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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9 views

Convergence of an Ornstein-Uhlenbeck Process

Given the stochastic differential equation: $dx=-xdt+dW$ and the initial condition $\left( t_{0},x_{0}\right) $, the solution trajectory $x\left( t;t_{0},x_{0}\right) $ can be derived by ...
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17 views

How to compute the quadratic variation of $\langle X,Y \rangle_t$?

Let $(\Omega,(\mathcal{F}_t)_{t≥0}, \mathcal{P})$ be a filtered probability space. Let $(B_1(t), B_2(t))_{t≥0}$ be a two-dimensional Brownian motion. Let $ θ\in \mathbb{R}$ and let $$ X_t^θ = B_1(t) \...
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17 views

Does Causation imply dependency in the Euler scheme discretization for SDEs?

I have a confusion concerning the following statement: Assume we have n-observations $(X_0,\dots,X_n)$ that are coming from a discretization of an SDE using the following Euler scheme : $$ X_i=X_{i-1}...
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15 views

what linear stochastic equation will fractional Brownian motion satisfy? [closed]

Is there a linear stochastic equation driven by Brownian motion of which solution is the fractional Brownian motion?
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1answer
24 views

Existence and uniqueness for SDE 2D linear system

I have the following SDE $$\ddot{x} +x = \dot{B(t)}$$ with some given initial condition $(x_0,\dot{x_0})$ and where $B(t)$ is a standard Brownian motion. I can reduce it to first order by ...
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10 views

Which of these equations is the backward Kolmogorov Equation?

I am reading Stochastic Processes and Applications by Pavliotis. At some point he defines Kolmogorov backward equation as the differential equations $$\frac{\partial u}{\partial t} = \mathcal{L}u$$ $$...
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29 views

Finding the drift coefficient

$X_t$ is a solution of SDL $$dX_t=b(X_t)dt+X_tdB_t.$$ What is the the drift coefficient $b(x)$ of this equation, if the square of its solution $$M_t=X^2_t, t\geq0,$$ is a martingale? So far, I ...
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55 views

Solve following SDE

Any ideas on how to solve the following SDE $$X_t=\int_0^te^{-X_s}dW_s+\frac{1}{2}\int_0^te^{-2X_s}ds$$ It does not appear there is a strong solution, but what about weak solution? Consider the ...
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1answer
19 views

Discrepancy between initial value and closed form solution of linear 2D SDE

Kloeden Platen Schurz 1994 states the linear 2D SDE: $$dX_t = A X_t dt + B X_t dW_t$$ where $a=5$ and $b=0.01$ and $$A = \Big( \begin{matrix} -a & a \\ a & -a \end{matrix} \Big)$$ $$B = \...
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1answer
28 views

Variance of $n$-dimensional Brownian motion (Oksendal)

I apologize if this is a trivial question but I am confused by eq. $2.2.9$ in Oksendal (see image below). If $B_t$ is $n$-dimensional Brownian motion then isn't $B_t\sim\mathcal N(\mathbf x,tI_n)$ a $...
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1answer
21 views

Preserving symplectic structure iff matrix has unitary determinant

Let us consider a linear stochastic oscillator system: $$ d \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} x(t) \\y(t)\end{bmatrix}...
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41 views

Multi-factor Ito Lemma & Girsanov Theorem

Assume $X(t)=X(0)+\int^{t}_{0}\mu X(h)dh+\int^{t}_{0}\sigma_1 X(h)dW_1(h)+\int^{t}_{0}\sigma_2 X(h)dW_2(h)$ FIRST PART OF THE QUESTION: Assume first that $\mu$ and $\sigma_1$ & $\sigma_2$ are ...
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1answer
110 views
+100

Show that a certain solution of an ODE is a stochastic process.

I am working on an exercise as follows: Let $v_{1}, v_{2}:\mathbb{R}^{2}\longrightarrow\mathbb{R}^{2}$ be $C^{\infty}$ vector fields over $\mathbb{R}^{2}$. Let $(X_{n})_{n\in\mathbb{N}}$ be an i.i....
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15 views

How can I prove that the solution to this SDE is a Markov process?

