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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30 views

Simplifying a Stochastic Equation

Let $$dG_t=\mu(t)dt+\sigma(t)dW_t$$ where $W_t$ is standard Brownian motion and define $$f(t,u)=E[e^{-r(u-t)}(\mu(u)-rG_u)|F_t]$$ where $(F_t)_{t\in [0,T]}$ is a filtration, $T\ge u\ge t$ and $r$ is a ...
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18 views

SDE of the Lévy process

I consider a Lévy process of the form $$ X_t = X_0 e^{(\mu-\frac{\sigma^2}{2})t+\sigma B_t-\sum_{i=1}^{N_t} Y_i}, $$ where $\mu$ and $\sigma$ are constants, $B_t$ is a Brownian motion, $N_t$ is a ...
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18 views

Vasicek Model Expectation Answer check

Vasicek model for the evolution of the spot interest rate is parametrised by the reversion rate $\gamma$ and so $\bar{r} = \frac{\eta}{\gamma}$ the average short rate to which the simulated rate ...
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30 views

Markov process and non-deterministic random variables

How do I show the following: If $Z_1$ and $Z_2$ are non-deterministic random variables and we define the process $(X_t)_{t\geq 0}$ by $X_t = Z_1 \cos(t)+ Z_2 \sin(t)$. I want to show that this is not ...
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20 views

1d stochastic differential equation with logarithm in diffusion

Consider the SDE $dX_t = X_t \sqrt{\ln(1/X_t)}\,d W_t$ with initial conditions $X_0 = a\in(0,1)$. Is there a closed form solution? If not, is there a drift $b$ which could be added $dX_t = b(X_t)\...
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15 views

Why do you need to assume a right continuous filtration for the Doob-Meyer decomposition?

I am reading "Stochastic Filtering Theory" right now which is the first book that didn't just assume a right continuous filtration (a "usual condition") all the time, since it doesn't come w.l.o.g. in ...
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46 views

Running Maximum of Brownian motion is singularly continuous?

Let $W_t$ be the standard Brownian motion, and define the running maximum of Brownian motion as $$M_t\doteq \max_{0\leq s\leq t} W_s.$$ Then $M_t$ is non-decreasing function. We can regard this non-...
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1answer
26 views

Ito's Lemma (Distribution of $S_t)$

If I am given that $$dS_t=S_t(\mu dt+\sigma dZ_t)$$ How do I find the distribution of $S_t$ by using Ito's Lemma? Thanks in advance.
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24 views

Why is Expectation[ $B_t^4] = 3t^2$?

Where $B_t$ is general Brownian motion. Also, what is the Expectation[$(B_s^2)(B_t^2)$] where $s < t$? Thank you for your help. I'm trying to wrap my head around these concepts, but I'm having ...
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22 views

How to show that $(e^{-rt}S_t)_{t\in [0, T]}$ is a $Q$ martingale?

We say $(W_t)_{t\in[0,T] }$ is a Brownian Motion on $(\Omega, F, F_W, P)$. For $\lambda\in \mathbb{R}$, we define $$Q(A):=E[1_AM_T], \quad A\in F,$$ where $$M_t:=exp(-\lambda W_t - \frac{1}{2}\lambda^...
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96 views

Quadratic Variation of a continuous function

I have got the following problem, but no idea where to start, so maybe one of you can help me out. We have got a continuous function $f:[0,1]\to \mathbb{R}$ with $f(0)\neq f(1)$. The task is to ...
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1answer
80 views

Why in SDE, we always consider the density $p(x,t|x_0)$ and never $p(x,t)$?

