Questions tagged [stochastic-differential-equations]

Stochastic differential equations (SDE) occur where a system described by differential equations is influenced by random noise. Stochastic differential equations are used in finance (interest rate, stock prices, …), biology (population, epidemics, …), physics (particles in fluids, thermal noise, …), and control and signal processing (controller, filtering, …).

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Matrix-valued stochastic DE to a vector-valued stochastic DE

Suppose we have a matrix-valued ODE: $$ i\tag{1} \frac{d}{dt}U(t) = HU(t) $$ where $U(t)$ and $H$ are complex square matrices. Given a time-independent vector $\psi$, we can convert (1) into a vector-...
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"Generalized" Geometric Brownian Motion as a SDE system

It is very well known that the equation $$d X_t = \mu X_t dt+\sigma X_tdW_t$$ has a solution $$X_t = X_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma W_t},$$ and we say that $X_t$ follows Geometric Brownian ...
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Mass matrix in underdamped Langevin dynamics and convergence

I have two doubts regarding the underdamped Langevin dynamic. The first one is that I noticed that the underdamped Langevin dynamic can be written in the following two ways: \begin{align}\label{eq:...
2 votes
1 answer
65 views

Comparison of Solutions to SDEs

Let $B(t)$ be a Brownian motion. If we assume that $b(t, x), b_i(t, x)$, $\sigma(t, x)$, and $\sigma_i(t, x)$ are Lipschitz functions for $i = 1, 2$ then it is known that if $X_i(t)$ is the unique ...
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Exact solution to a system of Ito SDE's in order to study convergence behaviour

I'm working on a simple stock pricing model described by the following model of Ito SDE's: $$ d S_t = \mu S_t dt + \sigma_t S_t dB^1_t \\ d \sigma_t = -(\sigma_t - \xi_t)dt + p \sigma_t dB^2 _t \\ d \...
2 votes
1 answer
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Brownian motion and geometric series convergence

I'm confused with the property of Brownian motion that it is of infinite variation. Consider the sequence $$\sum_{i=0}^{t-1} a^{t-i} (B_{i+1} - B_i)$$ where $a \in [0,1)$ and $B_t$ is Brownian motion. ...
1 vote
1 answer
42 views

Time-dependent drift in an SDE

Given an SDE $$dX_t = b(t,X_t)dt + dZ_t$$ where $Z_t$ is a Levy process. I am curious about the infinitesimal generator of this process. If the SDE was say $$dX_t = b(X_t)dt + dW_t$$ where $W_t$ was ...
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1 vote
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Bounds for the average solution of a stochastic differential equation

I have a general question regarding SPDE, but maybe even for SDE. Say for instance I have a Stochastic Burger Equation indexed by a constant $\epsilon>0$ and a variance function $\sigma_\epsilon$: $...
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29 views

Integrating exponential of Brownian motion wrt to time

Can you please explain which formula I have to use to integrate the exponential of Brownian motion w.r.t to time: $$\int_0^t e^{uB_s} ds$$ where $B_t$ is Brownian motion and $u$ is a real positive ...
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Approximating a stochastic difference equation with a stochastic differential equation.

I have a stochastic difference equation of the form, $$ x_t = x_{t-1} + \frac{1}{t}(- x_{t-1} + \varepsilon_t) \,, $$ where $\varepsilon_t$ are iid random normal variables, which I would like to ...
4 votes
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Why does Brownian motion cluster around singularities of the potential?

Suppose I give you the potential plotted on the left (with toroidal boundaries). On the right, I've plotted the associated Gibbs measure, which is how I'd naively expect a Brownian particle to spend ...
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Ito formula for non $C^{2}$ function

In the book Brownian motion and Stochastic Calculus by Karatzas and Shreve, page 215, there is a problem that says the following: Let $a_{1}, ..., a_{n}$ be real numbers and denote $D=\lbrace a_{1}, .....
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How to choose $g(t,x) $ when solving SDEs using Itô formula

I'm learning to solve SDEs through several examples and I understand how to use Ito's lemma once I have defined $ g(t,x)$. Nevertheless, I can't seem to find a way to derive the function $g(t,x)$. In ...
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58 views

Stochastic Integrals over infinite time horizon

The stochastic Integral w.r.t Semimartingales is defined for predictable Processes $\phi$ where $\int_{0}^T \phi_t dt < \infty$. How would I extend the Definition, when I want to integrate over $[0,...
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Are these conditions sufficient for the process to be continuous?

