Questions on finding solutions of stochastic differential equations (SDEs). For questions related to more theoretic aspects of SDEs such as existence of solutions, Stochastic-analysis may be a more appropriate tag.

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2
votes
1answer
19 views

Jump Diffusion Infinitesimal generator

I have this difussion process $dX(t)=\mu X(t)dt+\sigma X(t)dW(t)+u X(t) dN(t),\qquad X(0)=x > 0$ where $W(t)$ is a Brownian Motion and $N(t)$ is a Poisson process. And I need to know the ...
0
votes
0answers
17 views

To estimate the probability that a diffusion reaches a certain value

I have a diffusion process define by the following equation: \begin{equation} dX_t=X_t[\beta(N-X_t)-\alpha]dt+\sqrt{X_t(\beta(N-X_t)+\alpha}) { }dB_t \end{equation} and I proved that the solution ...
0
votes
0answers
8 views

How is the form of the transition functions of a process define as the solution of a stochastic differential equation?

I´m searching for a prove that solutions of SDE are markovian and i'm trying to find (or understand) in this case (SDE) what form have the transition functions associated to this kind of process. I´ve ...
1
vote
0answers
23 views

How can we evaluate the material derivative of the velocity of an particle by means of an Itō formula?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a $\mathbb R^d$-valued Brownian motion with respect to ...
1
vote
0answers
22 views

Existence and uniqueness of solution of a non linear SDE

I have the following SDE: $dX_t=(\mu+X_t^2) dt+e^t dB_t$. What can I say about existence and uniqueness of solutions? I would like to verify the usual conditions of sub-linear growth and Lipschitz, ...
0
votes
1answer
50 views

Why is the drift of an Itō process considered to be a Riemann integral even when it's not even Riemann integrable?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $(B_t)_{t\ge 0}$ be a real-valued Brownian motion with respect to $\mathcal ...
0
votes
1answer
33 views

Black-Scholes SDE solution help

I am trying to solve the Black Scholes SDE, but got really stuck. I have done most of the derivation but the integral seem intractable to me. The integrals look bit like the Normal Distributions PDF, ...
0
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0answers
11 views

4th order Runge-Kutta Scheme for Stochastic Differential Equations

In the book "Introduction to Stochastic Differential Equations" by Thomas C. Gard, they talk about higher-order runge-kutta type schemes. The SDE in question is a general Ito SDE of the form: $$dX = ...
1
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0answers
50 views

Why can the solution of a SPDE $\partial_tu(t,x)=\cdots$ be viewed as a stochastic process indexed by $t$ with values in a space of functions of $x$?

Please consider a stochastic partial differential equation of the form $$\partial_tu(t,x)=F(t,x,u(t,x),{\rm D}u(t,x),{\rm D}^2u(t,x))+G(t,x,u(t,x),{\rm D}u(t,x))\partial_tB(t,x)\tag 1$$ where ...
0
votes
2answers
69 views

Linear non-homogenous SDE

I'm struggling to understand how to resolve the following SDE: $$dX(t)=(\sin(t)-2X(t)) dt + (1+X(t))dB(t)$$ I understand that I should use the Ito formula but I have no idea how the $F(X(t),t)$ should ...
0
votes
0answers
16 views

Proving bounded expectation and switching order of stochastic integrals

I'm working with the following set of stochastic differential equations: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the ...
1
vote
0answers
51 views

Geometric Brownian Motion Properties

I was reading Oksendal's book "Stochastic Differential Equations", fifth Ed., pp. 62-63 and came across some counter-intuitive properties of the Geometric Brownian motion (GBM). Let $\alpha,r>0$ ...
1
vote
1answer
31 views

Solving equation with Wiener process

I want to show that if $E(f(X_{t}))=E(f(W_{t})e^{\lambda W_{t}-0.5*\lambda^2*t})$, where $W_{t}$ is a Wiener Process, then $X_{t}\sim N(\lambda t,t)$. Does anyone have a clue how to solve this ...
1
vote
0answers
26 views

