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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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Stochastic differential equation drive by dependent Brownian Motions

Let $b : [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^n, \sigma : [0,T] \times \mathbb{R}^n \rightarrow \mathbb{R}^{n \times m}.$ Consider an SDE $$ dX_t = b(t,X_t)dt + \sigma(t,X_t) dW_t$$ Then a solution exists if the coefficients satisfy…
White
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Is $$ when $r_t$ stochastic?

Let $D_t=e^{-\int_{0}^{t}r(s)ds}$ with $r_t$ being a stochastic process. Set $X_t = -\int_{0}^{t}r(s)ds$ By Ito : $$ \frac{dD_t}{D_t} = dX_t+\frac{1}{2} _t $$ $$ =-r_tdt+\frac{1}{2} $$ My question is : What is $$…
user30614
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Existence of a weak solution of a stochastic differential equation

I am currently studying stochastic differential equations and have no glue how to prove existence of a weak solution. For instance, consider $$ \begin{cases} dX_t=dB_t+\sqrt{X_t}dt\\ X_0=0 \end{cases} $$ on a compact interval $[0,T]$. Any help is…
user427574
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What does the following notation mean: $\tau = \inf \{n\in \mathbb{N} \mid X_n \in B \} $

Let $(\Omega, \mathbb{P})$ be a probability space and $B\subseteq \Omega$ an event. Let further be $X_1,...,X_n$ random variables. What does $\tau = \inf \{n\in \mathbb{N} \mid X_n \in B \} $ mean? For usual the notation $X\in M$ for a set $M$ means…
Sudix
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Question on weak convergence of random variables

Let $X_n, Y_n, X$ be real random variables such that $X_n \to X$ weakly and $\mathbb{P}_{Y_n} = N(0, 1/n)$ for all positive integers $n$. I am trying to prove that $X_n + Y_n \to X$ weakly as well. I have been trying to prove that $E[f(Y_n)] \to 0$…
eugen1806
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Covariance of integrated Brownian bridge

How to find the Covariance of $Z(p)$ and $Z(q)$, where $$Z(p)=\int_0^p B(u)du+\int_p^1 \frac{B(u)}{1-u}du$$ and $B(u)$ is the standard Brownian Bridge on $[0,1]$. My approach is that $Z(p)=\int_0^p B(u)du+\int_p^1 \frac{B(u)}{1-u}du$ is a Gaussian…
Dhrubasish Bhattacharyya
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Deriving Stochastic differential equations

I am having a difficulty in deriving stochastic differential equations from geometric Brownian motion dynamics. Assume S follows the geometric Brownian motion dynamics, dS = μSdt + σSdZ, with μ and σ constants. Derive the stochastic differential…
Tee
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Calculate $P(X \leq 1 \cap Y \leq 1)$ and $f_{Y|X}(y|x=0)$ of a density function

I have a density function with $f(x, y) = x^3+0.25y$, $0 \leq x \leq 1, 0 \leq y \leq 2$ and for x and y out of that range it is defined as 0. First i need to calculate $P(X \leq 1 \cap Y \leq 1)$. How can i do that? The variables X and Y are not…
MagikarpSama
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Ito's formula with jump process

Suppose I have $dP_t^i = (r^i + h_i^{\mathbb{P}})P_t^i dt - P_{t-}^i dH_t^i$ where $H_i(t) = \mathbb{1}_{\tau_t \leq t}$ denotes a default indicator process of i. $\tau_i$ is the default time and $h_i$ is some constant. Let $w_t^{i, \mathbb{P}} :=…
SinusK
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Quadratic Variation

Let $$ [X,Y]=\int_{0}^{\tau} (dX_t)(dY_t)$$ where $X_t$ is cadlag process and $Y_t$ is a Brownian motion. Can we say $$ \frac{\partial}{\partial\tau}\int_{0}^{\tau} (dX_t)(dY_t)= 0 $$
Dr. Kandy Junior
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What does $\langle dW_1, d W_2\rangle = \rho$ mean?

What does $\langle dW_1, d W_2\rangle = \rho$ mean? It shows up in stochastic differential equations involving brownian motions. I am told it's the "correlation", but why, how, and what does it mean? In particular, I don't know what the scalar…
Jaood
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Determine $h$ process so that the following stochastic representation result holds

I want to find a process $h$ such that $$m(T) = Em(T) + \int_0^T h(t) dW(t), $$ where $m(T) = e^{ \int_0^T t dW(t)}$. Here, $T$ is some positive constant, and $W(t)$ is Brownian motion. I get $Em(T) = \frac{1}{6}T^3$. But then I am unsure how to…
Imean H
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stochastic differential equations in matrix notation

How is Matrix K determined? $ dY_1 = \frac{-1}{2} Y_1dt - Y_2dB_t $ $ dY_2 = \frac{-1}{2} Y_2dt - Y_1dB_t $ In matrix notation, the above equations can be written as: $ dY(t) = \frac{-1}{2}Y(t)dt + KY(t)dB_t $ where K is equal…
flyhigh1
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Limiting behavior of sde

What can we say about the limiting behavior of Xt , as t goes to infinite , where Xt is the solution of the sde $$dX(t) = e^{-t}X(t)dB(t)$$ ?
amm
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How to solve the expected value of such integral

$$ V_t = E^\mathbb{Q} \left[\int_t^{+\infty} e^{-r(u-t)}X_u \, du|X_t\right] $$ with a given process $ X_t $ satisfied: $dX_t = (\mu-\sigma^2 \gamma) X_t \, dt + \sigma X_t \, dW_t^\mathbb{Q}$
Tony
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