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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0
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1answer
34 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
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0answers
48 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, ...
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0answers
13 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 ...
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0answers
13 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
54 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
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2answers
21 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
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1answer
11 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 + ...
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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 ...
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0answers
48 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. ...
1
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1answer
28 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
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1answer
33 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
26 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
41 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 ...
2
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0answers
48 views

Solving SDE:$\frac{dy(t)}{dt}=(c+\sigma_wW(t))y(t)+\epsilon(t) $

I want so solve the following SDE. Specifically, I want to know if $y(t)$ is a Gaussian Process and if so the corresponding mean and covariance function. ...
2
votes
1answer
43 views

Find $\mathbb{E}_{X_0 = x} X_\tau$ for an Ornstein-Uhlenbeck process $(X_t)_{t \geq 0}$ where $\tau = \inf\{t>0 \mid X_t \notin [a,b]\}$

Let $X_t$ satisfy the following SDE: $dX_t = X_t dt + \sigma dB_t$, $\sigma$ is a constant and $B_t$ is Brownian Motion. Find $\mathbb{E}_{X_0 = x} X_\tau$ where $\tau = \inf\{t>0 \mid X_t \notin ...
1
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0answers
21 views

Using Ito theory to decide whether $M^f$ is martingale or a local martingale

I came across the following while reading Ikeda & Watanabe book Stochastic differential equations and Diffusion processes, in page 163-164 At first the sentence $$f(X_t)- f(X_0) - ...
2
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1answer
25 views

Trying to understand Tanaka's example of SDE.

Consider the following Stochastic Differential Equation: $$dX_t = \sigma(X_t) \, dB_t \tag{1}$$ Where $$\sigma(x)= \begin{cases}1 & x \geq 0\\ -1 & x < 0 ...
3
votes
1answer
50 views

Solving a SDE / Finding expectation Value

I am working on a physics problem, and have come across the following stochastic differential equation: $dX(t) = \left( \frac{8}{3} X(t) - 3 X(t)^3\right)dt + dW$. I have tried all the methods to ...
1
vote
0answers
67 views

Solve simple system of SDEs

I want so solve the following SDE \begin{align}\dot{y}(t)=r(t)y(t)+\epsilon_1(t) \\ \dot{r}(t)=r(t)+\epsilon_2(t)\end{align} with $r,y$ both being stochastic processes and $e_1,e_2$ both being ...
4
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1answer
75 views

How to prove that this process is always positive?

I would like to ask is there any way to prove that following process $$ \mathrm dY_t=\left(a+\frac{b}{Y_t}\right)\mathrm dt +\mathrm dW_t, \ \ Y_0=y_0>0, $$ where $a\neq 0$ and $b\geq 1/2$, is ...
0
votes
1answer
28 views

Showing a process satisfies an SDE

The example of Ito and Watanabe in the following notes http://www.stat.uchicago.edu/~lalley/Courses/391/Lecture12.pdf is an SDE without unique solutions. $$dX_t = 3X_t^{1/3} dt + 3X_t^{2/3} dW_t$$ ...
0
votes
1answer
13 views

Negative volatility of Ito Diffusion?

This might be a silly question. But I wonder if the volatility or diffusion parameter in Ito diffusion must be positive or not. I.e. dX=$\mu dt$+$\sigma dz$, where z is a standard brownian motion. ...
1
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0answers
21 views

Continuous dependence on an initial condition (SDE)

Let's say I have a (one-dimensional) diffusion process $$dX=\mu(X_t)dt+\sigma(X_t)dW.$$ Assume we have fixed $\epsilon > 0$ and $t >0$ Under what conditions is $\mathbb{P}^x(X_t < ...
0
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0answers
18 views

Given a geometric Brownian motion, obtain a recurrence relation for moment $\mathbb{E}X_k^q$ and deduce $\mathbb{E}_k^q=\alpha^k(ah^{1/2},q)$

I am really confused about a step in the solution of this problem. I would really appreciate it if someone could explain to me what is in bold below. Part A is OK, but I don't understand something in ...
1
vote
1answer
24 views

