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

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4
votes
3answers
688 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 ...
9
votes
1answer
649 views

Calculate $\mathbb{E}(W_t^k)$ for a Brownian motion $(W_t)_{t \geq0}$ using Itô's Lemma

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 ...
12
votes
3answers
384 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 ...
7
votes
3answers
1k 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 ...
2
votes
1answer
893 views

Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
3
votes
1answer
304 views

Hermite Polynomials and Brownian motion

I am asked to prove the following : Let $B_t$ be a standard brownian motion. The $n$th Hermite polynomial is $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
1
vote
0answers
86 views

Estimation of a Ito's semi-martingale linear functional

Could someone check my solution for the following problem please? Or maybe propose a smarter/shorter solution. Consider a stochastic process $X=(X_t)_{t \in [0,1]}$ defined in a filtred ...
6
votes
1answer
523 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
0answers
162 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
1
vote
1answer
52 views

show that the solution is a local martingale iff it has zero drift

Most financial maths textbook state the following: Given an $n$-dimensional Ito-process defined by \begin{equation} X_t = X_0 + \int_0^{t} \alpha_s \,d W_s + \int_0^{t} \beta_s \,d s, \end{equation} ...
1
vote
2answers
207 views

Generating function of the stopping time

Let $X_t$ be a generalized Wiener process with drift rate $\mu$ and variance $\sigma^2$, and let $\tau$ be the stopping time $$\tau:=\inf \left\{ t\geq0: X_t= b\right\}, \quad b\geq0 $$ Can anyone ...
4
votes
1answer
212 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 ...
4
votes
1answer
1k 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 ...
2
votes
2answers
601 views

Matlab Code to simulate trajectories of Ito process.

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
2
votes
0answers
217 views

Difference of two convex functions

This is an exercise from a probability textbook on Ito's formula, basically Ito's formula extends to functions of this type. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f$ is ...
2
votes
1answer
138 views

solution of SDE: $dS_t=(\alpha S_t+f(t))dW_t$

does someone know how to solve the following SDE $$dS_t=(\alpha S_t+f(t))dW_t, S_0=s$$ where $f(t)$ is a deterministic function and $W_t$ is a standard brownian motion. Is there a explicit solution ...
1
vote
1answer
40 views

Lower bound on the probability of the maximum of a reflecting Brownian motion

Let $\{W_t\}_{t\geq 0}$ be a standard Brownian motion (starting at $0$). For $T$ large enough, I would like to prove that $P(\max_{t\in[0,T]} |W_t| \leq c T^{1/3})$ is bigger than a negative power of ...
1
vote
1answer
85 views

Evaluating Stratonovich integral from definition

$\bf 3.9.$ Suppose $f\in\mathcal V(0,T)$ and that $t\to f(t,\omega)$ is continuous for a.a. $\omega$. Then we have shown that $$\int\limits_0^T f(t,\omega)dB_t(\omega)=\lim_{\Delta ...
1
vote
1answer
127 views

martingale and stochastic Integral

Let ${W_t}$ be 1 dimension Brownian motion and $X_t:=\exp(t/2)\cos W_t$ $t\in[0,T]$. Show that $X_t$ is martingale. I understood $df(t,W_t)=-\exp(t/2)\sin xdW_t$ , but I don't know why it become ...
1
vote
1answer
144 views

american put option

For a perpetual american put option $v(s)$, satisfies the following problem: $$\frac12\sigma^2S^2\frac{\mathrm d^2V}{\mathrm dS^2}+(r-D)S\frac{\mathrm dV}{\mathrm dS} - rV = 0\quad\text{for ...
1
vote
1answer
393 views

Haar basis on $L^2(0,1)$ - proof?

I have the following problem. We defined $\mathbb{H}=\{f_0,\quad f_{j,n} \quad j=1,...,2^{n-1} \quad n=1,2,...\}$ where for all $t\in[0,1]$ we put $f_0(t)=1$ and setting $K=2j-1$, $$f_{j,n}(t)=\left\{ ...
0
votes
1answer
75 views

Prove directly from the definition of the Ito's integral

I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question. Prove directly from the ...
0
votes
1answer
52 views

Relation between autocorrelation and mean of a stochastic process

It is said that if the autocorrelation approaches zero as $\tau$ tends to zero, then the mean of the stochastic process is also zero. I am having trouble understanding the above concept. Say we have ...
0
votes
1answer
96 views

Finite expectation of renewal process

Let $T_n$ be a random variable with $T_n=X_1+...+X_n$ where the $X_i$'s are iid. Further we set $N(t)=max\{ n: T_n\leq n\}$ with the property $\Pr(N(t)<\infty)=1$. I want to prove that ...
8
votes
1answer
2k 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 ...
5
votes
4answers
4k views

Where to begin in approaching Stochastic Calculus?

