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

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1answer
25 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} ...
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17 views

Why are these processes indistinguishable?

My class notes on Stochastic Calculus says that processes in $\mathbb{M_c}^{loc}, \mathbb{A}_c $ and $\mathbb{V}_c$ where they have their usual meaning, are indistinguishable of continuous processes. ...
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21 views

Basic question about application of Ito's formula

I am a complete beginner in stochastic calculus, and I am looking at a calculation of $d(W_t^2)$ where $W_t$ is a Brownian motion, using Ito's formula $$ df(W_t) = f'(W_t)dW_t+ \frac{1}{2}f''(W_t)dt ...
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23 views

Usual augmentation filtration? (Sigma algebra generated by a descreasing family of sets)?

My aime is to understand the usual augmentation filtration. More pricesely, I want to understand the last identity in this PDF file. http://onlinelibrary.wiley.com/doi/10.1002/0470863617.app1/pdf ...
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1answer
28 views

Finding mean and variance of stochastic process

If I'm given a Stochastic Process Xt that satisfies a stochastic diff. equation, let's say fXt, what is the formula to find the mean and variance of Xt? I think it's: $mean = dE(X_t) = dX_0e^t$ ...
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2answers
56 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 ...
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1answer
46 views

Will this well enough to serve as a prerequisite to oksendal's book?

Will this well enough to serve as a prerequisite to oksendal's stochastic differential equations: an introduction with applications book? I refer to shiryeav's probability, but i guess it still miss ...
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1answer
18 views

Prove that $B \in \Lambda_\text{loc}^2 $ if $B=(B_t)_{t \in \mathbb{R_+}}$ is a real valued B.M

I know that $\Lambda_\text{loc}^2=\{\phi $ is progressive $: \forall t \geq 0,\int_0^t \phi_s^2 \, ds < \infty\text{ a.s.} \}$ Since B.m $B_t$ is almost surely continuous and ...
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1answer
44 views

Inequality for the expected values of norm of stochastic processes

Let $\underline{X}=(x_1, x_2, x_3), \; x_i \sim \mathcal{N(0,1)}$ i.i.d. For any fixed $t>0$ and $\underline{X}_0$ prove that the following holds ($\Vert\cdot\Vert$ is the Euclidean norm): ...
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39 views

Optional Sampling Theorem Application

Let x, y > 0. Define the first passage time of a Brownian motion $W_t$ as $\tau_a$ = min{t $\ge$ 0: $W_t$ = a}. I need to show that $$E[e^{-u\tau_x}1_{\tau_x < \tau_{-y}}] = ...
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1answer
46 views

Stochastic differential of Bessel process [closed]

Let $ \underline{B}_{t}=(B_1(t), \dots, B_d(t))$ be a $d$-dimensional Brownian motion. How to calculate the stochastic differential of $ \Vert{\underline{B}_t}\Vert$? $\Vert . \Vert$ denotes the ...
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2answers
32 views

Expected Value of product of Ito's Integral

Any idea on how to compute the expected value of product of Ito's Integral with two different upper limit? For example: $$\mathbb{E}\left[\int_0^r f(t)\,dB(t) \int_0^s f(t)\,dB(t)\right]$$ I only ...
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0answers
21 views

integral involving wiener process

Suppose $W_t$ is standard Brownian motion and define $$ R(x,y) = \int_{0}^{T} W_{t+x}\,W_{t+y}\,dt, $$ which is sort of the sample covariance function. What is the distribution of $R(x,y)$?
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20 views

Level sets of a Wiener process

Assume we have a Wiener process $W$ starting at $W_0=0$. What can one tell about the Lebesgue measure of "level sets" $A_y = \{t>0; W_t=y\}, y \in \mathbb{R}$? I actually need to estimate these ...
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1answer
29 views

A question on proving the existence of a martingle which has a deterministic square bracket

Let $g:\mathbb{R^+} \to \mathbb{R^+}$ be a non decreasing and continuous function . Show that there exists a continuous martingale M such that its square bracket $<M>_t=g(t)-g(0)?)$ I have ...
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0answers
25 views

non-additive noise?

I always hear about noise that is "additive" (as well as being Gaussian),and I guess I'm wondering what the opposite is - what kind of noise is not additive? What does the SDE with non-additive noise ...
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0answers
38 views

Expectation of Exponential of Stochastic Integral

Let $z$ be the standard Brownian motion, $\omega$ an element of the sample space. Is it true that $$ \mathbf E\bigg[\exp\Big(\int_0^t f(\omega,s)\,\mathrm dz(s)\Big)\bigg] = \mathbf ...
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2answers
44 views

What is the polarization identity?