Simulating “Nested” Stochastic Differential Equations

I haven't had much luck over in Stats SE, so I'm going to try over here. I doubt many people here have experience with R, so I would like to know what are some methods to simulating a set of "nested" ...
0
votes
0answers
22 views

Noise with heavy tails

The main type of noise I know other than white noise is a colored noise (Ornstein-Uhlenbeck) of the form: $$d\eta = \lambda \eta dt + \alpha dW_t$$ with exponential correlation. I'm interested in ...
4
votes
0answers
42 views

Methods of SDE calibration

There is somewhere summary of methods that can be used to estimate parameters of SDE? I currently using MLE and regression due to linear dependence between samples. I searching for something ...
1
vote
1answer
32 views

Non-linear SDE: how to?

$$ \newcommand{\mcl}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\avg}[1]{\langle#1 \rangle} \newcommand{\pth}[1]{\left( #1 \right)} \newcommand{\bck}[1]{\left\{ #1 \right\}} ...
1
vote
1answer
35 views

Check solution to the SDE $dX_t = - \mu X_t \, dt+ \sigma \, dW_t$

I get stuck in this problem. I just can't get the hang of how we need to "guess" a function first and almost everything along the process of solving depends on it; It's not entirely logical to me when ...
1
vote
0answers
25 views

differential stochastic equation and solutions

Let E(K):$Z_0=0$ and $dZ_t=K_t(B_t-Z_t)dt+\alpha K_td \beta_t$ with B and $\beta$ two brownian motions, $\alpha>0$ and K a continue function. 1)Show that E(K) has a solution Z with $E(\int_0^1 ...
1
vote
1answer
55 views

Show that stochastic process solves certain SDE

1) I want to know the mechanism how to: show that the process $X_t$ solves this SDE 2) know if my friends though, mine though below are correct/incorrect. I have the general linear stochastic ...
0
votes
2answers
42 views

Solution of an SDE

Can someone guide me on how to solve the SDE \begin{equation} dX_{t} = \gamma(a-\beta X_{t})dt + \delta X_{t}dW_{t} \end{equation} where $a,\beta,\gamma,\delta$ are all positive constants? I tried ...
0
votes
0answers
15 views

Does the law of an Itô process $X_t$ at each time has a density? [duplicate]

Let $X$ be a the strong solution of the SDE on $\mathbb{R}$ $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ $$X_0 = x$$ where $b,\sigma \colon \mathbb{R} \to \mathbb{R}$ are Lipschitz functions. Dose the law ...
1
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0answers
14 views

Which research groups use stochastic processes and/or stochastic differential equations in computer graphics/vision?

Some research groups use stochastic differential equations for mathematical image processing. Which research groups do use stochastic processes in general and/or stochastic differential equations in ...
0
votes
1answer
34 views

Euler-Maruyama simulation of an SDE

The Euler-Maruyama method for the following SDE \begin{align} dX_{t} &= -\lambda X_{t}dt + \mu dW_{t}\\ X_{0}&=x>0 \end{align} where $\lambda,\mu$ are given constants, is (according to ...
1
vote
1answer
19 views

Solve and prove uniqueness of the SDE $dY_t = tY_tdt + e^{t^2/2}dB_t$ without using the general linear SDE formula

Let $(B_t)_{t \in [0,T]}$ be standard brownian motion, and let $(Y_t)_{t \in [0,T]}$ be a stochastic process in $(\Omega, \mathscr F, \mathbb P)$. Without using the general linear SDE formula, solve ...
0
votes
0answers
24 views

What practical applications do SDEs and SPDEs have?