Consider the two-dimensional SDE \begin{align} dS_t &= \mu S_t dt + S_t \sqrt{V_t} \big(\rho dW_t^{(1)}+\sqrt{1-\rho^2}dW_t^{(2)} \big); \tag*{(1)} \\ dV_t &= \kappa(\theta - V_t)dt + \sigma ...
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26 views

$\int_0^t e^{a\cdot s + b\cdot W_s}\, ds$

I'm trying to solve the following integral, $$\int_0^t e^{a\cdot s + b\cdot W_s}\, ds $$ where $W_s$ is a standard brownian motion and $a$ and $b$ are constants. I've thought about using integration ...
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1answer
31 views

Solve the SDE $dX_t=dB_t+\frac{c-X_t}{T-t}dt$

How to solve the following SDE characterizing Brownian motion with fixed end point $c$ at time $T$? $$dX_t=dB_t+\frac{c-X_t}{T-t}dt$$ I do not believe a strong solution exists, by ito lemma it would ...
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27 views

Stochastic differential equation form

A few days ago, I disccussed my Phd thesis (thesis defense), one of the mathematical mistakes that the committe members alerted me to it, is how to write correctley an ordinary differential equation ...
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22 views

What is a Markov kernel of a Euler-Maruyama discretization?

Let $B_t$ be a $d-$dimensional Gaussian process and we consider being given the SDE, $dY_t = - \nabla U(Y_t)dt + \sqrt{2}dB_t$. A Euler-Maruyama discretization of this will look like the Markov Chain, ...
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12 views

Steady-state distribution for a 2-dimensional SDE

Consider the following 2-D SDE \begin{align*} \dot x_1(t) &= \frac{1}{1+x_2(t)}+x_1(t) + w_1(t)\\ \dot x_2(t) &= \frac{1}{1+x_1(t)}+x_2(t) + w_2(t) \end{align*} where $w_1(t)$ and $w_2(t)$ are ...
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13 views

What's the role of commutation relations in stochastic mechanics?

In a stochastic context, we can understand a term like $$ \int_0^T \frac{d q(t)}{dt} dq $$ either as the (Ito) limit $$ \lim_{N\to\infty} \sum_{i}^N dq(t_i) \frac{d q(t_i)}{dt} $$ or the (Anti-...
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40 views

quadratic covariation of SDE

consider the following SDE for the 2-dimensional stochastic processes $ X_t=( ( X_1(t) ),( X_2(t) ) ) $ driven by a 2-dimensional continuous standard Brownian motion $ B_t=( ( B_1(t) ),( B_2(t) ) ) $ ...
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32 views

Solution to the SDE

i'm confused with the first integral (not the stochastic part) of following SDE: $$dX_t = (X_t - X_t^3)dt + \int kX_t dB_t.$$ The solution can be written as $$X_t = x_0 + \int_0^{t} X_s - X_s^3 ds + \...
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1answer
38 views

Show that $\left(\phi\left(X_{t}\right)\right)_{t \geq 0}$ is a martingale

For some point $x \in \mathbb{R},$ we consider the SDE: $$ \forall t \geq 0, \quad X_{t}=x+\int_{0}^{t} \sin \left(2 X_{s}\right) d s+\int_{0}^{t}\left(1+\cos ^{2}\left(X_{s}\right)\right)^{1 / 2} d ...
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0answers
20 views

is $\int_{0}^{t} dB_s = B_t$?

$(B_t)_{t \geq 0}$ is a standard Brownian motion, first of all I know that they are equal in distribution but as I was solving the Geometric Brownian motion SDE I got a solution of the form : $$X_t =...
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1answer
25 views

Why is the expectation value of a stochastic integral equal to $\sum_k (\tau_k - t_k)$ where $\tau_k$ denotes the partitioning points?

In the book "Handbook of Stochastic Methods for Physics" by Crispin and Gardiner, I found the following calculation to show that stochastic integrals depend on the choice of partitioning points. ...
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1answer
50 views

Differential equation to Ito equation

I have a differential equation: $$\frac{dX}{dt}=f(X)+\epsilon X n(t)$$ where $f$ is a deterministic function, $\epsilon$ is a constant, $n(t)$ is a white Gaussian noise and $X$ is a random process. I ...
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2answers
67 views

Why do stochastic integrals depend on the choice of partitioning points?

