s) dW_s + \sum_{i=1}^n f_i(X_{t_i^-}) 1_{\{t > t_i\}}$$ where $t_1,\cdots,t_n \in [0,T]$ and $(f_i)_{1 \le ...
7
votes
1answer
497 views

Ito's Lemma and Brownian Motion

Show by using Ito's Lemma, for $k \geq 2$ the following result hold. $$E[W(t)^k] = \frac{1}{2} k(k-1)\int_0^t E[W(s)^{k-2}]ds$$ where $W(t) = N(0,t)$ is standard Brownian motion. I think ...
7
votes
1answer
395 views

Gradient Descent on Non-Convex Function Works But How?

For Netflix Prize competition on recommendations one method used a stochastic gradient descent, popularized by Simon Funk who used it to solve an SVD approximately. The math is better explained here ...
7
votes
1answer
931 views

What are some open research problems in Stochastic Processes?

I was wondering, what are some of the open problems in the domain of Stochastic Processes. By Stochastic Processes. Any examples or recent papers or similar would be appreciated. The motivation for ...
7
votes
1answer
224 views

Strictly positive martingales

Does the following property for martingales hold? Given a continuous martingale $(X_t)_{t\leq T}$ that is almost surely strictly positive at time T, i.e. $\mathbb{P}(X_T >0)=1$, we have $P(X_t > ...
7
votes
2answers
664 views

Ito integral of a Brownian Motion w.r.t. an independent Brownian Motion.

Let $B$ and $W$ be independent Brownian motions, let $\tau$ be a stopping time adapted to $\mathcal{F}^{W}$, do we always have $E[\int_{0}^{\tau}B_{s}dW_{s}]=0$? I know that $\int_{0}^{t}B_{s}dW_{s}$ ...
7
votes
1answer
148 views

Prove the density of this SDE is not smooth in a parameter

Consider the following, 1-dimensional, equation $$X_t^x = x + \int_0^t \mathbb{E} |X_s^x| \, ds + B_t , $$ where $B$ is a Brownian motion. This a McKean-Vlasov equation, sometimes called a nonlinear ...
6
votes
3answers
723 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 ...
6
votes
1answer
262 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 ...
6
votes
1answer
52 views

double area integrals over coherence functions on circles

I am having trouble showing the following, which shows up from coherence theory: $\frac{\pi b^2}{\alpha^2}(1-J_0^2(\alpha b)-J_1^2(\alpha b))=\int_0^{2\pi}\int_0^b\int_0^b r_1r_2\frac{J_1\left ...
6
votes
1answer
197 views

Very basic doubt about Itô's lemma

While trying obtain the dynamics of $X_t = \exp( \int_t ^T \phi_s ds)$, where $\phi$ is an Ito process following $$ d\phi_t = \mu dt+ \sigma dW_t \ ,$$ I had some doubt concerning the application of ...
6
votes
1answer
1k views

What are the prerequisites for stochastic calculus?

I am not a math student, and only kind of picking up something whenever I need it. After emerged in the field of machine learning, probability, measure theory and functional analysis seem to be quite ...
6
votes
1answer
429 views

Covariance of Gaussian stochastic process

Could someone help me to figure out solutions of following problems?: Let $X = (X_t)_{t \geq 0}$ be a Gaussian, zero-mean stochastic process starting from $0$, i.e. $X_0 = 0$. Moreover, assume that ...
6
votes
1answer
693 views

Expectation of an integral w.r.t. Brownian Motion

I know the following statement: if $f$ is a deterministic function and continuous, i.e. $f\in C^0([0,T],\mathbb{R})$, then $\int f(s)dW_s$ is normally distributed with mean zero and variance $\int ...
6
votes
0answers
348 views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and ...
6
votes
0answers
309 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 $H$. ...
5
votes
1answer
138 views

existence/uniqueness of solution and Ito's formula

Given the Ito SDE $$ dX_t=a(X_t,t)dt + b(X_t,t) dB_t $$ where $a(X_t,t)$ and $ b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions. Given a function $f(X_t,t) ∈ C^2$ ...
5
votes
2answers
201 views

$L^1$ bounded martingale

If $(M_t)_{0\leq t<\infty}$ is continuous martingale and it is $L^1$ bounded, does it imply that quadratic variation $\langle M\rangle_\infty$ is finite a.s. ?
5
votes
1answer
94 views

Is $t^{-\frac{1}{2}}B_{t^2}$ a Brownian Motion?

I think the title says it all. Let $X_t = t^{-\frac{1}{2}}B_{t^2}$, with $B_t$ being a brownian motion started at $0$. I think I have proved continuity at $0$ by doing the following: $$ X_t = ...
5
votes
1answer
69 views

Question concerning maximum increment of Brownian motion

Suppose $B$ is the standard Brownian motion, how to calculate $$\mathbb{P}\left((\max_{0\le s\le t}B(s))-B(t)<a\right)$$ I tried $B(t)-B(s)=B(t-s)$, ...
5
votes
1answer
230 views

generating set of predictable sigma algebra

I am solving an exercise in Rogers and Williams and want to ask if my solution is correct. Let me first introduce the notation. The space $b\mathcal{E}$ is the space of processes of the form ...
5
votes
4answers
156 views

Bell Numbers: How to put EGF $e^{e^x-1}$ into a series?

