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

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2answers
58 views

Inequality- Absolute Value general powers

Iam trying to understand the following inequality:$p>0$ Let $$T_{m}:=\sum_{i=1}^{m}\left(\left|\int_{\frac{i}{n}}^{\frac{i-1}{n}}g(s)dW_{s}\right|^{p}-\left|g ...
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1answer
135 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 ...
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1answer
42 views

Meyer's Theorem in Williams & Rogers

In Diffusions, Markov Processes and Martingales Volume 2 by Rogers and Williams they state the following theorem due to Meyer: $\mathbf{Theorem }$ Le $M\in\mathcal{M}^2_0$. Then there exists a ...
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1answer
36 views

Concepts: time homogenous and independent increments

Can someone give me an illustrative example for a time homogenous process without independent increments and for a process that is not time homogenous, but has independent increments?
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1answer
197 views

Will I have learned the prerequisites for self learning stochastic calculus and monte carlo method?

I'm an undergraduate econ major, and my main focus is in actuarial sciences, which as you may or may not know it's pretty mathematical. Some of the topics I will have to learn at some point on my own ...
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1answer
445 views

Distribution of stochastic integral w.r.t. to centered Poisson process

Let $N(t)$ be a Poisson process with intensity $\lambda$ and define the centered process as $N_0(t):= N(t)-\lambda t$. A stochastic integral can be properly defined w.r.t. to $N_0(t)$ (but not w.r.t. ...
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2answers
137 views

Conditional expectation of a functional of an Itô's semimartingale under its equivalent martingale measure

Consider a probability filtered space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfying the habitual conditions and is generated by $1 d $- ...
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1answer
171 views

Approximation of stochastic integral

Let $f \in C^2_C(\mathbb{R})$ and $$X_t = X_0 + \int_0^t \sigma(s) \, dB_s + \int_0^t b(s) \, ds$$ (1-dim.) Itô process where $\sigma,b: [0,\infty) \times \Omega \to \mathbb{R}$ progressively ...
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1answer
59 views

Does the domain of the generator of an Ito diffusion contain $C^2$ or just $C_c^2$?

Does the domain of the generator of an Ito diffusion contain $C^2$ or just $C_c^2$? From Wikipedia (For the generator $A$) One can show that $C_c^2$, i.e. any compactly-supported $C^2$ (twice ...
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0answers
64 views

Rate of increase of maximum process of Brownian Motion

Suppose $M_t=\sup_{0\leq s\leq t}\{B_s\}$, where $\{B_t\}_0^{\infty}$ is a standard Brownian Motion. I would like to know if it is true that $M_t e^{-t}$ converges to 0 almost surely? Thanks!
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1answer
122 views

Martingale inequality

Let $f: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}$ be a deterministic function, as nice as you want, $W$ a Brownian motion and define $$ Y^r_t := \int_0^t f(r,s) dW_s $$ For each fixed $r$, ...
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0answers
240 views

Spectral process for the Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process $X(t)$ is a centered, Gaussian process with covariance function $$B(s,t) = e^{-\vert t-s \vert /2}$$ The spectral measure is abs. cont. w.r.t. the Lebesgue measure ...
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0answers
61 views

When is the spectral measure absolutely continuous w.r.t. Lebesgue?

According to Bochner's theorem, the covariance function $b(t)$ of a centered, weakly stationary process $X(t)_{t\geq 0}$ can be written as $$b(t) = \int_{-\infty}^{\infty} e^{i t \lambda} ...
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1answer
243 views

Time integral over stochastic process depends on distribution only?

Let $X(t),Y(t)$ be two stochastic processes, integrable on $[0,T]$ with $X(t)\stackrel{d}{=}Y(t),\forall t\in [0,T]$. Does this imply $$\int_0^t X(s)ds = \int_0^t Y(s)ds, \qquad \forall t \in ...
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0answers
114 views

Quadratic variation process of $G$–Brownian motion

I would like to prove the inequality $$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$ where $\langle B ...
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1answer
60 views

Discontinuous Martingales on the interval $[0,T]$

Does there exist a Martingale on continuous time $[0,T]$ such that it is discontinuous for every $t \in [0,T]$?
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0answers
59 views

Probability computation $P(X_n/\log(n))$

Let $X_1, X_2, ...$ denotes a sequence of i.i.d. random variables such that $X_1$ ~ $exp(1)$ and c>0. What is $P( X_n/\log(n) > c$ for infinitely many $n$'s) ? Can I simply say that $P(X_n > c ...
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0answers
45 views

Can Ito's formula apply to $f(t, B_t)$ if $f(t,x)$ itself is random?

