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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 $\mathbb{R}^n$.

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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 returns a PDF for the integral of a stochastic process…
gmb
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Variance of the Cox Ingersoll Ross model

Consider the Cox-Ingersoll-Ross (CIR) interest rate model: $\displaystyle d r_t = \kappa (\theta - r_t) \, d t + \sigma \sqrt{r_t} \,d W_t$ where $\kappa$, $\theta$, $\sigma$ are positive constants and $W_t$ is a standard Brownian motion. A solution…
Barbados
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Use of stochastic calculus outside finance?

I have noticed most of the books about stochastic calculus are targeted fo finance and derivatives. Are there any other areas outside finance where stochastic calculus is applicable?
ZeroCool
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Elementary product rule for Ito formula

In "Baxter, Rennie" book, there is an explanation of product rule for Ito formula. They apply Ito formula (without any details) to $$\frac{1}{2}((X_t + Y_t)^2 - X_t^2 - Y_t^2) = X_tY_t$$ and obtain $$d(X_tY_t) = X_t\,dY_t + Y_t\,dX_t +…
Cron Merdek
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Stochastic calculus based on non-Brownian motion, or other-than-normal distributions

Based on the book "Financial Calculus" by Baxter and Rennie, I understand that stochastic calculus is based on the equation $$dx(t)=\mu(t) dt+\sigma(t) dW(t),$$ in the same way that standard calculus has the standard…
user56834
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Ito Quadratic Variation

In Ito's lemma for a Brownian motion $B_t$ the term in $dB_t^2$ is replaced with $dt$ without any averaging. It seems that higher moments are an order $dt$ smaller and that the term $dB_t^2$ is dominated by its expectation and it becomes…
Dom
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The most general version of Ito's lemma

Wiki gives a version of the Ito's lemma for the Ito proccess when we differentiate a function $f(t,X_t)$ of time and some diffusion process. In the general case of multivariate semimartingale (possibly discontinuous) it is written for the function…
paul
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Ito's lemma 2nd order term notation.

I have a notation question here. In the simplest form of Ito's lemma, we have this $ df(Y_t) = f'(Y_t) dY_t + \frac{1}{2} f''(Y_t) d\langle Y \rangle_t$ I know how to calculate the $ d\langle Y \rangle_t $ term, but I always want to ask what is the…
Paul
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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 always a conditional density ? The density $p(x,t)$…
John
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What are the norms in Ito isometry?

Itō isometry from Wikipedia: Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical real-valued Wiener process defined up to time $T > 0$, and let $X : [0, T] \times \Omega \to \mathbb{R}$ be a stochastic process that is adapted to…
Tim
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What is wrong with this stochastic calculus derivation?

Given stochastic processes $X_t, Y_t$ of form given below, we want to find $d(X_t Y_t)$. Assume $$X_t=\sigma_tW_t + \mu_t t$$ $$Y_t=\rho_tW_t + \nu_t t$$ I have two different answers depending on how I derive them, and both of them are wrong, but I…
user56834
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Let $Z_t=\int{W_s }ds $. Show that $Z_t=\int (t-s) dW_s$

Let $Z_t=\int_{0}^{t} W_s ds$. Use integration by parts to show that $Z_t=\int_{0}^{t} (t-s) dW_s$. I have tried and i can't get the answer.
Hing Yee
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Fokker Plank EQUATION

I would be grateful if you let me know an application of Fokker plank equation in a financial market or introduce a related paper to me. For example, when the price of stocks in our market satisfiy the Black- Scholes model then the solution of the…
Sara
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What is the stochastic integral of $\frac{dW_t}{W_t}$

Does anyone know the solution to the Ito integral with the scaling factor on $dW_t$ being $\frac{1}{w_t}$? In other words what is: $\int \frac{dW_t}{W_t}$ ? It looks dangerously close to what mathemattical finance people do when they look at…
foobar
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Independence of random variable and sum of iid random variables

Let $T_n=\sum_{i=1}^{n} X_i$ and $\{ X_i \} $ be a sequence of i.i.d. (strictly) positive random variables. So I know that $X_{n+1}$ is independent of $X_1,...,X_n$. Futher we have $T_{n+1}=T_n+X_{n+1}$. Is it then true, that $X_{n+1}$ is also…
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