The text book I’m working on considers the following SDE: $dX(t) = \mu(t)dt + \sigma(t)dW(t)$, where $\mu$ is defined to be a càdlàg predictable and finite variation process, while $\sigma$ is a ...
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Pushforward of Gaussian Measure by Solution Operator to SDE

Consider an n-dimensional Ito process $$ X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dW(s) $$ driven by an n-dimensional Brownian motion; where $\alpha,\beta$ are Lipschitz functions with ...
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Optimal Consumption Verification Theorem

I want to solve an optimal consumption problem for a infinite time horizon in a jump diffusion market. How ever i couldn't find a Verification theorem with proof so far. Does anybody have good ...
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Verify Langevin equation

Consider the Langevin equation: $$dX_t = -bX_tdt +adB_t, ~~~ X_0 = x_0,$$ where $a,b>0$. We know that the solution is $$X_t = e^{-bt}x_0 + ae^{-bt} \int_0^t e^{bs} dB_s.$$ Now I want to verify the ...
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Calculating the unconditional expectation of an Ito process

Suppose that: $$dX_t =\mu(t,X_t)dt+\sigma(t,X_t)dW_t,$$ where $X_t$ is a vector valued stochastic process, $W_t$ is a vector of Brownian motions, $\mu$ is a vector valued function and $\sigma$ is a ...
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Stochastic differential equation and generalization to more dimensions

Given deterministic functions of one variable $f(t)$ and $g(t)$, the stochastic differential equation $$ dX= f(t) X dt + g(t) X dW $$ has the following solution: $$ X(t) = \exp \left \{ \int_0^t f(s) -...
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Convergence of an integral of a stochastic process

I am not a mathematician so I apologize for the sloppy language in advance. I am dealing with a random variable $z(t)=\int_{0}^{t} r(\tau) d\tau$ where $r(t)$ is some hitherto unknown random variable. ...
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Is this a convention? Correlated Brownian Motions in SDE

Sometimes I see the following in certain papers for SDE in $\mathbb{R}^n$: $$dX = \mu dt + \sigma dB$$ But they specify $\mathbb{E}[B^i B^j] = D^{ij} \neq \delta^{ij}$ for some symmetric matrix $D$. I ...
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Differential Notation of SDE with Random measures

I often see in Papers SDE of the form: $dX(t)=X(t-)\alpha dt+ X(t-)\beta dW(t)+X(t-)\int_{E}\eta(e)N(dt,de)$ where $N(dt,de)$ is a Poisson Random Measure on $[0,T]\times E$. Does the last term mean $\...
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Evaluating multiple integrals after Itô-Taylor expansion

Consider an autonomous, scalar stochastic differential equation (SDE): $$ d[X(t)] = f[X(t)]\textrm{d}t + g[X(t)]\textrm{d}W(t) $$ Consider also a scalar function $U[X(t)]$ of the solution of the SDE. ...
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Solving Stochastic differential equation example

can sb please help me with these stochastic differential equations? "W" is the Wiener process or the Brownian motion. dX = Xdt + dW dX = (X+t)dt + 2dW dX = (X+2t)dt + tdW dX = (X+1)dt + (...
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Solving chemical master equation using Kolmogorov system

I am recently studying about chemical reaction networks. And I have problem about application of Kolmogorov system to actual examples. So, here's a chemical reaction $S_1 + S_2 \rightarrow S_3$ with $...
2 votes
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75 views

Find probability distribution of stopping time consisting of a two-sided barrior and a time constrained for Brownian motion with drift?

For the following Brownian motion with drift $X_t = X_0 + \mu t + \sigma B_t$ where $\mu \in \mathbb{R}$, $ \sigma > 0$ and $X_0 \in (a,b)$ which is a solution to the stochastic differential ...
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47 views

Existence Uniqueness theorem for SDEs

Is there a uniqueness and existence result for solutions to systems of Stochastic Differential Equations of diffusion type where coefficients are $\mathcal F_t$-measurable, i.e., could depend on the ...
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1 answer
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Calculating different expected values of a normal distribution

Be $X ∼ N$ $(µ, σ^2)$ a normally distributed random variable on a probability space $(Ω, \cal F, \cal P)$ Can someone help me to calculate the following expected values: i) $\mathbb E[X^{2k+1}]$ for $...
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1 answer
37 views

How to make program for five realizations of ito process X = exp(t + 0.2W(t)) [closed]

I want to plot the following figures in octave. I prepared the following program to plot one realization of ito process X =exp(t + 0.2W(t)) where W(t) is a Wiener process= $ \displaystyle\sum_{j=0}^{\...
2 votes
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67 views

Stochastic Lorenz model

Consider Lorenz model $$ \begin{align*} \frac{dx}{dt}&=\sigma(y-x)\\ \frac{dy}{dt}&=\rho x-y-xz\\ \frac{dz}{dt}&=xy-\beta z \end{align*} $$ with $\sigma=10$, $\rho=28$ and $\beta=\frac{8}{...
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1 answer
83 views

SDE driven by Poisson Process

Suppose that $(N_t)_{t\in \mathbb{R}^+}$ is a Poisson process with intensity $\lambda$>0 and that $a\in\mathbb{R}$ and $X$ being a stochastic process which solves the following SDE:$$dX_t=aX_t^-...
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Reference for Mckean-Vlasov stochastic differential equations

Could someone please provide me with some reference books on nonlinear stochastic differential equations in the sense of McKean-Vlasov, the propagation of chaos in corresponding systems of particles, ...
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Numerical schemes for simulating path dependent SDEs

Background: The Euler-Maruyama scheme for SDEs of the form $$dX_t = \mu(t, X_t) dt + \sigma(t, X_t) dB_t,$$ is well studied and easy to implement. It is given by $$X_{t_i}=X_{t_{i-1}}+\mu(t_{i-1}, X_{...
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2 votes
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Does this class of stochastic differential equations exist a unique solution?

