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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How to make sense of SDE notation

I have been trying to understand SDE notation and what it means but I cant seem to figure it out. Im sorry to post this question here, since this is probably elementary. It is known to me that $$ dX_t ...
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Non-linear backward Kolmogorov equation

The backward Kolmogorov equation (BKE): $$\frac{du}{dt} = A(x,t) \cdot \nabla_x u(x,t) -\frac12 \text{Tr}(BB^t(x,t) \text{Hess}_x u(x,t) - f(t,x,u, B,\nabla g), \;\;\; t<T$$ If $f\equiv 0$ then ...
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Compare a computer simulation (of a chemical reaction network) to an SDE

I am currently reading Anderson & Kurtz's Stochastic Analysis of Biochemical Systems. In this book, one studies chemical reaction networks and also how to simulate these with MATLAB. I know, that ...
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How to determine the reflecting horizontal hidden barrier for Ito diffusion

The first article at the link below talks about that the Bi-Directional Grid Constrained (BGC) stochastic processes for a random variable X over time t is one in which the further it departs from the ...
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When is an SDE solution differentiable in its starting value?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, $\mu,\sigma\colon [0;T]\times\mathbb R \to \mathbb R$ ...
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1 answer
34 views

Distribution of solution to SDE

Let $X_0$ be a standard normal random variable and suppose that $$dX_t=-\frac{1}{2}X_tdt+dB_t.$$ $X_0$ is independent of the Brownian motion. Find the distribution of $X_t$ for $t\geq0$ and find $\...
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Proving uniqueness of weak solution to SDE

This is the SDE: $$dX_t=\operatorname{sign}(X_t+1)dt + dB_t$$ This is a $\mathbb{Q}$-Brownian motion: $$W_t=B_t -\int_{0}^{t}\operatorname{sign}(B_s+1)ds$$ I've already shown that $B_t$ under measure $...
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Question on use of Ito's formula with integral in the function

This question is being asked in the context of the Feynman-Kac formula. Suppose the real-valued process $Z$ satisfies the SDE $$dZ_t=b(Z_t)dt+\sigma(Z_t)dW_t.$$ Suppose we have functions $f:\mathbb{R}\...
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Find the mean and the variance of $X(1)$ for stochastic differential equation: $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7$

Suppose that $X(t)$ satisfies $\hspace{5cm}$ $dX(t)=-1.5X(t)dt+0.85dW(t)$ with $X(0)=0.7.$ Find the mean and the variance of $X(1).$ I know that $E[X(1)]$ will result in mean and $E[(X(1))^{2}]$ in ...
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Black-Scholes model with a derivative with payoff $S_{T}^{3}$

Given a Black-Scholes Model and a derivative with payoff $S_{T}^{3}$ at time $T$. Check that the value of that derivative at time t is $V_{t} = g(t, T)S_{t}^{3}$, where $g(t, T)$ has to be determined. ...
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Can I get the following estimate for a family of solutions of SDE's?

Fix a time horizon $T>0$. Consider a fixed discretization parameter $\Delta>0$ and divide $[0,T]$ into intervals of the form $[K \Delta, (K+1)\Delta]$ for $K=0...\lfloor T/ \Delta \rfloor - 1$. ...
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4 votes
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Using SDE uniqueness in law to show that two processes have the same distribution

Let $B$ and $\tilde{B}$ be independent standard Brownian motions defined on the same probability space with $B_0=\tilde{B}_0=0$. Let $$X_t=e^{B_t}\int_0^te^{-B_s}d\tilde{B}_s,\hspace{1cm}Y_t=\sinh(B_t)...
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Finding a change of measure so that rescaled local martingale is a local martingale

Let $\mu,\sigma:[0,\infty)\to\mathbb{R}$ be deterministic continuous functions, assume that $\sigma$ is bounded below by a strictly positive constant and that $\mu$ has compact support. Suppose that $...
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2 votes
1 answer
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Finding a weak solution to an SDE

Consider the SDE $$dX_t=\text{sign}(X_t)dB_t$$ with $X_0=0$ and where $$\text{sign}(x)=\begin{cases}-1&\text{if }x\leq0\\1&\text{if }x>0\end{cases}.$$ I am asked to find a weak solution to ...
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Using the BDG inequality to show a process is uniformly integrable

For a BSDE : $$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$ Which has a fixed $\xi \in L^2(\mathscr{F}_T)$ and $g_0(\cdot)$ satisfying $E(\int_{0}^{T}|g_0(t)|dt)^2 < \infty$. There ...
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Can this system of stochastic differential equation be solved algebraically?

