The process is given by $$dU_t=-\gamma U_t\mathrm{d}t+\sigma\mathrm{d}X_t$$
where $U_0 = u$ and $\gamma, \sigma$ are constants. Can you help me out to solve the equation for $U_t$ and find the expectation and variance?
Thank you.
2026-03-26 01:07:22.1774487242
Stochastic Differential equation, expectation and variance
424 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ORDINARY-DIFFERENTIAL-EQUATIONS
- The Runge-Kutta method for a system of equations
- Analytical solution of a nonlinear ordinary differential equation
- Stability of system of ordinary nonlinear differential equations
- Maximal interval of existence of the IVP
- Power series solution of $y''+e^xy' - y=0$
- Change of variables in a differential equation
- Dimension of solution space of homogeneous differential equation, proof
- Solve the initial value problem $x^2y'+y(x-y)=0$
- Stability of system of parameters $\kappa, \lambda$ when there is a zero eigenvalue
- Derive an equation with Faraday's law
Related Questions in STOCHASTIC-PROCESSES
- Interpreting stationary distribution $P_{\infty}(X,V)$ of a random process
- Probability being in the same state
- Random variables coincide
- Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.
- Why does there exists a random variable $x^n(t,\omega')$ such that $x_{k_r}^n$ converges to it
- Compute the covariance of $W_t$ and $B_t=\int_0^t\mathrm{sgn}(W)dW$, for a Brownian motion $W$
- Why has $\sup_{s \in (0,t)} B_s$ the same distribution as $\sup_{s \in (0,t)} B_s-B_t$ for a Brownian motion $(B_t)_{t \geq 0}$?
- What is the name of the operation where a sequence of RV's form the parameters for the subsequent one?
- Markov property vs. transition function
- Variance of the integral of a stochastic process multiplied by a weighting function
Related Questions in DIFFERENTIAL
- In a directional slope field, how can a straight line be a solution to a differential equation?
- The Equation of Motion of a Snowball
- Supremum of the operator norm of Jacobian matrix
- Directional continuous derivative on vectors of a base implies differentiability in $\mathbb{R}^n$
- Need explanation for intuition behind rewriting $dy$ in terms of $dx$
- Does the double integrative of d^{2}x make sense from a mathematical point of view?
- Functional with 4th grade characteristic equation
- need to equate MATLAB and MATHEMATICA solutions
- Formula for Curvature
- Showing that $\Psi(f) = \int^{b}_{a}\phi(f(x))dx$ is differentiable.
Related Questions in STOCHASTIC-DIFFERENTIAL-EQUATIONS
- Polar Brownian motion not recovering polar Laplacian?
- Uniqueness of the parameters of an Ito process, given initial and terminal conditions
- $dX_t=\alpha X_t \,dt + \sqrt{X_t} \,dW_t, $ with $X_0=x_0,\,\alpha,\sigma>0.$ Compute $E[X_t] $ and $E[Y]$ for $Y=\lim_{t\to\infty}e^{-\alpha t}X_t$
- Initial Distribution of Stochastic Differential Equations
- (In)dependence of solutions to certain SDEs
- Expectation, supremum and convergence.
- Integral of a sum dependent on the variable of integration
- Solving of enhanced Hull-White $dX_t = \frac{e^t-X_t}{t-2}dt + tdW_t$
- Closed form of a SDE
- Matricial form of multidimensional GBM
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
You have $$dU_t + \gamma U_t dt = \sigma X_t$$ which is $$d(e^{\gamma t}U_t) = e^{\gamma t}\sigma dX_t$$ So $$e^{\gamma t}U_t - e^{\gamma 0}U_0 = \int_0^t e^{\gamma s}\sigma dX_s$$ i.e. $$U_t = e^{-\gamma t}U_0 + \int_0^t e^{\gamma (s-t)}\sigma dX_s$$
Can you go from here?