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and ...
3
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1answer
31 views

Girsanov Theorem Confusion

I'm getting completely bogged down by sign errors when using Girsanov's theorem. Keeping it simple, suppose $W_t$ is a standard Brownian motion under a probability measure $\mathbb{P}$, and we have a ...
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1answer
70 views

How to solve the SDE $dX_t = aX_tdt + (b(t)-X_t^2)^{1/2}dW_t$?

I need help on solve the following SDE: $\beta > 0$, $0<\gamma<1$, $X_0 = \frac{\sqrt{2}}{2}$ $$dX_t = -(\beta + \frac{1}{2}\gamma^2)X_tdt + \gamma\sqrt{e^{-2\beta t}-X_t^2}dW_t$$ I need ...
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1answer
20 views

Calculating the mass function of maximum of a sum

Find an expression for the mass function of $N(t)$ in a renewal process whose interarrival times $X_i$ are a) poisson distributed with paramter $\lambda$ and b) gamma distributed $\Gamma(\lambda,b)$. ...
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1answer
64 views

Evaluating Expectation of stochastic process

Say, for $u>t$ we have a stochastic process given by : $$ r_u=r_t + \int_t^u\theta_s ds+\sigma\int_t^udW_s, $$ where $W_t$ is a brownian motion, $\sigma$ is a constant and $\theta_t$ is some ...
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1answer
26 views

How can I make this computation of the expected value of a random variable formally correct?

Consider a time Interval $[0,T]$ and times $0<t_1 < t_2 < ... < t_n<T$ generated by a Poisson process. In my scriptum, the expected value of the function $$Y(t) = \sum_{t_i \leq t} ...
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0answers
27 views

Stochastic integral density of simple functions no1

I am trying to understand proposition 2.6 page p.134 from Karatza's book Brownian motion and stochastic calculus. If $M$ is continuous square integrable martingale on $(\Omega, \mathcal{F}, P)$ and ...
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0answers
47 views

Integral of a geometric Brownian motion [duplicate]

I would like to compute $G$ defined as follows $$G(t):= \exp(-\int _0^t h_s~ ds )$$ with $h$ being a geometric Brownian Motion. For that I would need first to compute $$\int_0^t ...
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3answers
49 views

Change of Variables Theorem

I am searching for a proof of the following theorem: THEOREM Suppose $(X_1, \ldots, X_n)$ is a random vector with joint density function $f_{X_1, \ldots, X_n}(x_1, \ldots , x_n)$ and $g$ is ...
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1answer
33 views

Differential of two geometric brownian motions

I am currently taking a finance course which includes some math that is currently above my level, it is however not a pure math class and we are just supposed to be able to apply the math to the given ...
2
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1answer
56 views

Ito Integral representation for bounded claims

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable and bounded. Does it ...
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1answer
51 views

Representation Theorem for functionals of Continuous Semimartingales

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be Probability space and let $\{(X_t) : 0 \leq t \leq T \}$ be a continuous semimartingale on it. Let $\xi$ be $\mathcal{F}_T^X$ measurable. Does it mean that ...
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1answer
66 views

How to solve the SDE: $dX_t = \frac{1}{X_t}dt + X_tdW_t$

I have difficulties in solving following SDE: $$dX_t = \frac{1}{X_t}dt + X_tdW_t$$ I tried the transformation method provided in the following link: Name of the formula transforming general SDE to ...
2
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1answer
57 views

Black Scholes PDE

How to show that $V_1(S,t)=S\frac{\partial V(S,t)}{\partial S} $ satisfies Black-Scholes PDE given as $\frac{\partial V}{\partial t} + \frac{\sigma^2 S^2}{2}\frac{\partial^2V}{\partial S^2} + ...
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0answers
13 views

Characterise expression involving white noise

I would like to characterise an expression, for example by finding its spectral density. The function is $\int_{-t_0}^{t}\mathbf{C}_s^t(\tau)\mathbf{q}_{\omega}(\tau)d\tau\cdot\mathbf{q}_{a}(t)$ ...
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19 views