Can the following system of stochastic differential equation be solved algebraically? I was trying to solve numerically but I wonder this can be done algebraically. $dX_t=(a-Y_t)X_tdt+\...
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Integrating factor and homogeneous equation for SDEs.

Can someone explain how an integrating factor is obtained when solving SDEs. An example would be when finding the solution for a general linear SDE: $dX_t = (a(t)X_t +b(t))dt + (c(t)X_t +d(t))dB_t, ...
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1 vote
1 answer
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Using BDG inequality to show the solution to a BSDE belongs to $S^2_{\mathscr{F}}$

For a BSDE: $$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$ Which has a fixed $\xi \in L^2(\mathscr{F}_T)$ and $g_0(\cdot)$ satisfying $E(\int_{0}^{T}|g_0(t)|dt)^2 < \infty$. There ...
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2 votes
1 answer
88 views

Stochastic Differential Equation after a Change in the Time Parameter

How does the stochastic differential equation for a stochastic process change under a change in the time parameter? For example, consider the Bessel Squared Process $$ dR_t = m \, dt + 2 \sqrt{R_t} \, ...
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2 votes
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Show that $\mathbb{E}(X|\mathcal{F}_t)$ is a square-integrable martingale

For a BSDE : $$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$ Which has a fixed $\xi \in L^2(\mathscr{F}_T)$ and $g_0(\cdot)$ satisfying $E(\int_{0}^{T}|g_0(t)|dt)^2 < \infty$. There ...
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3 votes
1 answer
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Question about Ito formula and BSDE

When I was reading the paper from Peng, I saw an equation which I had no idea about how to get it. The details are shown below: For a BSDE : $$y_t = \xi + \int_{t}^{T}g_0(s)ds - \int_{t}^{T}z_sdB_s$$ ...
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  • 141
2 votes
1 answer
83 views

The stochastic differential of $\cos (B_t^{(1)}B_t^{(2)})$

Let ($B_1$, $B_2$) be a bi-dimensional correlated Brownian motions Calculate the stochastic differential equation of the process $\cos(B_{1,t}B_{2,t})$. Attempt: Let $X_t$ be the stochastic process ...
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17 views

Showing that a composition of a non-explicit function is Lipschitz

Let $b$ be a bounded and continuous function and let $W$ be a scalar Brownian motion. Consider the SDE $$dX_t=b(X_t)dt+dW_t.$$ I was tasked with showing that there exists a strictly increasing ...
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4 votes
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61 views

Radon-Nikodym derivative of pushforward measures and Girsanov theorem

Let $\mu$ and $\nu$ be two measures on a measure space $(\Omega, \Sigma)$, and $\mu$ is absolute continuous w.r.t. $\nu$. Also let $X\colon \Omega \to H$ be a measurable functions mapping to another ...
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2 votes
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Solution of a Stochastic Differential Equation

I am trying to prove that the solution of the SDE: $$ Z_t = Y_t + \int_0^t Z_s dX_s $$ is: $$ Z_t = \mathcal{E}(X)_t \left(y_0 + \int_0^t \mathcal{E}(X_s)^{-1}dY_s - \int_0^t \mathcal{E}(X)_s^{-1}d\...
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3 votes
1 answer
37 views

How to show that $(X,B)$ and $(Y,W)$ satisfy the same SDE if their joint law is equal?

Let $(X,B)$, where $B$ is a standard BM and $X$ is process, satisfy the SDE $dX_t = b(t,X_t) + \sigma(t,X_t)dB_t$. Suppose that for some other standard BM $W$ and process $Y$ we have that the joint ...
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How can I apply the Euler–Maruyama approximation to the following SDE?

