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.

3,273 questions
16 views

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 ...
27 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 ...
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 ...
19 views

15 views

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 ...
41 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
33 views

10 views

49 views

31 views

Is an integrated Wiener process recurrent or transient?

Like the title says, if I take an integrated Wiener process / Brownian motion $\int ^t _0 W_s ds$, will it be recurrent or transient? Or, under what conditions will it be one or the other? I know ...
21 views

Space-time white noise on PDE

Suppose I want to solve the following PDE \begin{equation} \frac{\partial u}{\partial z} + \alpha \frac{\partial u}{\partial t} = f(z,t) u+g(z,t)\int_{0}^{t}e^{\beta(s-t)}\xi(z,s)ds \end{equation} ...
31 views

Bounding the exponential of a stochastic integral

Recently I found this question on bounding \begin{equation} \mathbb{E} \left[ \exp\left(\sup_{0\leq s\leq t} \left| \int_0^s f_u \mathrm{d}W_u \right| \right) \right], \end{equation} where $f_u$ ...
Let $W = (Wt)_{t\geq0}$ be a standard one dimensional Brownian motion. Prove that $$\int_{0}^{t} W_s^2 dW_s= \frac{1} {3} W_t^3-\int_{0}^{t}W_sds$$ $$\int_{0}^{t} sdW_s=tW_t-\int_{0}^{t}W_sds$$ I ...
Question Is it true in general that if you have a multidimensional process:  \begin{align} \mathrm{d}X_t &= \mu(X_t,Y_t)\,\mathrm{d}t + \sigma(X_t,Y_t)\,\mathrm{d}V_t \\ \mathrm{d}Y_t &= \...