I have experience in Abstract algebra (up to Galois theory), Real Analysis(baby Rudin except for the measure integral) and probability theory up to Brownian motion(non-rigorous treatment). Is there a ...
17
votes
2answers
971 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 ...
5
votes
0answers
179 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
2answers
883 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
274 views

Background for studying and understanding Stochastic differential equations

Assume I have back ground of the following knowledge based on the textbook as : ODE : ODE by Tenenbaum Baby probability : Ross 's baby probability Baby real anlysis : Bartle's introduction to real ...
2
votes
2answers
131 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
2
votes
1answer
177 views

How to solve $\mathrm dX(t)=B(t)X(t)\mathrm dt+B(t)X(t)\mathrm dB(t)$ with condition $X(0)=1$?

I want to solve the stochastic differential equation $$\mathrm dX(t)=B(t)X(t)\mathrm dt+B(t)X(t)\mathrm dB(t)$$ with condition $X(0)=1$.
2
votes
1answer
142 views

Proving that $T_t := S_t -\left| x \right| -\frac {n-1}{2} \int _0 ^t \frac {1}{S_u}~du$ is a brownian motion

Consider $B=(B_t)_{t\geq 0}$ $\mathcal F_t$ - brownian motion in $\mathbb R ^n, \ (n\geq 2)$ starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. ...
6
votes
1answer
69 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 ...
4
votes
2answers
585 views

Can we prove directly that $M_t$ is a martingale

Suppose we define the stochastic process $$M_t:=e^{\int_0^t\phi_s dW_s -\frac{1}{2}\int_0^t\phi_s^2ds}$$ where $\phi\in L^2[0,T]$, $t\in [0,T]$. Note that $M_t$ is just the stochastic exponential of ...
3
votes
3answers
232 views

When is a stochastic process defined via a SDE Markovian?

I was wondering when a stochastic process defined via a SDE is Markovian? The SDE may involved Ito integral, Lebesgue integral, jump component, and any other things. The reason I ask this question is ...
2
votes
1answer
234 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 ...
2
votes
1answer
143 views

Show that $dX_t=1_{X_t\not=0} dW_t$ does not have a pathwise unique solution.

Given the SDE : $$dX_t=1_{X_t\not=0} dW_t \qquad \text{with} \quad X_{0}=\xi $$ how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss Indeed Consider the ...
1
vote
2answers
130 views

Martingality Theorem: Solving expectation of a stochastic integral

I am trying to prove that: $$ \Bbb E\left[\int_s^t\sigma e^{-k(t-u)}\sqrt{V_u}dW_u\right] =0$$ Where: $$ dV_t=k~(\theta-V_t)~dt+\sigma\sqrt{V_t}dW_t $$ I have attempted to use Ito's formula on the ...
1
vote
1answer
248 views

Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
1
vote
2answers
202 views

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I'm in the last period of the undergraduate course and I have interest in this field, but my university don't have a specialist in ...
1
vote
1answer
323 views

About stationary and wide-sense stationary processes

I have just started with stochastical calculus, and I need some help with a pair of problems: $\bullet$If $X(t)$ is a mean square differentiable wide-sense stationary stochastical process then the ...
1
vote
1answer
103 views

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time $$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where ...
1
vote
2answers
125 views

Name of the formula transforming general SDE to linear

For SDE's of the general form $$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t \tag{1}$$ @saz taught me that there is a formula to transform it into a linear SDE, quoting from René L. Schilling/Lothar ...
1
vote
0answers
104 views

Characterization of hitting time's law. (Proof check)

Under the same assumptions of this early question, consider also a the random time $T_a := \inf\{ t > 0: B_t \geq a\}$ which is a stopping time. Since $M^\lambda$ is a continuous martingale, Doob's ...
0
votes
1answer
26 views

$p$-variation of a continuous local martingale

Here are two interesting statements stated from a book which I do not know how to prove: Let $\{V^{n}_t \}$ be a sequence of processes defined by $$ V^{n}_t = \sum_{k=0}^{\lceil 2^n t \rceil -1} \big| ...
0
votes
2answers
206 views

What is an alternative book to oksendal's stochastic differential equation: An introduction?

What is an alternative book to oksendal's stochastic differential equation: An introduction? But also An alternative that is over 300 pages and at the same level? Some professor refer that book as a ...
0
votes
1answer
57 views

Positivity of a stochastic process

I want to simulate the paths of a stochastic process $$ dS_t = r S_t dt + \sigma S_t dW_t$$ Using the Forward Euler method, we can write: $$ S_{n+1} = (1 + r \Delta t_n + \sigma \Delta W_{n}) S_n $$ ...
5
votes
2answers
770 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: $$\int_0^Tf(t)\;dB(t).$$ ...
4
votes
2answers
138 views

Is $\mathbb{E}\exp \left( k \int_0^T B_t^2 \, dt \right)<\infty$ for small $k>0$?

Suppose that $B$ is a Brownian motion. Does it hold that \begin{equation} \mathbb{E}\left[\exp\left(k\int_0^T[B(t)]^{2}\,dt\right)\right] <\infty\text{ ?} \end{equation} for some positive constant ...