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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16 views

Fokker Planck Equation

I was told in the lecture like the following: "The Fokker-Planck equation is a deterministic partial differential equation that in general has to be solved numerically. For vector systems ...
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13 views

Kolmogorov Backward and Forward Equations: Why are there different derivations for the forward and backward dynamics?

Dear knowledgeable people of math.stackexchange :) , In Stochastic Analysis and Diffusion Processes by Kallianpur on pages 218 to 221 the derivations for the forward and backward Kolmogorov equations ...
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30 views

Show Geometric Brownian motion is the unique solution to $\frac{dS_t}{S_t}=\mu dt+\sigma dW_t$.

Given $$ X_t=\sigma W_t+\mu t, $$ the SDE $$ \frac{dS_t}{S_t}=\mu dt+\sigma dW_t $$ has a unique solution $$ Z_t=\exp\Big(X_t-\frac{1}{2}\langle X\rangle_t\Big). $$ We know from various posts and ...
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21 views

Linear transformation of SDE

Say that I define an SDE as \begin{align} dX(t) = F(x)dt + \sigma dW_t \end{align} where $F(x)$ is the deterministic part, $\sigma$ is the noise intensity and $dW_t$ is the Winer process. Then I show ...
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47 views

Probabilistic recursion

I have the following recursion: $$p_{t+1} = \begin{cases} 1 \text{ with probability } 1-p_t \\ \alpha p_t \text{ with probability } p_t \end{cases}$$ for $0<\alpha<1$. Numerical simulations show ...
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18 views

Solving/Rewriting SDEs in Non-Matrix Lie Groups

I'm working on trying to solve a state estimation problem in a non-matrix Lie group. I have found some good resources for state estimation in certain matrix Lie groups. For instance, in this paper ...
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1answer
26 views

Weiner Process Markovian property

Wiener process definition: $W(t) \sim \mathcal{N}(0, \sigma\sqrt{t})$ $W(t)$ is Markovian $W(t+t')-W(t)$ and $W(t)$ are independent. I don't know of a case where $2$ is true but $3$ is not & ...
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A question about SDE and geometric Brownian motion.

In Bernt Oksendal's Stochastic Differential Equations, Chapter 4, one has the following stochastic differential equation (whose solution is geometric Brownian motion): $$dN_t=rN_tdt+\alpha N_tdB_t\;\;\...
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41 views

SDE books for non-mathematicians

I want to learn about SDEs. Most books on this topic are very dry & mathematical with no intuition at all. Can someone recommend a good book or courses for non-mathematicians? I am from computer ...
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29 views

Can a stochastic process correspond to multiple SDE's?

In this question, and this question, it is clear that the solution of the one-dimensional SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ is $X_t-X_0=B_t$. This is also the Bessel process. I am confused by the ...
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28 views

What kind of stochastic process could this be?

I have been studying the fascinating subject of stochastic processes and have constructed various equations I can think of and then I try to look up literature for that process. One process I have ...
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1answer
45 views

Finding the SDE

Let $X_t = e^{(\mu-\frac{\sigma^2}{2})t + \sigma B_t}$, where $B_t$ is a standard Brownian motion. How do I find the SDE satisfied by $X_t^{-1}?$ I know I must use Ito's formula but not sure how to ...
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48 views

Derivative of conditional expectation of integral of stochastic process

Let $T>0$. Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and let $\mathbb{F}=(\mathcal{F}_u)_{u\in[0;T]}$ be a filtration such that $\mathcal{F}_T=\mathcal{F}$. Let $(\alpha_u)_{u\...
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44 views

Dealing with SDE involving $\max\{X_t,0\}\mathrm{d}W_t$

I am considering the SDE $\mathrm{d}X_t=\ln(1+X_t^2)\mathrm{d}t+\max\{X_t,0\}\mathrm{d}W_t$, with the initial condition $X_0=a$. I am posed with the following daunting tasks: prove $\exists$ a unique ...
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70 views

Estimate to arrive at $\mathbb{E}[(|X^{i}_t|+\frac{1}{N}\sum_{j=1}^N|X^j_t|)^q\mid X_0^{i}=x]\leq C x^q$

Setup We have a system of $N$ diffusion processes described by $$X_t^{i}=X_0^{i}+\int_0^t \mu(s,X_s)ds+\int_0^t\sigma(s,X_s)dW^{i}_s,$$ with the Brownian motions $W^{i}$ as well as the drift $\mu$ and ...
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64 views

How to evaluate the integral of white noise multiplied by exponential (decaying) term

Consider a matrix ODE for the vector $\mathbf{y}$ of the form \begin{equation} \dot{\mathbf{y}}(t) = A \, \mathbf{y}(t) + \mathbf{b}(t) \, , \end{equation} where $A$ is a constant square matrix ...
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26 views

Divergence when calculating moments with a space-time Gaussian white noise

Consider a variable $M(x,t)$ driven by space-time Gaussian white noise: $$ \partial_t M(x,t) = -k M + \xi(x,t).$$ The noise has mean $\langle \xi(x,t) \rangle = 0$ and correlation function $\langle \...
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44 views

