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
1
vote
0 answers

calculation on Ito's Lemma

I have a question on the calculation on Ito's Lemma: ${{Y}_{t}}={{t}^{{{W}_{t}}}}$ solve for $d{{Y}_{t}}$ the following is my solution [\begin{align} & d{{Y}_{t}}=\frac{\partial Y}{\partial t}dt+\frac{\partial Y}{\partial…
gW xa
  • 11
1
vote
1 answer

proof of Feynman–Kac formula

the article given by wikipedia http://en.wikipedia.org/wiki/Feynman%E2%80%93Kac_formula#Proof states at some point of the proof that: (line 7) ''the third term is o(dtdu) and can be dropped'' Can anyone see why the cross-variation term can be…
aflous
  • 563
1
vote
1 answer

Question about the exponential Martingale

Question: In general, $Z_t := \exp(\int_0^t \phi_s dBs - \frac{1}{2}\int_0^t\phi_s^2ds)$ is the exponential martingale. Is $Z_t := \exp(-\int_0^t \phi_s dBs - \frac{1}{2}\int_0^t\phi_s^2ds)$ also an exponential martingale? Origin: Let $(B_t)_{t \geq…
Oskar
  • 812
1
vote
1 answer

Find the Itô representation of the following random variable

Let $(B_t)_{t \geq 0}$ be a Brownian motion and $T \geq 0$. Find a constant $z_T \in \mathbb{R}$ and $(s,\omega) \in \mathcal{V}([0,T])$ such that $F_T(\omega) = z_T + \int_{0}^{T} \phi_T(s,\omega)dB_s(\omega)$ when $F_T(\omega) = \int_0^T…
Oskar
  • 812
1
vote
0 answers

Quadratic Covariation of $X_t-[X]_t$

Let $X_t$ be a continuous semimartingale. I want to prove that $[X-[X]]_t = [X]_t$. I understand that $[[X]]_t = 0$. In my script, it says that $$[X-a[X]]_t = [X]_t-2a[X,[X]]_t+a^2[X]_t = [X]_t-2a[X,[X]]_t$$ for all $a>0$. Then it is argued that…
Rooibos
  • 179
1
vote
2 answers

Calculating covariance, with multiplication by stochastic variable.

As an exercise I'm supposed to calculate; $\text{cov}(X \cdot Y,X)$, where $X$ and $Y$ are independent discrete stochastic variables, with probability functions given by; $$ p\left(var\right) = \left\{ \begin{array}{ll} 0.1 & \text{ if } var = 0…
Skeen
  • 141
1
vote
1 answer

Proving that a discrete stochastic variable is binomial distributed.

Given a discrete stochastic variables, with the probability function; $$ p_{X}\left(x\right)=\left\{ \begin{array}{cc} \frac{1}{4} & \text{if } x = -1 \\ \frac{1}{4} & \text{if } x = 0 \\ \frac{1}{2} & \text{if } x = 1 \\ 0 &…
Skeen
  • 141
1
vote
1 answer

Dynkin's formula, $dX_{t}=b(X_{t})\,dt+\sigma (X_{t})\,dB_{t}$ or $dX_{t}=b(t, X_{t})\,dt+\sigma (t, X_{t})\,dB_{t}$?

Wiki page of Dynkin's formla says: Let $X$ be the $R^n$-valued Itō diffusion solving the stochastic differential equation $$\mathrm {d} X_{t}=b(X_{t})\,\mathrm {d} t+\sigma (X_{t})\,\mathrm{d} B_{t}.$$ Let $A$ be the infinitesimal generator of $X$,…
athos
  • 5,177
1
vote
1 answer

Basic application of Ito's formula during a proof

On pg 61 of Diffusions, Markov Processes and Martingales Volume 2 by Rogers and Williams, a proof of Ito's formula for continuous semimartingales involves demonstrating that the space of functions for which it applies is a multiplicative algebra.…
Christian Chapman
  • 4,763
1
vote
0 answers

Making an ansatz for solving a SDE

I have some struggles regarding solving SDEs. I know how the machinery works and all that, I just don't know how to make an ansatz when starting to solve a SDE. For example when solving the Langevin SDE it is used that the starting solution/ansatz…
Gaussen
  • 81
1
vote
0 answers

(reference request) Rigorous proof of Ito formula for time dependent function .

Could someone provide rigorous proof of Ito formula for time dependent function $f(t,x)\in C^{1,2}$ ? I can find some proofs in the books but they are for functions of the form $f(x)$ of $f(t,B_t)$, but not for $f(t,X_t)$, where $X_t$ is Ito…
Anton Sorokovskiy
  • 160
1
vote
1 answer

Does a Stochastic process integrated over the time has to be adapted?

I would like to know that: Is it necessary that a stochastic process $X$ has to be adapted to some filtration $\{\mathcal{F}_t|t\in[0,T]\}$ for define the next integral? \begin{equation} \int_0^TX_tdt \end{equation} or it is possible to calculate…
Don P.
  • 313
1
vote
0 answers

Problem about Ito isometry

The following question is from my professor, I have no clue about it at all. Let $(W (t))_{t\geq0}$ denote a standard Brownian motion. Consider a model of asset price given as $$\mathrm{d} S(t)=\mu S(t) \mathrm{d} t+\sigma(t) S(t) \mathrm{d} W(t),…
Ito_lover
  • 11
1
vote
1 answer

Itô's lemma on function composition

I have a question on how to apply Itô's lemma on a function composition: Let us consider a stochastic process $$ dX_t = a(X_t,t)dt+ b(X_t,t)dW_t, $$ where $W_t$ denotes the standard Brownian motion. If we know the function $Y=Y(X_t,t)$, we can…
user9865
  • 11
  • 2
1
vote
1 answer

Change stochastic integral on a null set

This is a bit vague but something I have been wondering about. Consider a Lebesgue integral $\int f \;\text{d}\mu$ - then we know that we can change $f$ on a $\mu$-null set without changing the integral. Now consider a stochastic integral with…
htd
  • 1,764
1 2 3
…
14 15