There is a stochastic differential equations: $$dX_t=a(t,X_t)dB_t+b(t,X_t)dt\quad t\in[0,T]$$ where $B_t$ is the one-dimensional Brownian motion.The "a" and "b" are functions: $$a,...
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Ito's formula and Feynman Kac formula for path-dependent SDEs

In Rogers and Williams Volume 2, Chapter V Section 2 Subsection 8, we have introduced there the notion of a general form of SDEs where the coefficients are taken to be previsible path functionals: $$...
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A SDE depending on its running maximum

Let $X_t$ be a process which satisfies the SDE $$dX_t=(aX_t+bM_t)dt+M_tdB_t$$ where $a,b$ are constants, $B$ is a standard Brownian motion and $M_t=\sup_{s\leq t} X_s$. Since the coefficients are ...
3 votes
1 answer
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Understanding HJB equation for the infinite horizon consumption control problem

Context Given the following maximization problem as well as wealth dynamics $$\max\mathbb{E}\left[\int_0^{\infty} \frac{1}{\gamma} e^{-\beta t} c_t^\gamma \mathrm{d} t\right]$$ $$\mathrm{d} X_t=X_t\...
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Ito rule in backward difference integrals

While defining the Ito integral, we generally take forward difference. Then we go on to prove many properties, one of them being Ito rule. Formally, we could write the Ito rule as $$ df(X_t) = f'(X_t)...
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Definition of stochastic differential equations in infinite dimensions without traceclass

In infinite dimensions, the main idea to define a stochastic differential equation is to consider a trace class (cylindrical) Brownian motion. However, I've been wondering whether there exists a body ...
3 votes
1 answer
92 views

Ito drift term of extrinsic Brownian motion aka Mean curvature of intersection of hypersurfaces

Suppose we have manifold in the form $M=f^{-1}(\{\vec{0}\})$, where $f:\mathbb{R}^d\to \mathbb{R}^p$ where $p=D-d$, and $f \in C^{\infty}$ ands its Jacobian, $J_f(x)$ has full rank on $M$. Here, we ...
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Estimates of supremum distance between diffusion and a given curve

Let $g\in C^1[0,1]$ be a given smooth curve. Consider a random path $X\in C[0,1]$ that solves the SDE $$dX_t = \mu(t) dt+\sigma dB_t,$$ i.e. $$X_t = X_0+\int_0^t \mu(s) ds +\sigma B_t.$$ We will be ...
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1 answer
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Textbook definition for path measure /probability measure over paths

I need a formal definition for the path measure for stochastic differential equations. Which textbook or paper should I consult?
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49 views

Non-trivial examples of mapping between SDEs and PDEs?

The Feynman-Kac formula (or Kolmogorov backward equation) describes a link between SDEs and PDEs. This answer introduces Ricci flows on Riemann manifolds which also seems to provide a mapping between ...
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Product rule for multi-dimensional Ito processes

Suppose we have d-dimensional Ito process $X_t$ with: $dX_t = b(t, X_t)dt + \sigma(t, X_t)dW_t$, where $W_t$ is d-dimensional standard Brownian motion. And one-dimensional Ito process $Y_t = f(X_t)$: $...
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Solving SDE $dx_t = (A - a x_t) dt + (b) dZ_t$

I am new to stochastic differential equations. I would like to solve something like this: $dx_t = (A - a x_t) dt + (b) dZ_t$ where: $A = \frac{ - \delta k }{\delta + a} $ The solution is: $x_t = e^{-...
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1 answer
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Solve the SDE: $dX = \cos(X)\sin^3(X)dt +\sin^2(X)dW$

Solve the SDE: $$ dX = \cos(X)\sin^3(X)dt +\sin^2(X)dW $$ where $W$ is a Brownian motion. The SDE has initial condition $X(0) = \frac{\pi}{2}$. I am given the hint that $\int dx/\sin^2(x) = -\cot(x) + ...
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Cross Covariance Matrix of Multidimensional Ornstein-Uhlenbeck Processes

The multivariate Ornstein–Uhlenbeck process is defined as the following \begin{equation} dX(t) = - I_p X(t) dt + \sqrt{2}I_p dW(t) \end{equation} where $I_p$ is an $p \times p$ identity matrix, ...
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1 answer
34 views

Variance of Wiener processes in Geometric Brownian Motion

The analytical solution to the Geometric Brownian Motion (GBM) SDE is given by $ S_t = S_0 \exp( (\mu - \frac{\sigma^2}{2})t + \sigma W_t ) $ where $W_t$ is a Wiener process. One of the properties of ...
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22 views

how to proof the stability of the solution of this SDE equation

I want to proof the stability of the following SDE equation: $$dx_t=(1-clnX_t)X_tdt+\sigma X_t dW_t$$ with the solution in the form $$ X_t=\exp\left[\frac{1}{c}+(\ln(x_0)-\frac{1}{c})e^{-ct}+\sigma\...

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