Intuition of Kolmogorov equations and infinitesimal generators

So if we have the infinitesimal generator of a Markov process $\{X_t\}_{t\geq0}$, the infinitesimal generator is given by $$Af(x) = \lim_{t\rightarrow 0} \frac{1}{t} \big(P_tf(x) - f(x)\big)$$ where $...
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42 views

On an example of SDE where strong and weak solutions are different

I was wondering about explicit examples of SDEs where strong and weak solutions are indeed different. I found online that the following is an example of and SDE where the two solutions are different: ...
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1answer
38 views

Have trouble understanding Matlab code of stochastic integrals

Hell, I have trouble understanding the matlab code of approximating stochastic integrals. ...
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26 views

Find an SDE given a probability density function p(x,t)

Usually, we start with an SDE for $X(t)$ and then try to solve it and find the corresponding probability density function $p(x,t)$. But what if we start with a pdf - can we find a corresponding SDE? ...
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27 views

Simplifying terms in Ito's Lemma like $Y_t dX_t$ and $dX_tdY_t$

Suppose that I have two stochastic processes that can be represented as $$dX_t= udt + \sigma dB_{1,t}$$ $$dY_t= vdt + \nu dB_{2,t}$$ I know by Ito's Lemma that $$d(X_tY_t)=X_tdY_t + Y_tdX_t + dX_tdY_t ...
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19 views

Diffusion in 1D with spatially varying diffusion coefficient, Fokker Planck equation

I am trying to find the steady state densities to the following problem. We have a random walk in 1D with spatially varying diffusion constant, which i numerically solve as $$ x_t = x_{t-1} + \sqrt{2 ...
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1answer
66 views

Solving SDE with sign function in drift term?

Consider the following SDE with $X_0 = 1$, $$ dX_t = X_t\operatorname{sign}(X_t) \, dt + X_t \, dW_t, $$ where $\operatorname{sign}(x) = \mathbb{1}_{\{x \ge 0\}} -\mathbb{1}_{\{x < 0\}}$. How am I ...
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1answer
14 views

Apply Ito's formula to $d(X_t)$, where $X_t = e^{\int_0^t(X_s-1/2)ds+W_t}$

I'd like to find $dX_t$, where $X_t = e^{\int_0^t(X_s-1/2)ds+W_t}$. I have no idea how to apply Ito's formula here.
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16 views

Expected Discount Factor with respect to First Passage Time

I am curious about and have been trying for a long time the following question. I was wondering if there will be any hints or references that could help. Assume that $B_t$ represents a standard ...
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26 views

Scaling of white noise from scaling of brownian motion

It is a well-known fact that $$\lambda^{1/2}B(\lambda^{-1}t)\stackrel{d}{=}B(t) \ ,$$ for $B$ a $1$-dimensional brownian motion. Also, we know that for $\xi$ a $1\text{D}$ white noise, we have $$\...
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1answer
36 views

Linearization of a SDE and comparison to deterministic setting

I am a bit puzzled having no experience in this area on the following problem. It is well know that if we consider a system as: \begin{equation} \dot x = f(x(t)), x \in R \end{equation} with $f$ of ...
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24 views

Write stochastic integral equation in differential form.

If I have a stochastic process given by $$Y(t)=x+\int_s^t\frac{a(\tau)-\sigma(\tau)\gamma(\tau)}{Z(\tau)}d\tau + \int_s^t\frac{\gamma(\tau)}{Z(\tau)}dW(\tau)$$ where $X(s)=x$, how would I write it in ...
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42 views

Itô's formula for an expectation result

I applied Itô's formula to $f(x)=x^q$ for $q>1$ with a process defined by the SDE \begin{align*} dX_t = \mu(t,X_t)dt + \sigma(t,X_t) dW_s, \end{align*} where $W_s$ is a Brownian motion, $\mu(t,X_t)$...
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1answer
138 views

Application of Dynkin's formula

Let $x\in\mathbb{R}^d, (W_t)_{t\geq0}$ be a $d$-dim. Brownian motion. I have the following processes $$ Z_{t, x} = b(Z_{t, x})dt + \sigma(Z_{t, x})dW_t, \qquad Z_{0, x} = x \\ V_{t, v} = \nabla b(Z_{t,...
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31 views

Independence of the components of an Ito process

If $X_t = (X^i_t)_{i \leq n}\in \mathbb{R}^n$ solves an SDE of the form $$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$ Does it hold that each of the real valued random variables $X^i_t$ and $X^j_t$ are ...
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12 views

Probability density function (PDF) of a two-dimensional linear stochastic differential equation (SDE) with additive noise

I am considering the same two-dimensional linear SDE as in a previous question: \begin{cases} dX_1 =\bigl(A_{11} X_1 + A_{12} X_2 \bigr)\,dt + \sigma \,dW_1, \\ dX_2 =\bigl(A_{21} X_1 + A_{22} X_2 \...
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1answer
36 views

Form of Ito's formula for Ito diffusions

Suppose we have an Ito diffusion ($Z_{t, z}$) satisfying $$ Z_{t, z} = z + \int_0^t b(Z_{s,z})ds + \int_0^t \sigma(Z_{s,z})dW_s $$ where $z\in\mathbb{R}^d, b:\mathbb{R}^d\to\mathbb{R}^d, \sigma:\...
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65 views