Let the SDE $$dX_t=\mu(X_t)dt+\sigma (X_t)dB_t,\quad X_0=x_0,$$ for $\mu$ and $\sigma $ nice enough, why do we always consider $p(x,t|x_0)$ for the density function and never $p(x,t)$ ? Why is it ...
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52 views

Conditional expectation of exponential Brownian motion

Let $f\in L^2[0,T]$. Show that conditional expectation $$ \mathbb{E}\bigg[ e^{\int_{0}^T f(s)dB_s}\bigg| \mathscr{F}_t\bigg] =\exp \left(\int_0^t f(s)\ dB_s + \frac{1}{2} \int_0^T |f(s)|...
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987 views

Expectation and Variance of Stochastic Differential Equations

Consider the SDE $$dr_t=\kappa(\theta-r_t)\,dt+\sigma dW_t,\ r_0=x,$$ where $\kappa$, $\theta$ and $\sigma$ are constants. You are given that the solution is $$r_t=\theta+(x-\theta)e^{-\kappa t}+...
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44 views

Expectation of the exponent of a constant times exiting time of a Brownian motion (i.e. $\mathbb{E}_x[e^{n\sigma}]$)

Suppose $a, n>0$ and $B_t$ is a Brownian Motion, define $$\sigma=\inf\{t:B_t\in\{-a,a\}\}.$$ I want to find $\mathbb{E}_x[e^{n\sigma}]$. (Notice since $n>0$ it is $\textbf{not}$ Laplace ...
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1answer
39 views

Brownian motion and Running Maximum

Take B$_t$ as a standard Brownian motion such that B$_0$ = 0. And M$_t$ is the corresponding running maximum. i.e. M$_t$ = max$_{0\leq s \leq t}$ B$_s$. My goal is to compute: (i) Quadratic ...
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1answer
416 views

Splitting a renewal process

This is a follow-up question of the question "When superposition of two renewal processes is another renewal process?". How can we split a renewal process $P$ into a renewal process $P_1$ and ...
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18 views

Book-recommendation: Numerical method for stochastic differential equations

Speaking of numerical stochastic differential equations, the book of Peter Kloeden 1992 Numerical Solution of Stochastic Differential Equations is a quite famous and standard reference. But when I ...
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1answer
14 views

Reconstructing From Conditionals

Let $X_t=f(Y_t,Z_t)$ be a stochastic process depending on $Y_t$ and on $Z_t$; all of which are Markovian. If I know $g,h$ where $$ E[X_t|Y_t]=h(Z_t,Y_t), $$ and $$ E[X_t|Z_t]=g(Z_t,Y_t) $$ then can ...
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815 views

Generated sigma algebra from Brownian Motion

Suppose that we have a Brownian motion and we define the P-augmented filtration by $$\mathcal{F}^W_t:=\sigma(\mathcal{F}^0_t \cup \mathcal{N})$$ where $\mathcal{F}_t^0:=\sigma(W_s;s\le t)$ and $\...
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28 views

Problems related to Ito lemma

This question is related to Ito lemma. I am trying to understand math in page 107, "Mortgage Valuation models by Andrew Davidson and Alexander Levin". Here is the setup: Let $\frac{dF(t,T)}{f(t,T)}=\...
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1answer
15 views

Characterization of Brownian Motion (Problem Karatzas/Shreve)

In the book "Brownian Motion and Stochastic Calculus" by Karatzas/Shreve, they state the following problem (chapter 5, problem 4.4): A continuous, adapted process $W= \{W_t,\mathcal{F}_t;0\leq t < ...
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16 views

Semi-martingale that is a martingale in its own filtration

Let $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ be a filtered probability space where $\mathbb{F}=(\mathcal{F}_t)_{0 \leq t \leq T}$ is a Brownian filtration. Consider a stochastic process $S$ that ...
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15 views

Convergence of reflected Brownian motion [closed]

Let $B_t$ be a standard Brownian motion process starting at $B_0=0$. For $c,\delta>0$ and some function $F(t)$, I am considering the following $\lim_{t\rightarrow \infty} \frac{e^{ct}P(|B_t|>\...
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1answer
58 views