I'm trying to apply the Euler–Maruyama discretization to a python code using this Wikipedia page Wikipedia page where it says that the SDE $$dX_t = a(X_t, t) dt + b(X_t, t) dW_t, \quad X_0 = x_0 $$ ...
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Stochastic difference equation selecting solution

I'm hoping someone can point me in the right direction for the following "problem": start with a difference equation using lag operator: $E_t[(1-\lambda_1^{-1}L)(1-\lambda_2L^{-1})\pi_{t+1} =...
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2 votes
0 answers
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Is a Riccati BSDE explicitly solvable?

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $C\in (0;\infty)$ and $\{a_t\}_{t\in[0;T]}$ be a ...
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2 votes
1 answer
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$dS_t=\mu S_t dt +\sigma S_t^{\alpha/2} d W_t$ and $Y_t=S^{2-\alpha}$, can one simulate exact paths for Y_t?

The task states that $dS_t=\mu S_t dt +\sigma S_t^{\alpha/2} d W_t$ and the question is if one can generate (simulate) exact paths for $S_t$ by taking the transformation s.t. $Y_t=S_t^{2-\alpha}$. I ...
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3 votes
2 answers
47 views

Solving a linear backward stochastic differential equation

Let $W=\{W_t\}_{t\in[0;T]}$ be a real-valued Brownian motion, $\{F_t\}_{t\in [0;T]}$ the filtration generated by $W$, augmented with the nullsets, let $\{B_t\}_{t\in[0;T]}$ be an adapted process and ...
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  • 495
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39 views

Correlation function in the Langevin equation

So the Langevin equation of Brownian motion is a stochastic differential equation defined as $$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$ where the noise function $\eta(t)$ has ...
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26 views

Stochastic Integral and Standard Brownian motion

Let $B_t (t\geq 0)$ be a standard Brownian Motion, where $B_0=0$. I have $d(B_s^2)= 2B_sdB_s$. I want to find the integral $ \int_0^tB_sdB_s$. Therefore, I get $ \int_0^tB_sdB_s= \frac{1}{2}\int_0^td(...
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2 votes
0 answers
41 views

How are anisotropic heat-equation and Fokker-Planck equation related?

Consider first a rescaled brownian motion $X_t$, in $\mathbb{R}^n$ which fulfills the SDE $$dX_t = \sqrt{2} dB_t,$$ where $B_t$ is brownian motion. Then the density $p(t,x,x_0)$ of the process ...
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3 votes
1 answer
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Properties of the time integral of Ito process

Consider the Ito process $X_t$ defined by $$ dX_t = a(t,X_t) dt + b(t,X_t) dW_t $$ where $W_t$ is the standard continuous-time Wiener process. Let's define the process $Y_t$ to be some integral of $...
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Verify process is an Ito process

Let $(\xi(t),t\ge 0)$ be an Ito process with $\xi(0)=\theta$ and $$ d\xi(t)=\kappa(\theta-\xi(t))dt+\sigma\sqrt{\xi(t)}dW(t)$$ for $\kappa, \theta, \sigma \in \mathbb{R}$. Show that $(\eta(t),t\ge 0)$,...
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0 answers
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Deriving Stochastic Differential Equations about Population Dynamics

This is somewhat of a high-level, I-hardly-know-much-about-the-topic-yet question, but here goes. I came across online a presentation in which part of it documented deriving an SDE for a population ...
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Calculating the limit of a Wasserstein distance of two SDE's

I am trying to prove that: $\lim_{t \to \infty} W_2(\mu_t, \nu_t) = 0 $ where we have that $\mu_t = Law(X_t)$ and $\nu_t = Law(Z_t)$ with $$dX_t = -h(X_t)dt + \sqrt(\frac{2}{\beta})dB_t$$ $$dZ_t = -h(...
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2 votes
1 answer
66 views

Expectation of modified Ornstein-Uhlenbeck process with random long run mean

I would like to compute the expectation of a modified Ornstein-Uhlenbeck process of the form $$ dx_t = \theta(\mu_t-x_t)dt + \sigma x_t dW_t \ ,$$ where $\kappa, \sigma>0$ and $W_t$ is a Brownian ...
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1 vote
0 answers
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Is there any way to “analytically” compute the following conditional expectation?

