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

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31 views

Definition of Itô process

Let $\lambda_t$ and $r_t$ be predictable processes and suppose that $\int_{0}^t | \lambda_s |^2 \,ds < +\infty$, for all $t>0$. We define \begin{equation} Y_t = Y_0 \,\text{exp} \bigg\{ ...
2
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1answer
22 views

Quadratic covariation of Martingales

I was succesful at showing that the quadratic covariation $\langle\cdot ,\cdot \rangle_t$ is a positiv semidefinit, symmetric and bilinear form for each $t$ on the set of local martigales. So the ...
2
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0answers
16 views

Question on perpetual American put

Define $u(x):=\sup_{\tau \in T_{0,\infty}}E[e^{-r\tau}(K-S_{\tau})_{+}1_{\tau<\infty}$]. $T_{0,\infty}$ the set of stopping times taking values in $[0,\infty)$ and ...
2
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1answer
42 views

Requirements for square integrable in the Doob-Meyer-Decomposition

Hey i have given a non negative supermartingale $(J_{t})_{t\in[0,T]}$ of Class D. So there exists a Doob meyer decomposition $J_{t}=M_{t}-A_{t}$ where $M_{t}$ is uniformly integrable since $(J_{t})$ ...
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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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1answer
27 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 ...
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6 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 ...
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1answer
86 views

Reversing a diffusion bridge.

Suppose I have an $n$-dimensional Itô SDE $$dX_t = \sigma(X_t) dW_t + \lambda(X_t)dt$$ and I'm interested in diffusion bridges from $X_0=a\in\mathbb R^n$ to $X_T=b\in\mathbb R^n$. Now let $Y_t$ be a ...
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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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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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2answers
59 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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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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0answers
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
906 views

Easy proof of Black-Scholes option pricing formula

I use this Book to read the option princing in Black-Scholes model in pages 93-99, The poof of the formula given by $$c(s,t)= N(d_1(s,t)- Ke^{-rT}N(d_2(s,t)))$$ where $$d_{1,2}=\frac{\ln(s/K)+(r\pm ...
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2answers
164 views

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I'm in the last period of the undergraduate course and I have interest in this field, but my university don't have a specialist in ...
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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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0answers
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
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
1k views

Questions and Solutions in Brownian Motion and Stochastic Calculus?

I am currently studying Brownian Motion and Stochastic Calculus. I believe the best way to understand any subject well is to do as many questions as possible. Unfortunately, I haven't been able to ...
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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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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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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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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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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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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
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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40 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 ...
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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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 ...
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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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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
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 ...
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
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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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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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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1answer
29 views

A Property of the Ito Integral

Let $f,g \in \mathcal{V}(0,T)$ and let $0 \leq S < T.$ Then $E[\int^{T}_{S}f dB_t]=0$ Apparently this holds clearly for elementary functions, (Im not so sure), and can be obtained by taking ...
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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 ...
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1answer
65 views

Stochastic Integration and Ito Calculus

Before reading this I must not I think I am a little behind on some of the prereq for this topic but I really want to be able to understand it in a relatively meaningful way. I am having trouble ...
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3answers
50 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
132 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 ...
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1answer
55 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
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86 views

How to solve the SDE $dX_t = \frac{b-X_t}{T-t} \,dt + dW_t$?

SDE: $$dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T, \qquad X_0 = a$$ Answer: Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$, then $$\begin{align*} ...
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
112 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 $[\alpha,\beta]$ ...
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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 ...
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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 $\int_{-t_0}^{t}\mathbf{C}_s^t(\tau)\mathbf{q}_{\omega}(\tau)d\tau\cdot\mathbf{q}_{a}(t)$ ...