When we integrate a function, we must make some choice about how we approximate it before we take the limit. In principle, we can choose $\tau_i$ to be any value between $t_{i-1}$ and $t_i$. But for ...
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0answers
22 views

Centre the solution of an SDE

In the following $W_t$ is a Brownian motion say in $\mathbb{R}$ or $\mathbb{R}^d$ and $y_0$ is deterministic. If you have an SDE say $Y_t\in \mathbb{R}^d$ $$ dY_t=b(Y_t)dt+dW_t,~~Y_0=y_0$$ you can ...
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1answer
18 views

how to compute the cross variation process here?

let $C = (C_t = C_0 e^{\alpha W_t})_{t\geq 0}$ where $C_0 \geq 0$ (constant) and $W_t$ is a standard Brownian motion let $X_t$ be a stochastic process such that : $dX_t = \mu X_t dt + \sigma X_t d ...
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12 views

Solution of a Simple Discrete-time SDE

I'm looking for the solution of the following SDE: $p_{t+1} = a \cdot p_t + b + c \cdot w_t$, where $a, b$ and $c$ are constant scalars, $w_t \sim N(0, \sigma^2)$ is a white noise and $p_t$ is ...
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0answers
29 views

Solve $dX_t=(a+bW_t)dt+cdW_t.$

Question: If $a,b,c$ are constants and $W_t$ is a Brownian motion, solve the following SDE $$dX_t=(a+bW_t)dt+cdW_t.$$ First of all, this is not a linear SDE due to presence of $W_t$. I attempted ...
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0answers
28 views

How to solve the following SDE: $\mathrm{d}X_t = a(b-X_t) \mathrm{d}t + c X_t \mathrm{d}W_t$

Disclaimer: I am a total beginner in stochastic calculus Let $\mathrm{d}X_t = a(b-X_t) \,\mathrm{d}t + c X_t \, \mathrm{d}W_t$ be a stochastic differential equation (this SDE is the variance ...
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22 views

Prerequisite for “Limit Theorems for Stochastic Processes”

Just wondering if anyone had read this book: "Limit Theorems for Stochastic Processes" by Albert Shiryaev and Jean Jacod. For one, I know this book is not at beginner level. I have introductory to ...
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132 views

First hitting time density of Ornstein-Uhlenbeck model

Let $x(t)$ be an standard Wienner process defined by the SDE: $$dx(t) = \mu dt + \sigma \, dW $$ where the constants $\mu$ ,$\lambda$ and $\sigma$ are non-negative and constants. Let be $\tau$ the ...
3
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0answers
67 views

PDE for holding a stock

Let's suppose we owe an asset and its path is given by the following SDE: $$ S(t) = A(S, t) dt + B(S, t) d\widetilde{W}$$ where $\widetilde{W}$ is a Brownian motion (under risk neutral measure). What ...
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0answers
21 views

Moments of Multivariate SDEs

Consider the following SDE: $$ \mathbf{y}_{t} = \mathbf{F}_{t,s} \mathbf{x}_{s} + \int_{s}^{t} \mathbf{B}_{t,u} d\mathbf{W}_{u} $$ where the subscripts indicate dependence on $t$, $s$, and $u$ ...
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1answer
42 views

What sort of qualitative behaviour does a stock following a process of the form $dS_t=α(μ-S_t )dt+S_t \sigma dW_t$ exhibit?

The following is a question taken from Mark Joshi's Concepts and Practice of Mathematical Finance, second edition, Exercise $5.5$ Question: What sort of qualitative behaviour does a stock following ...
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0answers
39 views

Why do we use $\mathbb P^x(X_t^0\in A)$ to denote $\mathbb P(X_t^x\in A)$.