I'm working on exponential generating functions, especially on the EGF for the Bell numbers $B_n$. I found on the internet the EGF $f(x)=e^{e^x-1}$ for Bell numbers. Now I tried to use this EGF to ...
5
votes
3answers
1k views

Itō Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itō integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
5
votes
2answers
355 views

Almost sure convergence of stochastic process

Suppose that we have a (almost surely) continuous stochastic process $\{ X_{t} \}_{t \geq 0}$ on $[0,1]$ with non-stochastic initial value $X_{0} = x_{0} \in [0,1]$ and exponentially decreasing ...
5
votes
3answers
509 views

Why isn't the Ito integral just the Riemann-Stieltjes integral?

Why isn't the Ito integral just the Riemann-Stieltjes integral? What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral: $$\int_0^Tf(t)\;dB(t).$$ ...
5
votes
1answer
123 views

lower bound of expectation of stochastic differential equation

I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
5
votes
1answer
306 views

Computing the limit of the expectation of a function of a stochastic process (phew!)

I state my problem in a few lines then describe what I have already done. I have a quite simple stochastic differential equation (SDE): $dx=-2x \, dt+\sqrt{1-x^2} \, dW$ with $W$ a brownian. I ...
5
votes
2answers
254 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 ...
5
votes
1answer
76 views

In stochastic calculus, why do we have $(dt)^2=0$ and other results?

I'm doing actuarial problems of Exam MFE and it covers some of the stochastic calculus (like Ito's Lemma). One of the frequently used results are the so-called "multiplication rules": $(dt)^2=0$ ...
5
votes
1answer
197 views

Girsanov: Change of drift, that depends on the process

Known: If I am looking at an SDE like: $dX_t = b(t,\omega) dt + dW_t$ with $W_t$ a Brownian motion under a measure $P$. I know that I can change the drift by using Girsanov to $dX_t = ...
5
votes
0answers
96 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
2answers
69 views

Show $ \int _0^t \frac{\left|B_u \right|}{u}du < \infty \ a.e.$

How to show that for all $t\geq 0$ $$ \int _0^t \frac{\left|B_u \right|}{u}du < \infty \ a.e.,$$ where $ \left( B_t \right)_{t\geq 0}$ is the real standard brownian motion starting from zero ?
4
votes
1answer
264 views

Stopping time and Brownian motion (specific example)

Let $B$ be a Brownian motion. I want to show that $$ \inf\{t\geq0 \mid B(t)=\max_{x\in [0,1]}B(s)\} $$ is not a stopping time w.r.t. the standard filtration. How can one intuitively see that this ...
4
votes
2answers
227 views

Intuition for random variable being $\sigma$-algebra measurable?

Is there some sort of intuition or a good ilustrative example for random variables being $\sigma$-algebra measurable? I understand the definition, but when looking at martingales, the meaning of ...
4
votes
1answer
124 views

Computation of basic stochastic integral.

I am trying to compute the covariance of a 1 dimensional Ornstein-Uhlenbeck process $dx_t=-\theta x_t dt+ \sigma dW_t$, $\theta>0$ and I am at the stage, $$\text{Cov }(x_s,x_t)=\sigma^2 ...
4
votes
2answers
523 views

Physical meaning of Ito integrals

I'm having trouble getting my head around the meaning of the stochastic Ito integral. Specifically: the intuitive meaning of "Stochastic Integral" to me is a function that takes a time $t$ and ...
4
votes
1answer
108 views

How to compute $E\left[W_t \int_0^t s \, dW_s\right]$?

I want to compute $E\left[W_t \int_0^t s \, dW_s\right]$ where $W_t$ is a Brownian motion. My attempt below is based on some very shaky mathematics; in particular I have no justification of the 4th ...
4
votes
1answer
876 views

Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
4
votes
1answer
187 views

Exponentials of stochastic processes and Brownian motions

This is my first time looking at problems in stochastic calculus, so please bare with the simplicity of the question. As always, any help is greatly appreciated. 1) Given $X_t=\int_0^ur_sds$ for a ...
4
votes
1answer
206 views

Tricky a.e. limit question, studying the $ \text{a.e.-lim}_{t \rightarrow \infty} \frac {1}{t} \int_0 ^ t W_s ~ds$

Could someone give some advice in order to study $$\underset{t\to\infty}{\operatorname{a.e.-lim}} \frac {1}{t} \int_0 ^ t W_s ~ds,$$ where $(W_t)_{t\geq 0}$ is a standard brownian motion starting at ...
4
votes
1answer
92 views

How to show that the following process is a submartingale

Suppose we have a filtration $(\mathcal{F}_t)$ satisfying the usual conditions. Let $W$ be a Brownian Motion with respect to that filtration. We define the two processes $X_t:=W^2_t$ and ...
4
votes
1answer
490 views

covariance of integral of Brownian

What is the covariance of the process $X(t) = \int_0^t B(u)\,du$ where $B$ is a standard Brownian motion? i.e., I wish to find $E[X(t)X(s)]$, for $0<s<t<\infty$. Any ideas? Thanks you very ...
4
votes
3answers
535 views

Wiener Process $dB^2=dt$

Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...
4
votes
1answer
108 views

Ornstein-Uhlenbeck process: increments

I'm new to the forum so I hope this first question goes well. Let the Ornstein-Uhlenbeck process be defined as: $$ dV_t = - \beta V_t dt + \sigma dW_t $$ with $V_0 = v$, where $W_t$ is a Wiener ...
4
votes
1answer
49 views

Resource for Stochastic Calculus and Ito processes

May someone please recommend a book or website where one can learn Stochastic Calculus and Ito processes from scratch.
4
votes
1answer
111 views

Karatzas and Shreve Problem 3.3.38

Let $X$ be a continuous process and $A$ a continuous, increasing process with $X_0 = A_0 = 0$, a.s. Suppose that for every $\theta \in \mathbb{R}$, the process $$Z_t^{\theta} = ...