Mathematics Stack Exchange
Mathematics Stack Exchange

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.

The main flavors of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of , such as semimartingale processes, but the related Stratonovich integral is frequently useful in problem formulation.

The Stratonovich integral can readily be expressed in terms of the Itô integral. The main benefit of the Stratonovich integral is that it obeys the usual chain rule and therefore does not require Ito's lemma. This enables problems to be expressed in a coordinate system invariant form, which is invaluable when developing stochastic calculus on manifolds other than $\mathbb{R}^n$.

5617 questions
2
votes
0 answers

Itô's lemma and Feynman-Kac theorem for Lévy processes?

I'm facing the problem to try to extend some financial way of reasoning in the case we do not live in the platonic Brownian Motion world. I come from an economic background so I'm stuck on this: the idea is to use, say, an infinitively divisible…
Vincenzo
  • 21
2
votes
2 answers

Compute expectation of the cube of stochastic integral

I want to compute: $$\mathbb{E} \left( \left(\int_0^tudW_u \right)^3 \mid \mathcal{F_s} \right),$$ hence I write $$\int_0^t \text{as} \int_0^s + \int_s^t.$$ Then I need to compute: $$\mathbb{E} \left( \left(\int_s^tudW_u \right)^3 \mid…
Ievgenii
  • 413
2
votes
0 answers

Stochastic exponentials

Let $X$ be a good integrator with $X_0=0$, then the process \begin{equation*} Z_t=\exp(X_t-\frac{1}{2}[X,X]_t)\prod_{0\leq s \leq t}(1+\Delta X_s) \exp(-\Delta X_s + \frac{1}{2}(\Delta X_s)^2) \end{equation*} satisfies $Z_t=1+(Z_{-} \circ X)$ and is…
math student
  • 63
2
votes
0 answers

Generator of a stochastic process

I have a question about the generator of a stochastic process. $T>0$: fix Let $b: \mathbb{R} \to \mathbb{R}$ be a bounded measurable function. Let $\left( (X_{t})_{t \in [0,T]}, \left(P_{x} \right)_{x \in \mathbb{R}} \right)$ be a…
sharpe
  • 928
2
votes
1 answer

Stationary distribution for Kolmogorov Forward Equation

Given $X_t$ which satisfies the following SDE, $$ dX_t = f'(X_t)dt + \sigma dW_t $$ where f is an infinitely differentiable function, and $f'$ above is the first derivative of $f$. We know that the Kolmogorov Forward Equation is, $$ \frac{\partial…
Danny
  • 533
2
votes
0 answers

Show $\mathbb{E}Xf(X)=m\mathbb{E}f(X)+\sigma^2\mathbb{E}f'(X)$, for any function $f$, where $X$ is a Gaussian random variable$

I have the following problem which I am struggling to solve. I have the solution, but I think I am using the formula wrong. Any help would be really appreciated, thanks a lot in advance! QUESTION: Show…
s1047857
  • 891
2
votes
0 answers

On Borell's Theorem (Gaussian processes)

Let ${X(t):t \geq 0}$ be a Gaussian process with mean $0$ and bounded (with probability $1$) sample paths. Borell's Theorem states then that for all $u>0$ we have $$P(\sup_{t \geq 0} X(t)>u) \leq 2 \Psi \left(\frac{u-m}{\sigma_T}\right)$$, where…
Ganesh Ujwal
  • 171
1
vote
1 answer

Mean of stochastic exponential

Suppose $X_t$ solves an SDE. Is it true to say that the identity, $$ \mathbb{E}\left[e^{X_t}\right] = e^{\mathbb{E}[X_t]+\frac{1}{2}\text{Var}[X_t]} $$ holds only when the drift and volatility of $X_t$ are deterministic? i.e. $$ dX_t = \mu_t\, dt…
AlmostSurely
  • 255
  • 1
  • 7
1
vote
1 answer

Stochastic integral for local martingale

I have a question about local martingale. I am reading Karatzas and Shreve's Brownian Motion and Stochastic Calculus(Second Edition). On page 146, it wrote ($M$ is a local continuous martingale $M_{0}=0$ a.s. $P$ ) " For $X \in…
s34
  • 55
1
vote
1 answer

Conditional Expectation of the Vasicek Model

The solution to$~~~~ dr_t=\alpha(\mu-r_t)dt+\sigma dW_t $ is given by: $$ r_t=r_0e^{-\alpha t} +\mu(1-e^{-\alpha t})+\sigma \int_0^t e^{-\alpha (t-s)}dW_s $$ I have been able to show that: $$ r_t\sim N(~~r_0e^{-\alpha t} +\mu(1-e^{-\alpha…
WeakLearner
  • 5,982
1
vote
1 answer

Sequence solves inequalities

Suppose $Q(D)$ is a Markov chain with state space $E= \{0,1,...\}$. Further the transition matrix of $Q(D)$ is given by: $$P_D=\begin{pmatrix} \delta_0 & \delta_1 & \delta_2 & \delta_3 & ...\\ \delta_0 & \delta_1 & \delta_2 & \delta_3 & ... \\ 0 &…
mr_T
  • 303
1
vote
1 answer

is $(x-6)^2$ in $C_0^2$?

My math problem involves using a theorem that requires $f(x)=(x-6)^2$ to be in $C_0^2$. I'm trying to understand what $C_0^2$ means and how to check whether a function belongs to it. The course I'm taking is focusing on something very different and…
dark blue
  • 265
1
vote
0 answers

Reference or Intuition on a Stochastic Equation (Klyatski-Tatarski formula)

I am working through an already not so easy to find paper from the 70s, which in turn uses an even older result that i can not find at all. Im refering to http://iopscience.iop.org/0305-4470/24/17/010, unfortunately not freely availible as far as i…
ckrk
  • 235
1
vote
1 answer

Riemann integral over Itô integral?

let's say I have the Itô integral $I(t) = \int_{0}^{t} f(s)dW_{s} $ How do I then calculate $I_{2}(u) = \int_{0}^{u} I(v)dv = \int_{0}^{u} (\int_{0}^{t} f(s)dW_{s})dv$ ? Is it going to become $0$ because $dW_{t} \cdot dt = 0$, or is there some kind…
stoopid
  • 11
1
vote
0 answers

Stochastic Increments

Can anybody help me generate the increments $\Delta$$W_n$ in mathematica. I Know $W_{i+1}=w_i+Z_{i+1}\sqrt{\Delta t}$ where the $Z_i$ are independent and standard normal. But I cant make any code to generate the increments on Mathematica. There are…
user119079
  • 11
  • 6
1 2 3
…
14 15