Transforming $dX=-X^2dt+2X\circ dW$ (a Stratonovich SDE) to Ito form

As the title says, I need to transform Stratonovich SDEs to Ito form. I get similar results for some, but very different results in others. How do I do this? Thanks a lot! A) Stratonovich ...
2
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0answers
59 views

Properties of Stochastic Differential Equations

Suppose I have an SDE of the form: $$dx_i = x_if(x_1,\cdots,x_n) + \sigma_ix_idW_t $$ By defining $y_i = \log x_i$, I can change the SDE to: $$dy_i = y_i g(y_1,\cdots,y_n) + \sigma_idW_t $$ Both ...
3
votes
0answers
64 views

Bounded L2 increments for an Ornstein Uhlenbeck type process

Let $Z$ be an increasing Levy process (i.e. a subordinator). Let $\lambda>0$ and consider the Ornstein Uhlennbeck type SDE $$ d V_t = - \lambda V_t dt + d Z_{\lambda t } $$ where the integral can ...
2
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0answers
28 views

Infinitesimal Generator for Stochastic Processes

Suppose one has the an Ito process of the form: $$dX_t = b(X_t)dt + \sigma(X_t)dW_t$$ The infinitesimal generator $LV(x)$ is defined by: $$\lim_{t\rightarrow 0} \frac{E^x\left[V(X_t) ...
2
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0answers
23 views

What is a stochastic differential equation of the form $dZ = f(Z_{prev}, X_{prev})dt + CdW_t$ called?

At every time step I can approximate the change in $Z$ using the following equation: $$ dZ = f(Z_{prev}, X_{prev})dt + CdW_t, \quad(1)$$ $$dW_t = r\sqrt{dt}$$ where $C$ is some constant, and $r$ is ...
3
votes
1answer
28 views

Expected Value and Variance of a GBM Function

What is the the expected value of the process $Y = X^{3}$, where X satises the SDE $$ dXt = −X_tdt + σX_tdB_t $$ $(σ > 0)$ and $X_0 = 1$ I have two different answers: 1) I know that $X_t$ is a ...
2
votes
0answers
69 views

Derivation of Backward Kolmogorov Equation

I'm following Kallianpur-Gopinath's textbook "Stochastic analysis and diffusion processes" to study Kolmogorov equations and I got stuck in a step of the derivation of the backward equation. In ...
1
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0answers
22 views

Covariance between random variables in a stochastic differential equation

Suppose I have a SDE of the form: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda ...
3
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0answers
65 views

Finite Moments of complicated Stochastic Differential Equation

Suppose I have a SDE of the form: $$dx_i = x_i\left(b_i-\sum_{j=1}^n a_{ij}x_j\right) \,dt + \sigma_i x_i \, d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda ...
0
votes
1answer
21 views

SDE solution using Itô formula

I'd like to solve the Langevin SDE $$dX(t)=-bX(t)dt+\sigma dW(t),\\X(0)=X_0,$$ $W(t)$ being a standard Brownian motion, using the Itô formula $$du(t,X(t)) = \frac{\partial u}{\partial t}dt + ...
2
votes
0answers
16 views

How to solve SDE that looks like OU process

I'm trying to figure out how to solve the following SDE, $$ dZ_t = -\kappa(Z_t-\mu)dt + Z_tdW_t $$ It looks really similar to the OU process but applying the integrating factor approach which ...
0
votes
0answers
29 views

How to calculate variance - SDE

I have the SDE $$\large dS_t = \mu S_t dt + \delta S_t^{\beta/2}dB $$ where $\delta, \beta$ and $\mu$ are constants. I need to calculate the variance of $dS_t/S_t$ (the returns) I have the following ...
1
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0answers
17 views

What is the Euler Lagrange condition for SDEs?