I will study from the probability thory to its application to stochastic differential equations with my friend. Of cource I'm looking forward to study them but would be a littel discouraging because I ...
1
vote
1answer
32 views

Solve SDE $dX_t = (e^{-\gamma t} - \gamma X_t) dt + 2 e^{-\gamma t/2} \sqrt{X_t} dW_t$

Solve the SDE given by: $dX_t = (e^{-\gamma t} - \gamma X_t) dt + 2 e^{-\gamma t/2} \sqrt{X_t} dW_t$. My attempt Following the hint of my professor: suppose $X_t = e^{\gamma t} g(W_t)$. Then we ...
2
votes
1answer
89 views

Solving StochasticDifferential Equation

Please help me in solving this Stochastic Differential Equation for $Y_t$ $$dY_t = a Y_t dt+ b dX_t \qquad Y(0) = c $$ where $a$ and $b$ are constants. Also find the $\mathbb{E}[Y_t]$ and ...
0
votes
0answers
53 views

Expectation of a strongly anticipating Ito like integral

I am using some Ito stochastic differential equations (SDEs) of the form $$dx=A(x)dt+B(x)dW$$ where $dW$ is an increment in a Wiener process and the above are to be interpreted as stochastic ...
2
votes
1answer
77 views

How to approximate an AR(1) with an Ornstein-Uhlenbeck Process

I have a discrete stochastic process that can be described with the following equation $$y_{(i+1)\tau}=k y_{i\tau}+w\xi_{i+1}$$ where $i\in\Bbb{N}$, $(\xi_i)_i$ is an IID sequence with $\xi_i \sim ...
0
votes
0answers
18 views

derivation of anisotropic diffusion equation from SDE

I have read this article in wikipedia. I'm interested in the reference for the anisotropic diffusion equation or, maybe better, how to derive the anisotropic diffusion equation, with diffusion ...
1
vote
0answers
14 views

SLE theory, showing that $T_u$ is a.s. finite

I am currently reading this article, page 7 (enumerated 889). Let $g_t(z)$ be the solution for $t\in\mathbb{R}$ for $$\partial_tg_t(z)=\frac{2}{ g_t(z)-\xi(t)},\quad g_0(z)=z.$$ Where $\xi$ is a ...
2
votes
1answer
25 views

SDE solution with $\mu_t$ a bounded integral

Consider the following SDE: $$ dX_t = X_t (\sigma dW_t + \mu_tdt), \text{with } \mu_t \text{ a bounded integrable function of time} $$ The way I would solve this would be to use that: $$ X_t = ...
1
vote
0answers
16 views

Distribution of $(X(t_1),X(t_2))$ of a diffusion process $X(t)$

I am very new to SDE's and diffusion processes, I came across this diffusion process given by $dX(t)=[\alpha-(\alpha + \beta)X(t)]dt + \sqrt{2X(t)(1-X(t))}dB(t)$ where $B(t)$ is a continuous Brownian ...
1
vote
0answers
45 views

Limit of the solution to a SDE

I am trying to learn about SDEs and I don't seem to understand this claim: Assume $X_t$ is the strong solution to the following SDE: \begin{equation} dX_t=c\tanh X_t \, dt +dB_t \end{equation} Where ...
1
vote
1answer
32 views

existence of the strong solution to SDE .

The following text are mostly from Shreve's "Brownian Motion and Stochastic Calculus",GTM113,chapter 5.2. Consider a the following SDE: $$X_t=X_0+\int_0^tb(s,X_s)\,ds+\int_0^t\sigma(s,X_s)\,dW_s$$ ...
1
vote
0answers
45 views

A martingale associated with a solution of a stochastic differential equation

Assume $B_t$ is a standard real brownian motion defined on $\{\Omega,\mathcal{F},P\}$ and $X_t(y)$ satisfies the following SDE: \begin{equation} dX_t=-\cot X_tdt+dB_t \text{ with }X_0=y ...
0
votes
1answer
45 views

Markov processes: Hitting times for a point form an i.i.d. sequence

Say that I have a recurrent time-homogenous diffusion process X (continuous strong markov process) and two points $x,y$. If $X_t$ goes from $y$, to $x$ and then back to $y$ again we denote it as a ...
2
votes
0answers
61 views

one dimensional SDE with zero drift

I was trying to prove that the solution $X$ to the one dimensional SDE $dX_t = \sigma(X_t)dW_t$ (where $\sigma$ is a real valued Borel measurable function, $W$ is a 1d Brownian Motion) cannot explode, ...
0
votes
0answers
20 views