Let me define the following SDE: $$\begin{cases} dX_{t}=X_{t}dW_{t}\\ X_{0}=1 \end{cases},$$ where $W$ is an appropriate Wiener process. My question is how we can compute $$f\left(z\right)=\mathbb{E}\...
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3 votes
1 answer
36 views

Infinite variation of Brownian motion and Continuity

Let $C>0$ be a constant. Brownian motion is Hölder continuous for $\alpha=1/2$: $$| B(t+h) - B(t) | \leq C \sqrt{h \log(1/h)} \leq C h^\alpha,$$ for every sufficiently small $h$. But Brownian ...
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3 votes
1 answer
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A definition of Stochastic integral that is both a martingale and preserves chain rule?

my question is straightforward: is there any definition of Stochastic integral (so I assume at least some kind of Riemann sum compatibility and linearity) that is both a local martingale and preserves ...
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1 vote
0 answers
25 views

Confusion about the covariance of the Wiener process

When studying the Wiener process, I learned that the variance of this process is $Var(W_t) = t$, (which can be proven by calculating the quadratic variation) and furthermore that $\mathbb{E}W_tW_s = \...
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23 views

the mean of a stochastic differential equation

Let $(X_{t} , t \geq 0)$ be a processus and solution of the sifferential equation : \ $ X_{t}= X_{0} + \int_{0}^{t} \mu (1-2X_s)ds+ \int_{0}^{t} \sqrt{X_s(1- X_s)} dB_s$ with $ \mu > 0$ and B a ...
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2 votes
0 answers
31 views

Calculate the expectation of random differential equation?

I have a differential equation $$\frac{dx(t)}{dt}=F(g(x(t),\omega)),\;\;x_0=x,$$ where $\omega$ is standard normal variable. Given that for each fixed $y\in\mathbb{R}$, $$\mathbb{E}_\omega[g(y,\omega)]...
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1 vote
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31 views

Showing that a diffusion process is positive recurrent

Consider an one-dimensional diffusion, with values in $(0,\infty)$, solution of the SDE $$dX_t = \mu(m-X_t)dt + \sigma X^{\psi}_tdB_t$$ Where $B_t$ is a standard one-dimensional brownian motion, $m>...
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19 views

Finding the stochastic differential equation using multidimensional Ito's formula

I've been working on this question and have been having a little difficulty with it. I'm trying to find the SDE for Y, where: $$ Y_t=X_te^{W_{2,t}} $$ and $$ X_t=X_0+\int_0^{t}a_{1,s}ds+\int_{0}^{t}b_{...
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1 vote
2 answers
32 views

Solving SDE $dY = Adt + BYdW, Y(0) = Y_0$?

I am trying to solve the following SDE: $dY = Adt + BYdW$, where $Y(0)=Y_0$. Thus far, I have done the following. First, from Ito's Lemma, we have: $d(ln(Y))=\frac{1}{Y}dY-\frac{1}{2Y^2}dY^2$. ...
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1 vote
1 answer
27 views

Integrability of linear SDE

Consider a linear SDE of the form $$dX_t = X_t(\alpha_tdt + \beta_t dW_t), \ X_0=x, $$ where $\alpha_t$, $\beta_t$ are $L^p$ integrable stochastic process: $$\mathbb{E}\int_0^t |\alpha_s|^p ds , \...
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0 votes
1 answer
52 views

Ornstein-Uhlenbeck process with random long run mean

I am considering an OU process of the form $$ dx_t = \theta(\mu-x_t)dt + \sigma dW_t $$ where $\kappa, \sigma>0$ and $W_t$ is a Brownian motion. I know that $x_t$ has expectation given by: $$ \...
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