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

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1answer
69 views

finding the probability density function of $ dY_t = - Y_t X_t dW_t$

Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$: \begin{align} dY_t &= - Y_t\ ...
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1answer
88 views

Continuous time Stochastic Process stopping time measurability

Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
0
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1answer
74 views

Brownian motion and convergence in probability of step functions

For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
0
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1answer
42 views

Integrating a Poisson Process with respect to itself

I am just learning about Poisson Processes and I feel somewhat comfortable with the basic concepts, but I am a little stuck with the following problem: Let $N(t)$ be a Poisson process with intensity ...
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0answers
100 views

$dX_t=1_{X_t\not=0} dW_t$

Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $ how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss Indeed Consider the stopping time $$ ...
2
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1answer
97 views

Orthogonal projections for minimization problem

I have trouble to understand the existence of a minimization problem in a Hilbert space. Let $(\Omega,\mathcal{F}_T,P)$ be a filtred probability space with filtration $(\mathcal{F}_t),0\le t\le T$. We ...
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0answers
64 views

Product of predictable process and a characteristic function is integrable

Suppose the time parameter $t\in[0,T]$, $S$ is a Semimartingale and $\theta_t$ a predictable $S$-integrable process such that $$\int_0^T\theta_u dS_u\ge -a$$ for a $a>0$. Furthermore ...
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0answers
25 views

Submartingale bounds

Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
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20 views

A (notational) question on composition of maps involved in time change

This comes from a paper: we have $X_t = x+\int^t_0 a(X_s)Y_sdB_s$ If we let $M_s = \int^t_0 Y_sdB_s$. But time change $X$ by the inverse of $\langle M\rangle$, we have $G_t=x+\int^t_0 a(G_s)dW_s$ ...
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1answer
149 views

Black-76 pde hedging argument wrong

I want to obtain the PDE for the Black-76 model. I believe it has to be the following PDE: $$\left(\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}F^{2}\frac{\partial^{2} V}{\partial ...
0
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1answer
43 views

Integral: Is there a closed form?

I wonder whether there is a closed form or way to compute explicitly: $$\int_0^t e^{\alpha s} dB_s$$ where $\alpha$ is just a real number and the integral is in the Itô sense. Thank you very much!
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1answer
78 views

Solve a special non-linear Backward SDE

It is straigtforward to solve a linear Backward SDE. i.e. $dY_t=Z_tdW_t+ aY_tdt+bZ_tdt$ with $Y_T=\xi$ (where a and b are constants, $\xi$ is bounded Randon Variable.) How can I solve $dY_t=Z_tdW_t+ ...
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2answers
196 views

Diffusion process. Distribution vs transition probability.

I need confirmation on the following problem: Take a SDE of the form: \begin{equation} dX_t=a(X_t,t)dt+b(X_t,t)dW_t \end{equation} where all the conditions, such that the solution $X_t$ is defined ...
0
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0answers
41 views

Help solving the (degenerate) SDE: $X_t =\int_0^t |X_s|^\alpha ds$

In a homework exercise I am, as an example of non-uniqueness of SDE's with drift only Hölder continuous of index in (0,1) , asked to show that both the zero process and $X_t=C\cdot t^p$ where ...
6
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1answer
830 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 ...
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2answers
51 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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0answers
32 views

Find the distribution of the increments from Langevin equation?

Given a Langevin eq. of a stochastic process: X[I+1]=X[I]-F(X[I])+W[I] - where F(X[I]) is a position dependent force, and W[I] is the Wiener process term (i.e. Gaussian, white-noise). How do I ...
1
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1answer
128 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
39 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
34 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
177 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 ...
0
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1answer
309 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
125 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
129 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
45 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
58 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!
1
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1answer
98 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 $r$, ...
3
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0answers
161 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
53 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} ...
1
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1answer
149 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 ...
2
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0answers
95 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 ...
1
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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
57 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 ...
2
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0answers
37 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
101 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
99 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 ...
2
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1answer
108 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|) ...
7
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1answer
427 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 ...
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0answers
136 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
116 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
141 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
213 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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0answers
92 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 ...
3
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1answer
100 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
108 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
491 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 ...
0
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1answer
310 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 = ...
3
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1answer
670 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 ...
4
votes
1answer
110 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 ...
1
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1answer
132 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]} ...