Exact solution to nonlinear backward SDE

I have read a paper about numerical SDE. After deriving the method, it uses the method to calculate the following nonlinear cases: $$\begin{cases} dX_t=ud\tau+\sigma ...
0
votes
0answers
16 views

Finding a function to use for Ito's Lemma

The original problem was to show the following stochastic process has a global solution: $$dx_i = x_i\left(b_i-\sum_{j=1}^4 a_{ij}x_j \right)dt+\sigma_ix_idW_t$$ To do so, they considered the ...
4
votes
1answer
62 views

Deduce $\partial_tp=-\partial_x(b(x)p)+(1/2)\partial_{xx}(\sigma^2(x)p)$, for $p(x,t|y)$ of $X(t)$ and $dX=b(X)dt+\sigma(X)dW$, $X(0)=y$

I am stuck in this proof... I almost got it, but I must have made a mistake. It is part B that I am getting wrong. Thanks in advance for your help! QUESTION: Let $X$ satisfy the autonomous SDE ...
1
vote
2answers
25 views

Show the stationary distribution of $\partial_tp=\partial_x(bp)+(1/2)\sigma^2\partial_{xx}p$ (forward Kolmogorov) is $p=Ce^{-2\int b/\sigma^2dx}$

I am trying to understand this proof so that I can do the exercises without having to actually memorise the formula and plug in numbers, like a lot of people do. Thanks a lot in advance! So if we ...
0
votes
1answer
13 views

Solution to sde with specfic mean

Goal: I'm attempting to work backwards to recover an SDE as follows: Example: $e^{\mu t}$ is the mean of the geometric Brownian Motion, which solves the SDE: \begin{equation} dS_t = \mu S_t dt + ...
1
vote
0answers
26 views

Show $Y(t)=X^{(1)}(t)-X^{(2)}(t)$ and $\lim_{t\to\infty} \mathbb{E}Y^2(t)=0$ , for $dX^{(i)}=\mu X^{(i)}dt+\sigma X^{(i)}dW$

I am trying to solve this exercise which my professor has "solved" (he says what the result but not how he gets it). This is in a problem sheet which is about the Euler-Maruyama scheme. What I get ...
0
votes
0answers
63 views

Ito's Lemma / Expected Value / Variance - Mathematical Finance

Assume an asset price $S_t$ follows the geometric Brownian motion $$\Bbb dS_t = \mu S_t\Bbb dt + \sigma S_t\Bbb dWt,$$ where $\mu$ and $\sigma$ are constants and $r$ is the risk-free rate. ...
2
votes
1answer
49 views

Is there a way to estimate moments of strong solution to SDE

Suppose the SDE $$\mathsf dX_t =b(t,X_t)\mathsf dt + \sigma(t,X_t)\mathsf dW_t,\; X_0 = x$$ where $t\in[0,T]$ has a strong solution. I know in general we can't find an explicit formula for the ...
2
votes
1answer
39 views

Calculate the distance $d_{\mathcal{H}}(X_n,X):=\mathbb{E}\left(\int_0^{\infty}(X_n(t)-X(t))^2dt\right)$ for all $n\ge 1$

I have done this exercise but I have done something wrong because I don't get the correct result for the next part of this exercise (this is part B). I posted something earlier that is related to ...
1
vote
1answer
34 views

Derive the unique solution of $dX_t=\alpha X_t dt+2dW_t,\quad X_0=0$

I have the following question which makes sense when taking into account that $X_0=0$, but I don't get the same result if we use the variable. QUESTION: Consider the SDE $$dX_t=\alpha X_t ...
2
votes
1answer
45 views

Compute $\mathbb{E}[\tilde{X}_t]$, where $\tilde{X}_t=X_t=(1-t)\int_0^t\frac{1}{1-s}dW_s$ for $0\le t<1$ and $\tilde{X}_t=0$ for $t=1$

I have the following exercise and I don't really understand the answer. I am going to write my professor's answer first, then a question about what I don't understand about my professor's answer and ...