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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17 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 ...
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0answers
36 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 ...
3
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1answer
63 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 ...
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1answer
27 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. $$dX_t = 3X_t^{1/3} dt + 3X_t^{2/3} dW_t$$ ...
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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. ...
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0answers
17 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 < ...
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12 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 ...
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1answer
21 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
58 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 ...
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62 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 ...
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22 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) ...
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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
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1answer
26 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
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0answers
58 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 ...
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0answers
21 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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61 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
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1answer
18 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
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0answers
14 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 ...
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22 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 ...
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0answers
14 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
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1answer
93 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 ...
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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
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1answer
69 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
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2answers
19 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 ...
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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
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1answer
47 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 ...
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1answer
75 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 ...
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0answers
25 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. ...
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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) ...
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1answer
37 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 ...
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0answers
20 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, ...
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1answer
79 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 ...
0
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0answers
22 views

Variance in a system of SDEs

I have the following system of SDEs $$\begin{cases}\mathrm{d}X = - \beta XY \,\mathrm{d}t+ \sigma_\beta XY \, \mathrm{d}B_1 \\ \mathrm{d}Y = \beta XY \,\mathrm{d}t - \gamma Y \, \mathrm{d}t- ...
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1answer
45 views

Explosion time of $dX_t=X_t(adW_1+bdW_2)$

I found in Karatzas & Shreve (1991), $dX=\sigma(X_t)dW_t$ cannot explode. But what about $dX_t=X_t(adW_1+bdW_2)$? Here $W_1$ and $W_2$ are independent. Feller's test for explosion seems to work ...
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0answers
68 views

Covariance of Stochastic Differential Equation

What is the general expression for the covariance $cov \left[ X_s X_t \right]$ of a stochastic process given by \begin{equation} dX_t = f(X_t,t)dt + g(X_t,t) dW_t \end{equation} for some general ...
0
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1answer
34 views

Milstein scheme for stochastic differential equation with constant drift

The Milstein scheme to approximate the solution of an SDE is $$ Y_{n+1} = Y_n + a\Delta_t + b\Delta W_t + \frac{1}{2} bb' ((\Delta W)^2 - \Delta) $$ where $\Delta_t$ is the time step size (usually ...
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1answer
96 views

Using Markov Property in solving PDE/SDE

I am solving the PDE I used Feynman-Kac and eventually arrived at $F(t,x)$ $ = E[X_T^2|X_t = x]$ $ = E[(X_t \pm \sigma (W_T -W_t))^2|X_t = x]$ (iirc) So, I try to evaluate $E[(X_t \pm \sigma ...
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1answer
129 views

Expected value and variance of a stochastic process

Having trouble finding expected value and variance of a stochastic process defined by SDE: $dX_{t} = a X_{t} dt + b dB_{t}$ $X_0 = x$, $a$ and $b$ are constant values, $B_t$~$N(0,t)$ Thank you for ...
3
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0answers
79 views

quadratic SDE solution

I have this SDE $dX_t=[a+bX_t+sX_t(1-X_t)]dt+\frac{1}{2}X_t(1-X_t)dW_t, \, X_0=0,$ where $a,b \in(0,1)$ and let's say that $s$ is a real constant (it's actually a function of $X$, but I think I can ...
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0answers
24 views

Clarifications about SDEs, Differentials & Derivatives

A general SDE look like the following: $$ \mathrm{d}\psi=a\mathop{}\!\mathrm{d}t+b\mathop{}\!\mathrm{d}W,\tag{1} $$ where $\psi:t\mapsto y = \psi(t)$ is the solution, while $a$ and $b$ can be both, ...
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0answers
26 views

Strictly local martingales: what is the intuition behind them?

I did post this on the Quant Finance exchange a while back, but without any luck A process $X_t$ is a local martingale if for each increasing sequence of stopping times $\{τ_k,\ k=1,2,...\}$ the ...
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66 views

What is some reason that there are no book bridge the gap of these three books

I am referring to the (beginner's text- Stochastic Calculus by Mircea Grigoriu and Introduction to Stochastic Calculus by klebaner.) and the advanced texts - stochastic differential equation by ...
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1answer
109 views

Solve the SDE $dX_t = \frac{1}{2 X_t} dt + dB_t$ [closed]

Solve the following stochastic differential equations $ dX_t = \frac{1}{2 X_t} dt + dB_t$ or equivalently with a transformation $Y_t = X_t^2$ $ dY_t = dt + 2 \sqrt{Y_t} dB_t$ with $Y_0 = y_0 > ...
1
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1answer
65 views

Do we need Feller condition if volatility jumps?

Consider the SDE: \begin{equation} dv_t = k(\theta - v_t) dt + \xi \sqrt{v_t} dW^{v}_{t} \end{equation} It describes a process $v_t$ which is a strictly positive if the drift is stronger enough, i.e. ...
5
votes
1answer
76 views

Bounding an expected hitting time

Consider a stochastic differential equation: $$dX_t = dW_t + \sin(X_t) dt, \, X_0 = x$$ where $W_t$ is a Wiener process. Define $$\tau_1 = \inf \{ t : X_t \in 2 \pi \mathbb{Z} \} \\ \tau_2 = \inf ...
1
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1answer
79 views

Itô formula + SDE

I have a problem with solving the following problem: I.e. I want to show that $X_t$ is a solution to the SDE by employing the Itō formula. Now the problem is I don't get how I should set the ...
0
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0answers
101 views

Expectation of geometric mean reversion process

The second part to a question I asked here in which I had to show that the solution to $dX_t = \kappa\left(\alpha-\ln X_t \right)X_t dt + \sigma X_t dB_t$ was $ X_t = \exp \left( \mathbb{e}^{-k ...
2
votes
0answers
51 views

How to solve system of stochastic differential equations?

I have the following two SDEs $$dN_1=(2a-1)pN_1dt+\alpha_1 N_1dW_1$$ $$dN_2=(2pN_1-\mu N_2)dt+\alpha_2 N_2dW_2$$ $W$ is the standard Brownian motion/Weiner process. This isn't homework, I'm just ...
3
votes
1answer
93 views

conditional expectation of some solution of SDE

Let $(M_t)$ be a nonnegative martingale in a probability space $(\Omega, \mathcal{F}, \{ \mathcal{F}_t \}, \mathbb{P} )$ given by \begin{equation} dM_t = M_t \sigma_t dW_t \end{equation} for some ...