Prove that the first derivative of expectation is increasing

Let us define $$ X_t = x e ^{\mu t+\sigma B_t}, \quad x>0, \ t\in[0, T] $$ where $\mu, \ \sigma$ are some constant values and $B_t$ is the standard Brownian motion. I want to show that $v'(x)$ is ...
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1answer
17 views

Derivation of the expectation of an exponential

I would like to demonstrate the formula which states that if $X$ is a random variable such that $\forall \lambda >0, E[e^{-\lambda^2.\frac{X}{2}}] = e^{-\lambda.b}$ Where $b$ is a positive ...
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1answer
858 views

Density of the Absorbed Process

The curiosity arose while reading the Ch.18 of Arbitrage Theory in Continuous Time 3/ed, dedicated to pricing Barrier Options. Definition 18.1 For any $y\in R$, the hitting time of y, $\tau(X,y)$, ...
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42 views

$E[(\int_{0}^{\infty}f(t)dW_t)^2]$ for $f(t)=(W_2-W_1)1_{[2,3)}(t)+(W_3-W_1)1_{[3,5)}(t)$

Let $f(t)=(W_2-W_1)1_{[2,3)}(t)+(W_3-W_1)1_{[3,5)}(t), t \ge 0$. $(W_t)_{t\ge0}$ is Brownian motion. What's the 'best' method to calculate $\mathbb E[(\int_{0}^{\infty}f(t)dW_t)^2]$? I would ...
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300 views

How to calculate analytical solution to fokker planck equation in finite domain?

I am looking for the solution to the following fokker planck equation. $\frac{\partial f(x,t)}{\partial t}= (k_1 x - v)\frac{\partial f(x,t)}{\partial x} + (k_2 x + D) \frac{\partial^2 f(x,t)}{\...
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1answer
27 views

First variation on Brownian motion

The first variation of function $f(t)$ on interval $[0,T]$ is defined as $$ FV(f) = \lim_{\|\pi\| \rightarrow 0} \sum_{i=0}^{n-1} |f( t_{i+1} ) - f( t_i)|.$$ How can we estimate the first variation ...
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14 views

Showing Ito's formula gives a semimartingale decomposition for brownian motion

I'm trying to show that for a standard Brownian motion and some twice continuously differentiable function $f$ that $f(B_t)$ is a local martingale iff $f'' = 0$. Applying Ito's formula gives, and ...
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2answers
172 views

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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13 views

Fokker Planck equation for reflected stochastic process

I have that $X_t\in \mathbb{R}^d$ is a Ornstein-Uhlenbeck process with $X_0=x$. I define the reflected process $R_t=|X_t|$ and a real valued $C^2$-function $f$. I now want to find the Fokker Planck ...
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1answer
25 views

Show a stochastic process is a Martingale

B$_t$ is a Brownian motion starting from 0. For any fixed constant $\sigma$ $>$ 0, X$_t$ = e $^{\sigma B_t - \sigma^2t/2}$, t$>$0 is a martingale w.r.t. the filtration generated by Brownian ...
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26 views

Mixed Poisson process $\sim\operatorname{Poisson}(p(t)\lambda)$

The capital $C$ of a bank grows proportionally with time: $$dC = adt$$ Therefore at time $t$, the bank has $at$ units of capital. The bank has to undergo stress tests, which occur according to a ...
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1answer
400 views

Probability distribution of $\int_0^t \frac{W_s}{s} \,ds$

I am currently working on an exercise that requires the knowledge of the distribution of $\int_0^t \frac{W_s}{s} \,ds$, where $W$ is a Brownian motion. I can compute the distribution of $\int_{0}^T ...
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2answers
51 views

Expected value of general diffusion

lets take measurable $b,\sigma:\mathbb{R}^+\times \mathbb{R}\to\mathbb{R}$ and consider the SDE $$dX_t=b(t,X_t)dt+\sigma(t,X_t)dB_t$$ with $X_0=x$. How can i use Itô's Lemma to show $$E_x[X_t-x]=t\...
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21 views

