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

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2answers
72 views

$X_1,X_2,…$ real independent random variables $\not\implies S_n = \frac{1}{n}(X_1+ \cdots + X_n)$ are independent

Is it possible to show the following Lemma: Given independent real random variables $(X_i)_{i\in \mathbb{N}}$, then $(S_n := \frac{1}{n}(X_1+\cdots + X_n))_{n \in \mathbb{N}}$ are independent as ...
3
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0answers
117 views

Measurability of number of upcrossing $U_I(\alpha,\beta; X)$ in continuous time

These definitions come from Karatzas and Shreve, Brownian Motion and Stochastic Calculus. We may take for granted that $U_F(\alpha,\beta; X(\omega))$, the number of upcrossings over $[\alpha,\beta]$ ...
3
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2answers
154 views

Simple stochastic differential equation

Solve the following stochastic differential equation: $$ dX_t=X_t\,dt+dW_t. $$ Thank you very much for help! I even don't know where to start...
3
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1answer
181 views

question about Ito's formula

I'm currently learning about the Ito's lemma / formula In my textbook, a direct application of the formula is to compute quantities like that : (W is a Brownian motion) While trying to prove these ...
4
votes
1answer
514 views

Brownian bridge

Let $W = (W_t;F_t)$, $t \leq 0$ be a standard Wiener process, and let $(X_t)_{0 \leq t \leq 1}$ satisfy the stochastic differential equation $$ dX_t =- \frac{X_t}{1-t}dt+dW_t,\quad 0 \leq t \leq ...
2
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1answer
186 views

When is the following local martingale strict local martingale?

By Section 5.5 of the book [Karatzas and Shreve 1991], the following 1-d SDE has unique weak solution in the form of \begin{equation} d X_{t} = X_{t}^{\gamma} \cdot I_{\{X_{t}\ge 0\}} dW_{t}, \ ...
2
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1answer
150 views

Ito differential equation

Define $$X_t := \left( \begin{matrix} \cos W_t \\ \sin W_t \end{matrix} \right).$$ where $W = \left( W_t,\mathcal F_t \right) _{t\ge0}$ is a standard Wiener process. Find the Ito differential of X ...
2
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1answer
82 views

Stochastic differential equation problem and applying ito formula

I am given that for $b,a,\sigma >0$ and $x \in (-a,b)$ and $\nu \in \mathbb{R}$, I have the following stochastic differential equation: $$ dZ_t = \nu \,dt + \sigma\, dW_t$$ $$ Z(0) = x$$ and ...
3
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1answer
116 views

Basic doubt about stochastic integrals over general local martingales

Consider $M = (M_t)$ is a continuous square integrable local martingale and $$ \mathbb H ^2(M):= \left \{ \psi =(\psi_t)\ \text{is a real previsible process s.t.,} \forall t\geq 0, \ \mathbb E\left ...
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2answers
30 views

Algebraic problem for satisfying a given equation

I'm trying to solve the following exercise: My backward equation looks like: $P_{i,j}'(t) = i\lambda P_{i+1,j}(t) - i\lambda P_{i,j}(t) $ So i started with differentiating $P_{i,j}(t)$: ${j-1 ...
2
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0answers
209 views

Good books on “advanced” stochastic analysis

Any good books suggestion for studding advanced features of stochastic analysis ? Thank's in advance
1
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1answer
86 views

Tricky question about Ito's stochastic integral and continal law

Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real ...
0
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1answer
44 views

Prove that $Z_k := \int_{T_{2k+1}}^{T_{2k+2}}f^2(B_s) ~ds$ are independent and have same law

Consider $B=(B_t)_{t\geq 0}$ a real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider for $a,\ b \ ...
2
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1answer
136 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 $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. ...
1
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1answer
91 views

Proving that brownian motion pass almost ever by zero in $]0,t[$

Consider $B=(B_0)_{t\geq 0}$ a real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider $$ \Phi_t(x) ...
4
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1answer
224 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 ...
1
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2answers
49 views

determine whether an integral is positive

Given a standardized normal variable $X\sim N\left(0,1\right)$, and constants $ \kappa \in \left[0,1\right)$ and $\tau \in \mathbb{R}$, I want to sign the following expression: \begin{equation} ...
1
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0answers
25 views

The impact of jump on the returns of portfolio and asset pricing

There exsits jumps in financial market. What will be the impact of jump on the returns of portfolio and asset pricing? Please explain it both academically and plainly. If you can give some excellent ...
1
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1answer
822 views

How do you make dependent Brownian motions independent?

Can someone explain to me how to take 2 correlated Brownian motions and make them independent? I can't seem to grasp this process. Just assuming $dB_1(t)dB_2(t) = \rho dt $ From what was explained ...
0
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1answer
90 views

Continuous Non negative martingale converging to 0

Is there any (non trivial) continuous non negative martingale which converges to 0?
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2answers
121 views

Generalization of Doob Dynkin for Stochastic processes

Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
6
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1answer
151 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 + ...
2
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1answer
305 views

How is Brownian motion predictable?

Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has ...
1
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2answers
305 views

Moment generating function of a stochastic integral

Let $(B_t)_{t\geq 0}$ be a Brownian motion and $f(t)$ a square integrable deterministic function. Then: $$ \mathbb{E}\left[e^{\int_0^tf(s) \, dB_s}\right] = \mathbb{E}\left[e^{\frac{1}{2}\int_0^t ...
0
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1answer
363 views

Proving weak existence of CIR process

Consider the following SDE $$ X_t = x + \int_0^t \theta (\mu -X_s) ds + \int_0^t\kappa \sqrt{|X_s|} dW_s $$ where W is a brownian motion. I'm trying to show a weak solution exists, does anyone have ...
0
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1answer
40 views

For which p>0 does $S_t=W_t+t^p$ admit an equivalent martingale measure?

Let W be a brownian motion and p>0. For which p does $S_t=W_t+t^p$ admit an equivalent martingale measure? I recently saw at my lectures that NFLVR cond: There does not exist a sequence $\{H_n\}_{n ...
1
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1answer
73 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\ ...
0
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1answer
110 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
votes
1answer
89 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
55 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 ...
2
votes
1answer
135 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 ...
2
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1answer
138 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 ...
0
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0answers
75 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 ...
4
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1answer
208 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
votes
1answer
50 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
100 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+ ...
1
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2answers
396 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 ...
14
votes
2answers
2k 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 ...
1
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2answers
60 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 ...
1
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1answer
136 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
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1answer
44 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
38 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?
1
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1answer
200 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
votes
1answer
455 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 $- ...
1
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1answer
178 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 ...
1
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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 ...
1
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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!
1
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1answer
126 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
votes
0answers
250 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 ...