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

learn more… | top users | synonyms

0
votes
1answer
24 views

Inequality of an expectation (here: perpetual put of an american option)

for a given function $u(x):=\sup_{\tau \in T_{0,\infty}}E[(Ke^{-r\tau}-xe^{\sigma B_{\tau}-(\sigma^{2}\tau)/2})_{+}1_{\tau <\infty}]$ and $x \in [0,\infty)$, K a positive real number, $(B_{t})$ a ...
-2
votes
1answer
24 views

When does $\int_0^t dX_s = X_t-X_0$ hold for a stochastic process?

So I am learning stochastic calculus and I have seen this relationship be used many times: $$ \int_0^t dX_s = X_t-X_0 $$ where $X_t$ is some stochastic process. It looks like some sort of ...
0
votes
0answers
4 views

Is $f \in \mathbb{C(R)}$ in $\Lambda^2_{\text{loc}}$

I know that $\Lambda^2_{\text{loc}}=\{\Phi $ progressive $\mid \int_0^t \Phi^2 ds <\infty$ , $\forall t \geq0 \}$ Now since any process which is right/left continuous and adapted is progressive ...
0
votes
1answer
14 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} ...
0
votes
0answers
15 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. ...
0
votes
0answers
19 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 ...
0
votes
0answers
20 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 ...
0
votes
1answer
15 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: $mean = dE(X_t) = dX_0e^t$ ...
0
votes
2answers
38 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 ...
1
vote
1answer
39 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 ...
1
vote
1answer
16 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 ...
3
votes
1answer
42 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): ...
1
vote
0answers
29 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}}] = ...
1
vote
1answer
34 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 ...
1
vote
2answers
29 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 ...
0
votes
0answers
20 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 $R(x,y)$?
1
vote
0answers
19 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 ...
1
vote
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 ...
0
votes
0answers
8 views

Discritization of fractional brownian motion with Euler method

I m trying to discretize this SDE with Euler method. $S_t$ is stock price process $dS_t=S_t\mu dt+S_t\sigma dB_t^\epsilon $ where $B_t^{\epsilon}=\int_{0}^{t}(t-s+\epsilon)^{H-\frac{1}{2}}dW_s$ and ...
0
votes
0answers
24 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 ...
1
vote
0answers
32 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 ...
0
votes
2answers
41 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
votes
1answer
22 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 ...
1
vote
1answer
58 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 ...
0
votes
1answer
19 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 $\Gamma(\lambda,b)$. ...
1
vote
1answer
60 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 ...
1
vote
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} ...
1
vote
0answers
24 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 ...
2
votes
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 ...
0
votes
3answers
47 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 ...
-2
votes
1answer
25 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
votes
1answer
53 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 ...
4
votes
1answer
48 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 ...
0
votes
1answer
53 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
votes
1answer
51 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} + ...
0
votes
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 $\int_{-t_0}^{t}\mathbf{C}_s^t(\tau)\mathbf{q}_{\omega}(\tau)d\tau\cdot\mathbf{q}_{a}(t)$ ...
0
votes
0answers
18 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\, ...
3
votes
1answer
98 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
votes
1answer
27 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 + ...
6
votes
1answer
89 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 ...
2
votes
1answer
37 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 ...
1
vote
1answer
48 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: ...
0
votes
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? ...
0
votes
1answer
18 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= ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
1answer
23 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 ...
0
votes
0answers
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 ...
0
votes
0answers
21 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 ...
1
vote
1answer
27 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 ...