BDT model and the $\theta(t)$ function

Question about the answer in this question: Black Derman & Toy Model Where $$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t)}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$ is rewritten as $$dr=A\,dt+B\, ...
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1answer
103 views

Ito's Lemma for negative exponential

I'd been reading on Hull-White model, when I encountered the bond-pricing formula, that is if $$ dr(t) = (\alpha(t)-\beta(t)r(t))dt + \sigma(t)dW(t)$$ for some deterministic function $\alpha, \beta, ...
2
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1answer
32 views

Black Scholes Differential Form

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that $$ S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu\, dt + ...
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1answer
95 views

Converge to Brownian Motion problem

Consider the following sequence of SDEs: $dX^n_t = \sin(nX^n_t)dt + dW_t, X^n_0 = 0\,\,\,$ Show that the solutions $X^n$ converge in finite dimensional distribution to Brownian Motion. I have been ...
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1answer
40 views

Proof of the Début theorem

I was reading the stochastic calculus notes on this website and I read the following in the proof of the Début theorem but I could not understand what does it mean. Can someone explain it to me ...
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1answer
56 views

Why is the Stochastic Process in the HJM model non-Markovian?

I want to understand exactly what my title asks "Why is the Stochastic Process for the short rate in the HJM model of interest rates non-Markovian?" That process is the following: ...
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1answer
31 views

Can someone help me understand the following?

I was reading George Lowthers notes on Stochastic Calculus and , he says the following but I cannot figure out what it exactly means? ...
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1answer
20 views

question on a stopping time problem.

I borrowed some lecture notes on stochastic calculus, which contained the following exercise: Let $(X_n)_{n>0}$ be a sequence of random variables with $X_n: \Omega \to [0,\infty)$. We set $S_n= ...
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17 views

Random starting point for Brownian motion

The hitting probability for balls centered at origin is $P_{x}(T_{B_{r}(0)}<\infty)=\frac{r^{d-2}}{|x|^{d-2}}>0$ where $|x|>r$. Now consider hitting time $T_{A}$ of sphere A disjoint from ...
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29 views

Processes adapted to time changes

I have a question regarding a passage in Chapter X of "Calcul Stochastique et Problèmes de Martingales"J.Jacod(1979). In (10.13) they define an adapted process $X$ to the time change $\tau(t)$ as a ...
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1answer
31 views

Transition density of Brownian bridge using generators

Let $X_{t}:=(1-t)\int_{0}^{t}\frac{1}{1-s}dB_{s}$. This satisfies SDE: $$dX_{t}=-\frac{X_{t}}{(1-t)}+dB_{t}$$ So the generator will be $A(f)=\frac{-x}{1-t}f'+\frac{1}{2}f''$ and so I think the pde ...
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21 views

Estimate on the Positive probability of not hitting finite measure sets in $\mathbb{R}^{d}$

In $d\geq 3$, we have that BM is transient a.s. i.e. $\lim_{t\to \infty}|B_t|=\infty$. But does this imply $1-P_x(T_A<\infty)>0$ for Borel sets $A\subset \mathbb{R}^d$ with ...
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25 views

Computational rules for expectations of functions of wiener processes.

What are some general rules that are helpful for computation/calculation of expectations such as $$ E(X_t | \mathcal{F_s} ), $$ where $X $ is a function of Brownian motions $W_t$ and $\mathcal{F}$ is ...
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1answer
33 views

Joint convergence in distribution

I've one question concerning convergence in distribution of random variables: Let $X_n \rightarrow X$ and $Y_n \rightarrow Y$ for $n \to \infty$ where $\rightarrow$ denotes convergence in ...
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21 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
2
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1answer
120 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 $\displaystyle H_n(t,x)=\frac{(−t)^n}{n!} e^{x^2/(2t)} \frac{d^n}{dx^n}e^{-x^2/(2t)}$. ...
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1answer
61 views

Integral with respect to brownian motion

Let $f$ be a continuous function on $[0,\infty)$ and $B_t$ be a standard Brownian motion. Define $X_t=\int_0^t f(s) dB(s).$ a) Show that $X_t$ is Gaussian and computer its covariance $C(X_s, X_t)$ ...
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29 views

Measure Preserving tranformation of the space of brownian paths

Let $O$ be an orthogonal transformation of $L2_{[0,\infty)}$. Let $1_{[0,1]}$ be the indicator function for $0 \leq s \leq t$. Also let $B(t)$ be a standard brownian motion. Define $W(t) = ...