Can Ito's formula/lemma apply to $f(t, B_t)$ if $f(t,x)$ itself is random? I asked this, because in Ito's formula, $f$ is assumed to be a deterministic function? For example, define $f$ as $$ f(t, ...
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1answer
107 views

How to compute $\int_0^t s d B_s$ and $\int_0^t B_s ds$?

Consider the Itō integral $X_t := \int_0^t s \,dB_s$. Here is my attempt. Let $f(t,x) = tx$. By Itō's formula, $$ d f(t, B_t) = B_t dt + t dB_t $$ so $$ t B_t = \int_0^t B_s\, ds + \int_0^t s \,dB_s. ...
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1answer
143 views

Questions about existence and uniqueness theorem for stochastic differential equations in Oksendal's SDE book

In Oksendal's SDE book, Theorem 5.2.1. (Existence and uniqueness theorem for stochastic differential equations) assumes $Z$ is a random variable which is independent of the sigma algebra $\mathcal ...
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1answer
122 views

Are these two some kinds of generalized Ornstein–Uhlenbeck processes?

An Ornstein–Uhlenbeck process is $$ d X_t = (\mu - X_t) dt + d W_t $$ We try to build a model using some generalized Ornstein–Uhlenbeck processes. The first one is $$ d X_t = \exp(-|X_t- \mu|) ...
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1answer
588 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 ...
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0answers
177 views

Intuition: integration of function with respect to stochastic process

Let $X(t),t\in [a,b]$ be a stochastic process with $\mathbb E[X(t)]\equiv 0$ and uncorrelated increments, $f$ a continuously differentiable function. With the above conditions, the following equality ...
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1answer
163 views

Joint distribution of Gaussian process and its derivative

Let $X(t)$ be a Gaussian process with zero mean and covariance function $B(t,s) = 1/(1+(t-s)^2)$. Let $X'(t)$ be its $L^2$-derivative. I am looking for the joint distribution of $X(t)$ and $X'(t)$. ...
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1answer
184 views

Backward martingale property of quadratic variation

Let $\pi_n$ denotes a refining sequence of partitions of a finite closed interval (refining means $\pi_n\subset\pi_{n+1}).$ And we denote $\pi_n B = \sum_{t_i\in \pi_n}(B_{t_{i+1}}-B_{t_i})^2$, where ...
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1answer
260 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 ...
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1answer
138 views

Intuitive meaning of Lévy-Khintchine triplet

Let $\varphi$ be the characteristic function of an infinite divisible distribution. It can be expressed in the form $\varphi = e^\psi$ with $$\psi(\lambda) = i \lambda a - \frac{\sigma^2 ...
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1answer
127 views

Is this stochastic integral well defined?

Motivation: I want to prove that the existence of a $\sigma$-martingale implies NFLVR (No Free Lunch With Vanishing Risk). This comes from arbitrage theory in mathematical finance and was proved by ...
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1answer
187 views

Ito's formula for non smooth functions like Tanaka's formula

Does there exist an Ito's formula for function of Brownian Motion which are once differentiable but not twice differentiable like Tanaka's formula?
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3answers
827 views

Stochastic process with delta correlation in time

I am trying to learn stochastic calculus and when they talk about the Langevin equation they say that the correlation of the gaussian white noise (which i believe is the covariance between two random ...
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1answer
344 views

Solving Stochastic Differential Equations

Can anyone help me with the following SDE? Solve the following stochastic differential equation: $$dY_t=aY_tdt+(b(t)+cY_t)dB_t$$ with $Y_0=0$. Hint: Try a solution of the form $Z_tH_t$ where $Z_t = ...
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1answer
1k views

