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.

learn more… | top users | synonyms

2
votes
0answers
13 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) ...
-1
votes
0answers
23 views

integrate the exponential of brownian motion [closed]

I want to solve this integral to use it in the solution of the linear inhomogeneous SDE: $$ \int_0^t \exp\left(-as-bW_s\right)ds $$ with $a>0$, $b>0$. How can it be solved? Just a reference ...
2
votes
0answers
21 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
21 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 ...
1
vote
0answers
42 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
vote
0answers
20 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
votes
0answers
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
votes
1answer
15 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
13 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
17 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
vote
0answers
12 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 ...
0
votes
0answers
11 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
90 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
17 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
63 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
18 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
42 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
71 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
22 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
36 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
18 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
72 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
votes
0answers
21 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- ...
0
votes
1answer
42 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 ...
1
vote
0answers
59 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
votes
1answer
30 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 ...
1
vote
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 ...
0
votes
1answer
121 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
votes
0answers
78 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 ...
0
votes
0answers
23 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, ...
0
votes
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 ...
0
votes
0answers
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 ...
-2
votes
1answer
103 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
vote
1answer
59 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
vote
1answer
75 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
votes
0answers
90 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
50 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
92 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 ...
0
votes
1answer
31 views

Determine the dynamics (SDE) of X_t^4

Suppose X and Y evolve according to: dXt= (2 + 5t + Xt)dt + 3 dz_1t dYt= 4Ytdt + 8Ytdz_1t + 6dz_2t where z_1t and z_2t are Brownian motions with (dz_1t )(dz_2t )=0.1dt Can you give me some ...
1
vote
1answer
95 views

How to solve the SDE $dX_t = aX_tdt + (b(t)-X_t^2)^{1/2}dW_t$?

I need help on solve the following SDE: $\beta > 0$, $0<\gamma<1$, $X_0 = \frac{\sqrt{2}}{2}$ $$dX_t = -(\beta + \frac{1}{2}\gamma^2)X_tdt + \gamma\sqrt{e^{-2\beta t}-X_t^2}dW_t$$ I need ...
0
votes
1answer
80 views

How to solve the SDE: $dX_t = \frac{1}{X_t}dt + X_tdW_t$

I have difficulties in solving following SDE: $$dX_t = \frac{1}{X_t}dt + X_tdW_t$$ I tried the transformation method provided in the following link: Name of the formula transforming general SDE to ...
3
votes
3answers
124 views

1-dimentional stochastic differential equation

I would like to solve this SDE $$dX_{t}=\left(\sqrt{1+X^{2}}+\dfrac{1}{2}\right)dt+\sqrt{1+X^{2}} dB_{t}$$ I've tried to solve first the homogeneous equation ...
2
votes
2answers
151 views

Connections between SDE and PDE

I have encountered a number of situations where the solution of a PDE and a certain expectation associated to a Markov process are equal. Two examples include: The heat equation $u_t = \frac{1}{2} ...
0
votes
1answer
65 views

Question about Girsanov theorem

Tn the book "Stochastic Differential Equation" from Oksendal one can find the following theorem(6th edition, Theorem 8.6.8): Let $X(t)=X^x(t)$ and $Y(t)=Y^x(t)$ be an Itô diffusion and an Itô ...
0
votes
1answer
60 views

Stochastic Differential equation, expectation and variance

The process is given by $$dU_t=-\gamma U_t\mathrm{d}t+\sigma\mathrm{d}X_t$$ where $U_0 = u$ and $\gamma, \sigma$ are constants. Can you help me out to solve the equation for $U_t$ and find the ...
1
vote
0answers
24 views

Infinitesiman generator of Time dipendent process

I'm trying to find the infinitesiman generator of this process $dY_{t}=\dfrac{b-Y_{t}}{1-t}dt+dB_{t}$ $0\leq a <1$, $Y_{0}=a$ where $B_{t}$ is a brownian motion; and I've found the solution: ...
0
votes
0answers
44 views

Numerical solution of SDEs with fractional Brownian motion

I am trying to numerically solve some SDEs representing a nonlinear circuit (possibly chaotic) driven by noise: $$ dX = f(X) dT + \sqrt{P_{w}} dW + \sqrt{P_{f}} dC $$ where $X$ is my circuit state, ...