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. Consider the following SDE $$dX_t=\mu(X_t)dt+\sigma (X_t)dB_t.\tag{E}$$ Suppose that $\mu$ and $\sigma $ are nice enough so that it has a ...
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32 views

Derive $\frac{d R_{xx}(\tau)}{d\tau} = -\alpha R_{xx}(\tau)$ from the SDE $dX{t} = -\alpha X_{t} dt + \beta dW_{t}$

I want to derive the following relation: $$\frac{d R_{xx}(\tau)}{d\tau} = -\alpha R_{xx}(\tau)$$ from the following SDE: $$ dX{t} = -\alpha X_{t} dt + \beta dW_{t} $$ Motivation: In the paper, "...
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2answers
67 views

Find the covariance of $x_{t} = x_{0}e^{-\alpha t} + \rho \int_{0}^{t} e^{-\alpha(t-s)}dW_{s}$

I want to find the covariance of the following SDE: $$x_{t} = x_{0}e^{-\alpha t} + \rho \int_{0}^{t} e^{-\alpha(t-s)}dW_{s}$$ To start, I find the mean, it is simply: $$ E[x_{t}] = E[x_{0}] e^{-\...
2
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1answer
77 views

Prove that portfolio is self financing?

Let assume we have a portfolio with strategy described by $θ_t = \int_0^t S_udu$ (position in stock) and $ψ_t = -\int_0^t \frac{S_u^2}{B_u}du$ (position in bond). How to prove that this strategy is ...
4
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1answer
64 views

Show that a SDE has no solution

Let $B$ be a standard Brownian motion. I want to prove that the SDE \begin{equation} X_t = \int_0^t 1_{\{ X_s \geq 0 \}} \, dB_s \end{equation} has no solution on any set-up. A hint I was given was ...
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0answers
15 views

Comparison theorem for the following linear SDE

Let $W$ be a two-dimensional Brownian motion. Let $A^a$ and $A^b$ be two adapted (w.r.t the natural filtration of $W$), cadlag, non-decreasing processes with the convention that $A^a_{0-}=A^b_{0-}=0$. ...
2
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1answer
89 views

Martingale representation of European option.

Let stock price $S$ satisfy $$S(t)=S(0)e^{(\int_0^t\sigma(s)dB_s-\frac{1}{2}\int_0^t\sigma(s)^2ds)}$$ I want to calculate the Martingale representation $V(t)=E(F|F_t)$ of European option with strike ...
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0answers
37 views

Sample Second Moment of SDE

Consider a simple sde $$\mathrm{d}X_t = \mu \mathrm{d}t + \sigma \mathrm{d}W_t.$$ Assuming $X_0 = x$, $$\mathbb{E}\left[X^2_t\right] = \left(x +\mu t\right)^2 + \sigma^2 t.$$ I have been seeking to ...
4
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0answers
44 views

Ito Derivative of White Noise

We know that white noise $w_{t}$ is given by the time derivative of Brownian motion $\beta_{t}$, ie that: $$ w_{t} = \frac{d \beta_{t}}{dt} $$ Now I want to define a new process, called blue noise $...
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0answers
21 views

Simulating a simple SDE (geometric Brownian motion) - problems with Wiener Process

I am trying to simulate a simple SDE $$dS = \mu Sdt + \sigma S dW_t$$ with $\mu,\sigma \in\mathbb{R}$. When I use the scheme \begin{align} S_{n+1} &= S_n + \mu h S_n + \sigma S_n N(0,h) \end{...
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1answer
75 views

multivariate Ito isometry

I wonder whether there exists a straightforward extension of the Ito isometry to multidimensional processes. In the one-dimensional case the Ito isometry can be written as $\mathbb{E}[ (\int_0^T X_t ...
1
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0answers
43 views

Deriving Stochastic Differential Equations from Autocorrelation

Suppose I know my stationary stochastic process has the following autocorrelation function: $$ R(\tau) = \sigma^{2} e^{-\alpha |\tau|} \cos(\omega \tau) $$ How can I derive the stochastic ...
1
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0answers
10 views

What resources can I read to learn how the norm of the distance to a stationary distribution evolves over time in a Fokker-Planck solution?

Let's say we have an initial distribution $p_0$, which we evolve according the Fokker-Planck equation with stationary distribution $p_\infty$ (which need not be log-concave). $$ \frac{\partial p_t}{\...

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