Variance of the Cox Ingersoll Ross model

Consider the Cox-Ingersoll-Ross (CIR) interest rate model: $\displaystyle d r_t = \kappa (\theta - r_t) \, d t + \sigma \sqrt{r_t} \,d W_t$ where $\kappa$, $\theta$, $\sigma$ are positive constants ...
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1answer
141 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 ...
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1answer
185 views

Brownian Motion and the Functional CLT

Suppose we have a time series $(x_t\mid t\in \mathbb{Z})$ for which the partial sum process $X_T$ defined on the unit interval by $$ X_T(\xi)=\omega_T^{-1}\sum_{t=1}^{[T\xi]} ...
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0answers
76 views

Negative moments of a functional of Wiener process

At the moment I am reading D. Nualart's The Malliavin Calculus and Related Topics. The problem I am trying to solve is the following: Show that the random variable $\int_0^1 s^2\arctan W_s\, ds$ ...
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1answer
213 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 ...
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1answer
96 views

Some preliminaries for the canonical construction of a Brownian Motion, help needed.

I have a lecture in stochastic analysis and I was given some facts, which are completely new to me and I do not really understand hot to understand/proof them. I would very happy if somebody could ...
2
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0answers
159 views

When are two operators simultaneously diagonalizable?

I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...
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1answer
628 views

Joint density $X^2+Y^2$

Let's say we have a point $(x,y)$ in the unit circle. I've read (without proof :( ) that the joint density of $z$, where $z^2=x^2+y^2$, is: $$f_{X,Y}(x,y) = ...
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1answer
196 views

Distribution of integral with respect to Brownian motion

Let $B$ be a Brownian motion and define the complex sequence $(X(n))_{n\in \mathbb Z}$ as $$ X(n) := \int^\pi_{-\pi} e^{inx} dB(\pi + x)$$ What is the distribution of $X(n), n\in \mathbb Z$?
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1answer
355 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\{ ...
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2answers
76 views

Identity for exponential of Brownian motion using scaling relation

Let $B$ be a Brownian motion and $s\wedge t := \min\{s,t\}$, $s\vee t := \max\{s,t\}$. I stumbled over the following identity: $$ \mathbb E[\exp(B(s\wedge t) + B(s\vee t))] \\=\mathbb ...
4
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1answer
290 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 ...
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1answer
57 views

What is $1_{\{\tau_n>0\}}X^{\tau_n}$ process saying?

As title says, what is $1_{\{\tau_n>0\}}X^{\tau_n}$ process? I do have understanding of what stochastic processes are, but not sure what is this specific process saying.
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1answer
613 views

what is F-previsible process? And what would be F?

What is F-previsible process? I tried to search in the Internet but I couldn't find it... Also what is F here? context: http://en.m.wikipedia.org/wiki/Martingale_representation_theorem#section_2
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1answer
334 views

What is the rationale of solving SDE by Ito's formula?

When solving a SDE by Ito's formula, we have to find a function $f(t, X_t)$ of index $t$ and the process $X$ to be solved for. I was wondering what is the criterion of choosing $f$? Is it to make ...
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1answer
308 views

Expectation of a stochastic exponential

In class a while ago we used the following simplification: $$ \mathbb E \left[ \exp\left(\langle \boldsymbol a,\mathbf W_t\rangle \right) \right] \quad =\quad \exp\left(\frac12 |\boldsymbol a|^2 ...
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2answers
147 views

Questions regarding filtration - more information

So for stochastic process $X_k$, We can define probability space, and filtration $\mathcal{F}_k$. As far as I know, as $\mathcal{F}$ is sigma algebra, filtration represents sequences of events that ...
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1answer
50 views

what does $X_{s-}$ mean in the integration by parts formula for the Ito integral?

The integration by parts formula for the Itō integral is If $X$ and $Y$ are semimartingales then $$ X_tY_t = X_0Y_0+\int_0^t X_{s-}\,dY_s + \int_0^t Y_{s-}\,dX_s + [X,Y]_t $$ where $[X, ...
7
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1answer
1k 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 ...