Questions tagged [stochastic-calculus]

Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. It is used to model systems that behave randomly.

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Wiener Process $dB^2=dt$

Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...
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1answer
936 views

Prove directly from the definition of the Ito's integral

I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question. Prove directly from the ...
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Angle bracket and sharp bracket for discontinuous processes

The question is quite simple actually. I am trying to understand the differences between the angle bracket $\left<X,Y\right>$ of two processes with jumps $X,Y$, and the sharp bracket of $[X,Y]$. ...
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404 views

Solving the SDE $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$

How to solve $dX(t) = (c(t) + d(t)X(t))dt + (e(t) + f(t)X(t))dW(t)$ together with the initial condition $X(0) = X_0$.
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Thinning a Renewal Process - Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a ...
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Stochastic calculus book recommendation

I'm a quantitative researcher at a financial company. I have a PhD in math, but I'm an algebraist, so I only took the two required analysis courses in grad school (measure theory for the first, and I ...
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464 views

Using Markov Property in solving PDE/SDE

I am solving the (boundary?) value problem (from Bjork I think, see below) By Feynman-Kac, any solution has the form of a conditional expectation $$F(t,x) = E[\psi(X_T)|X_t = x]$$ where $$\psi(x) =...
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Distribution of $\max_{t \in [0,1]} |W_t|$ for Brownian motion

For a standard Brownian motion $\{W_t, t\geq 0\}$, find $\mathbb{P}(\max_{ t \in [0,1]}|W_t| <x)$. Page 79-80 of Billingsley, P., Convergence of probability measures, New York-London-Sydney-...
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Itō Integral has expectation zero

I have a question about the following property, which I didn't know so far: Why does the Itō integral have zero expectation? Is this true for every integrator and integrand? Or is this restricted ...
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1answer
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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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Limit of a Wiener integral

How to show that $$ \lim _{\alpha \rightarrow \infty } \sup_{t \in \left [0,T \right]} \left | e^{-\alpha t} \int _ 0 ^t e^{\alpha s} ~ dB_s \right | =0, \ \ \text{a.e.} $$ where $\left (B_s \right)...
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Physical meaning of Ito integrals

I'm having trouble getting my head around the meaning of the stochastic Ito integral. Specifically: the intuitive meaning of "Stochastic Integral" to me is a function that takes a time $t$ and ...
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Expected value of average of Brownian motion

For a standard one-dimensional Brownian motion $W(t)$, calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$ Note: I am not able to figure out how to approach this problem. All ...
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1answer
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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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Could someone explain rough path theory? More specifically, what is the higher ordered “area process” and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, $(X,\mathbb{...
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Integral of a Gaussian process

Let $(\Omega,\Sigma,P)$ be a probability space and $X: [0,\infty) \times \Omega \to \mathbb{R}$ be a Gaussian process (i.e. all finite linear combinations $\sum_i a_i X_{t_i}$ are Gaussian random ...
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What are some easier books for studying martingale?

What are some easier books for studying martingale? They are defined to be comprehensive but easier than Roger and William's martingale book. For example, to study Q and F martingales? It should ...
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1answer
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Rigorous Book on Stochastic Calculus

I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been ...
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3answers
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Exchange integral and conditional expectation

I know that if we have $E[\int_0^1 |X_t|dt] < \infty$ we may apply Fubini's theorem and compute $E[\int_0^1 X_tdt] = \int_0^1 E[X_t]dt$. Is there a similar version that allows the exchange of ...
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How to calculate the PSD of a stochastic process

Say we have a stochastic process described by a stochastic differential equation (in the Itô sense), and maybe we are able to find an explicit solution of it in terms of deterministic and Itô ...
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506 views

Name of the formula transforming general SDE to linear

For SDE's of the general form $$dX_t = b(X_t) \, dt + \sigma(X_t) \, dW_t \tag{1}$$ @saz taught me that there is a formula to transform it into a linear SDE, quoting from René L. Schilling/Lothar ...
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1answer
562 views

What is a good book for learning Stochastic Calculus?

I am in search of a good book for learning Stochastic Calculus from a purely mathematical/statistical point of view. Almost all the books I see are based on Finance. Also, please specify the pre-...
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Is continuous L2 bounded local martingale a true martingale?

I can prove it briefly, but I found a "counter" example. (There must be a mistake in the following words...) I can prove: X is a continuous local martingale, with $X_0=0$ a.s, then X is $L_2$ bounded ...
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Easy proof of Black-Scholes option pricing formula

I use this Book to read the option pricing in Black-Scholes model in pages 93-99, The proof 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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1answer
382 views

(Ito lemma proof): convergence of $\sum_{i=0}^{n-1}f(W(t_{i}))(W(t_{i+1})-W(t_{i}))^{2}.$

The purpose of this question is to complete my personal exposition on the rigorous proof of Ito's lemma. I have consulted more than half a dozen mathematical finance texts and not a single one, for ...
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1answer
837 views

Probability Brownian motion is positive at two points

Let $0<s<t$ and $(B_r)_r$ is Brownian motion. Does anybody know what $P(B_s>0,B_t>0)$ is? I think I remember it was some $arctan$-law but I don't know the exact form. So I do not need a ...
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1answer
297 views

Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?