Does the Euler Lagrange condition... $$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0$$ ...have a meaningful extension to Stochastic Differential ...
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0answers
12 views

Sample variance matlab geometric brownion motion

I have a question about the geometric Brownian motion. I want to sample many paths and then showing that the sample variance equals the exact variance: $$\mathrm{Var}\left[S(t)\right]=S_{0}^2 e^{2 \mu ...
3
votes
1answer
94 views

Reversible Ito Diffusions

I have given a diffusion equation $$ dX_t = -\nabla V(X_t) \, dt + \sigma dB_t.$$ I found here(1) a characterization when $X_t$ is reversible, aslong as $\sigma=1$. Is this also true for $\sigma ...
0
votes
0answers
18 views

Expectation of Modified Stochastic Integral

Suppose one has a stochastic differential equation: $$dX_t = f(X_t) dt + g(X_t)d\eta(t)$$ where $\eta$ solves the Ornstein-Uhlenbeck process: $$d\eta(t) = \lambda \eta(t) dt + \sigma dW(t)$$ Suppose ...
3
votes
1answer
73 views

Ornstein–Uhlenbeck SDE.

I am trying to understand the solution to the following exercise, however it is kind of poorly written. Can someone please explain it to me? For $V = (V_t)$ the solution to the Ornstein-Uhlenbeck SDE ...
2
votes
2answers
20 views

Using Ito's formula, write down a stochastic diferential equation satiesfied by $Y_t:=X_t^2$, given both $Y_t$ and $X_t$

I am trying to solve this exercise and I am stuck in the third part of it. I checked the solution and it makes no sense to me, so I would really appreciate it if someone could explain to me how Ito's ...
0
votes
0answers
12 views

Large Deviation Theory

Consider a differential equation of the form: $$dX_0 = f(X_\epsilon) dt$$ and it's perturbed form: $$dX_\epsilon = f(X_\epsilon) dt+ \epsilon dW(t)$$ It's well-known that if one assumes $f$ is ...
3
votes
1answer
51 views

Solving an SDE: $dX=-Xdt+e^{-t}dW$

I have the following problem which comes with the solution, but I am unable to obtain the solution... Any help would be greatly appreciated - I am preparing for finals :( Thanks a lot! The SDE that I ...
1
vote
1answer
82 views

Strong solutions SDE inequality with an application of Gronwall's inequality

Suppose that we have a general SDE on a probability space $(\Omega,\mathcal{F},P)$ defined by: $$ dX_t = b(t,X_t) dt + \sigma(t,X_t) d W_t, $$ where $W$ is a Brownian motion and $b$ and $\sigma$ are ...
1
vote
0answers
30 views

Extension of Law of Iterated Logarithms

Suppose I have a stochastic differential equation ($X_t$ is a vector) $dX_t = f(X_t) dt + \sigma g(X_t) d\eta(t)$ and define $V = \sum_{i=1}^{n} x_i$. Here, $\eta(t)$ is an Ornstein-Uhlenbeck process. ...
1
vote
0answers
15 views

Confusion with indexes in this Stochastic D.E

I need to solve for $dS_n = 2S_ndt + 3S_ndB_t$ with $S_0 = 2$ If I were to substitute Ito's formula, would it appear in this form:? $d \ln S_n = f'(S_n)dS_n + \frac{1}{2} \sigma ^2 (S_n) ...
1
vote
1answer
38 views

Stochastic Integral Question

I'm reading a paper on noise and had a question about the stochastic integral. In the paper, they consider the SDE: $$dX = \lambda Xdt + \epsilon dW$$ which has the solution $$ X(t) = \epsilon ...
0
votes
0answers
21 views

Functions of Brownian Motion and Time

Sorry, this will be a little long. I'm currently working on a problem where I basically have an SDE logistic equation: $$dX_t = diag(x_1,\cdots, x_n)[b+Ax-\lambda \eta(t)] dt + diag(x_1,\cdots, ...
-1
votes
1answer
89 views

How do I solve this SDE (stochastic differential equation)?

I am stuck in trying to solve this equation \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} Here, $b$ is a constant. I am trying to apply my usual methods for ...