Conditional expectation on the product of squared Brownian motion with Doléans-Dade exponential (Radon-Nikodym derivatives)

I try to compute the conditional expectation $E_{t}^{P}\left[B_{T}^{2} \ e^{-\frac{1}{2} \int_{0}^{T} \left( a + b B_{u}^{2}\right)^{2} d u+\int_{0}^{T} \left( a + b B_{u}^{2}\right) d B_{u}}\right]$ ...
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1answer
33 views

Stochastic calculus of Gaussian white noise

I see a strange stochastic integral and don't know how to proceed? Denote $\xi$ a Gaussian white noise, Denote $$I(n)=\int_t^{t+n\Delta t}ds\int_t^sd\xi$$ How to prove that $<I(1)I(1)>= \frac{1}{...
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30 views

Name for class of functions from Øksendal's Introduction to SDEs

From Øksendal's Introduction to Stochastic Differential Equations we have the following: Definition $3.1.4.$ Let $\mathcal{V} = \mathcal{V}(S, T)$ be the class of functions $f(t, ω): [0, ∞) × Ω → R$ ...
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13 views

Power spectral density (PSD) of a two-dimensional linear stochastic differential equation (SDE) with additive noise

I am considering a two-dimensional linear SDE of the form $$\begin{cases} dX_1 =\bigl(A_{11} X_1 + A_{12} X_2 \bigr)\,dt + \sigma \,dW_1, \\ dX_2 =\bigl(A_{21} X_1 + A_{22} X_2 \bigr)\,dt + \sigma \,...
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1answer
34 views

What is the expression $E[X_t dt]$ using stochastic calculus?

I am trying to finish a calculation using Itô calculus but do not know the final step. What is the expression $E[X_t dt]$ equal to? Note that the $X_t$ is drifted Brownian motion $X_t = \mu_x t + \...
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40 views

Backward Stochastic Riccati Differential Equation

Let's say $(P,\Lambda)$ is an adaptive solution to the following backward stochastic riccati differential equation: $$ \begin{aligned} dP_t &= \left[a - \frac{2}{\kappa_t} \left(P_t-\lambda a^2\...
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80 views

Dynkin Operator explained to a finance student

Can somone gives me an introduction about the so-called Dynkin Operator ? (finance student means that I know stochastic calculus only at an introductory level without a solid background in funtional ...
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1answer
27 views

moments of a Geometric brownian motion using Ito's lemma

Can someone explain me what is wrong in my derivation of the formula for the moments of a GBM using Ito's lemma (I am not interested in other methods) ? \begin{equation} dX=\mu Xdt+\sigma XdW_t \end{...
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35 views

Distributions of stopping/hitting times of SDE

Let $W_t$ be an $m$-dimensional Wiener process on a complete probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following SDE: \begin{equation} dX_t = b(X_t,t)dt + \sigma(X_t,t)dW_t \...
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10 views

Girsanov theorem for bounded variation process

I have the path: $$X_t = B^H_t + H_t $$ where $(B^H_t)_{t \geq 0}$ is the fractional Brownian Motion (if necessary, we can consider ordinary Brownian Motion, that is $H=1/2$) and $H_t$ is a bounded ...
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25 views

Infinitesimal generator of smooth SDE

Suppose we have the SDE $dX_t = \mu_t dt + \sigma_t dS_t$ where $S_t$ is a.s. continuously differentiable. More precisely, I want to put $S_t = \sum_{i = 1}^N X_ib_i(t)$ where the $X_i$ are i.i.d ...
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39 views

How to differentiate this exponential process?

I understand how to apply Ito's formula to differentiate the Radon-Nikodym density $Z_t = \exp \bigg(-\int_0^t \theta_sdW_s - \frac{1}{2} \int_0^t \theta^2_sds \bigg)$ and get the SDE $dZ_t = -\...
2
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0answers
70 views

Expected value of stopping time

Let $X = (X_t)_{t\geq 0}$ be a non-negative stochastic process solving $$ d X_t = 3dt + 2\sqrt{X_t}d B_t, \quad\quad X_0=0 $$ with $B = (B_t)_{t\geq 0}$ is a standard Brownian motion. I want to ...
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13 views

Weak error versus strong error

Currently I'm learning about SDE's. I understand the conceps weak and strong error mathematically, but I find it hard to see in which context each error is used. The strong error tells something about ...
2
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0answers
35 views

Why are SDEs given in the form $dX_t=\mu dt+\sigma dB_t$?

All the resources I have found state that a Stochastic Differential Equation has the form: $$dX_t=\mu(X_t,t)dt+\sigma(X_t,t)dB_t$$ Where $B_t$ is Brownian motion. My question has two parts. The first ...
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1answer
90 views

$M = (M_t)_{t\geq 0}$ obtained by Itô's formula is a martingale

Here I defined a non-negative stochastic process. Now, taking $F(t,x) = tx^2$, I would like to find that the continuous local martingale ``part'' obtained by Itô's formula is indeed a Martingale. To ...

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