Limit of expectation of stochastic process

I am considering a bounded, continuous function $f$ on $\mathbb{R}^d$. For $X_t$ being a $d$-dimensional Uhlenbeck Ornstein process with $\theta=\sigma=1$, and for $x\in \mathbb{R}^d$, I want to show ...
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33 views

Intuitive reason on how predictable processes are defined

In discrete-time, we say that a process X defined in $\left(\Omega, \mathscr{F}, \lbrace\mathscr{F}_n\rbrace, \mathbb{P}\right)$ given by X $ = \lbrace X_n, n \in \mathbb{N}\rbrace$ is predictable if $...
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22 views

the product of a martingale and process with finite variation. [closed]

If $V_t$ is a process with finite variation and $M_t$ a martingale with $M_0=0$, Do we have $\mathbb{E}[V_t M_t]=0$ ? Thanks for your help
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1answer
59 views

Solution to Dirichlet is necessarily given by Brownian Motion (problem with stopping times)

Let $D \subset \mathbb{R}^n$ be a bounded open set and $f: \partial D \rightarrow \mathbb{R}$ a continuous function. The Dirichlet problem consists of finding a continuous function $U: \overline{D} \...
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1answer
26 views

A question on the application of Ito's lemma

I am reading a paper in which the author applies Ito's lemma to show that a particular process is a local martingale. Given an SDE $dZ(t)=\frac{a_i^+(Z(t))}{Z(t)}dt+\sqrt{a_i^+(Z(t))}dW(t)$, the ...
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1answer
46 views

Quadratic Variation of Brownian Motions

Let $B$ be a Brownian motion. The following statement is well-known: Let $(\pi_n)$ be a sequence of partitions of $[0, \infty)$ satisfying $\pi_n \subseteq \pi_{n+1}$ and $\text{mesh}(\pi_n) \...
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10 views

Concept of the “rate of change” in a stochastic sense

Let $X_t$ be an Ito process(or more general stochastic process). I wonder what concept is most suitable to explain the rate of change of $f\left(t,X_t\right)$ for $X_t$. I mean, some kind of ${df\...
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25 views

HJB when optimal control responds partially to Brownian motion

I cannot formulate the HJB $V$ in a simple problem where the state $a$ follows a Brownian motion, and the control $\ell$ is (for some states) such that the derivative $V_\ell$ stays constant. In ...
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1answer
27 views

Itô integral multiplied with random Riemann integral

I have seen the following equation I can't follow: $$ \mathbb{E}\left[\int_0^Tf(s,X_s)ds B_T\right] = \int_0^T\mathbb{E}(f(s,X_s)B_s)ds$$ where $(B)_t$ is a standard Brownian motion and $(X)_t$ is an ...
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1answer
41 views

Proof of Blumenthal's 0-1 law for Brownian Motion

I am currently reading the book "Brownian Motion, Martingales, and Stochastic calculus" by Jean-François Le Gall and am stuck at understanding the proof of Blumenthal's 0-1 law for Brownian Motion. ...
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1answer
37 views

Brownian motion remains nonnegative for some interval with length $1$ almost surely

Let $B_t$ be a continuous Brownian motion. I'm having a really difficult time to prove that the Brownian motion stays nonnegative for some interval with length $1$ almost surely. The reason for this ...
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20 views

Proving that $\int^\infty_0 e^{2B_s} ds=\infty$ [duplicate]

I am struggling to prove that $$\int^\infty_0 e^{2B_s} ds=\infty$$ where $B_t$ is a regular Brownian motion. We know that $\limsup_{t\rightarrow\infty}B_t=\infty$, so I figured this would be a quick ...
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22 views

Intuition behind local martingales

I know the definition that $X_t$ is a local MG w.r.t. $\mathcal{F}_t$, if $\exists$ stopping times $T_n \uparrow \infty$ a.s. so that $X^{T_n} = X_{t\wedge T_n}$ is a MG $\forall n$. The stopping ...