Suppose we have an sde of the form: \begin{eqnarray} dX_t=b(X_t)dX_t + \sigma (X_t)dB_t \end{eqnarray} where $b$ and $\sigma$ are Lipschitz. Then we have existence and uniqueness of the solution $X_t$...
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$E[e_te_s\Delta B_t\Delta B_s]$ for $\Delta B_t$ Brownian motion increments and $e_t(\omega)$ a measurable function.

Let $\Delta B_j=B_{t_{j+1}}-B_{t_j}$ where $B_t$ is Brownian motion, and $e_i(\omega)$ measurable with respect to $\sigma(B_{t_i})$. In Oksendal's 'Stochastic Differential Equations' he states: $$ E[...
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Change of measure of conditional expectation

How can I prove that: $E_π [ (dQ_X/dπ) S (T)| F_t ]= E_{Q_X} [S(T) | F_t]E_π [ dQ_X/dπ | F_t ]$. Obviously $E_π [(dQ_X/dπ) S(T) ]= E_{Q_X} [S(T)]$ I know that much, but how to prove when it is ...
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1answer
100 views

On “for all” in if and only if statements in probability theory and stochastic calculus

1 In my friend's Probability Theory long test there was this question: Let $(\Omega, \mathfrak{F}, P)$ be a probability space on which is defined all sub-$\sigma$-algebras, events and random ...
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How to show stochastic differential equation is given by an equation

I I tried using substitution and I got an extra integral at the end and do not know how to proceed. Can anyone help me to break this down?
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Implementing Ornstein–Uhlenbeck in Matlab

I am reading this article on Wikipedia, where three sample paths of different OU-processes are plotted. I would like to do the same to learn how this works, but I face troubles implementing it in ...
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1answer
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Why predictable processes?

So far I have seen two approaches for a theory of stochastic integration, both based on $L^2$-arguments and approximations. One dealt with a standard Brownian motion as the only possible integrator ...
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1answer
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What is the difference between stochastic calculus and stochastic analysis?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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1answer
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Solving SDE: $dX(t) = udt + \sigma X(t)dB(t)$

Solve the SDE: $dX(t) = udt + \sigma X(t)dB(t)$ Provided Question The SDE is $dX(t) = udt + \sigma X(t)dB(t)$. Find $X(t)$, where $X(t)$ is some stochastic process and $B(t)$ is a Wiener process. ...
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Expectation value of a product of an Ito integral and a function of a Brownian motion

this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated. Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
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Why do people write stochastic differential equations in differential form?

I am trying to teach myself about stochastic differential equations. In several accounts I've read, the author defines an SDE as an integral equation, in which at least one integral is a stochastic ...
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1answer
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(Elementary) Markov property of the Brownian motion

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion on a probability space $(\Omega,\mathcal A,\operatorname{P})$, i.e. $B$ is a real-valued stochastic process with $B_0=0$ almost surely $B$ has independent ...
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proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
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When is a stochastic process defined via a SDE Markovian?

I was wondering when a stochastic process defined via a SDE is Markovian? The SDE may involved Ito integral, Lebesgue integral, jump component, and any other things. The reason I ask this question is ...
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1answer
634 views

Show that a process is no semimartingale

_Hello everyone! I got a little question about how to show that the process $X_t:=|B_t|^{\frac{1}{3}}$ is NOT a semimartingale. So far I tried to apply Ito. Since if $X_t$ was a semimartingale so is ...
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1answer
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Hölder Continuity of Fractional Brownian Motion

I would like to prove the following theorem: Let $H\in (0,1)$. The fractional Brownian motion $B_H$ admits a version whose sample paths are $a.s.$ Hölder continuous of order strict less than $H$. ...
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Why isn't the Ito integral just the Riemann-Stieltjes integral?

Why isn't the Ito integral just the Riemann-Stieltjes integral? What I mean is, given a continuous function $f$, some path of standard brownian motion $B$, and the integral: $$\int_0^Tf(t)\;dB(t).$$ ...
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1answer
380 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
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1answer
850 views

Stopping times and hitting times for càdlàg processes

I can't find the proof of the following lemma in any book: LEMMA: If $X=\{X_t\}_{t\in T}$ is adapted and right continuous, then for every closed set $C \subset E $, the variable $\tau_{C}:=\inf\{t\...
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1answer
804 views

$\sin(W_T)$ and Ito / Martingale Representation Theorem

I've been solving some exercises which require a function to be represented as an adapted stochastic process such that $$ X = \mathbb{E}[X] + \int_0^T \Theta(s)\,dW(s) $$ For example, $X = W(T)$ ...
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2answers
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Brownian bridge [closed]

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 1,\...
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0answers
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No drift brownian motion problem

Given two same brownian motion with no drift and different variances: $$dG_1= \sigma_1 G_1 dW $$ $$dG_2= \sigma_2 G_2 dW $$ and two barriers $P_1 > P_2$ assuming that $ \sigma_1 > \sigma_2 $ ...
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1answer
236 views

Proving Brownian Motion and the Time Integral of Brownian Motion form a 2d Markov Process

I want to show the following: Let $W_t$ be a 1 dimensions brownian motion and $V_t= \int_{0}^{t} W_sds.$ Prove that the pair $(W_t,V_t)$ is a two-dimensional Markov process. I know that the ...
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1answer
192 views

Superposition of renewal processes: Variance of lifetimes

I've a question concerning the superposition of renewal processes. Assume we have $n$ independent renewal processes with the same lifetime distribution (especially mean $\mu$ and variance $\sigma^2$). ...