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, …), ...

learn more… | top users | synonyms

0
votes
1answer
27 views

Question about using Ito Chain Rule to Understand an SDE

Given $X_t = (x^{1/3}+ \frac{B_t}{3})^3, t \ge 0$, $X_0 = x$, why does $$dX_t = \left(\frac{X_t^{1/3}}{3} dt + X_t^{2/3}dB_t\right)?$$ I am having trouble because differentiation of the given ...
0
votes
0answers
21 views

Stochastic Differential Equation Time-Independent

I know that a generic 1-D SDE can be written in Ito form as: $dX_{t} = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$. I was curious as to how such an SDE is written when modelling time-independent processes. I ...
1
vote
0answers
22 views

Proof about solution to SDE with Lipschitz condition:

$X_t$ follows the Ito process as described by the following stochastic differential equation $$dX_t=b(X_t)dt+dB_t\quad , \quad X_0=x$$ and $b(X)$ satisfies Lipschitz Coditions.I want to show for ...
0
votes
1answer
50 views
+50

How to prove a solution of SDE with Lipschitz condition is unbounded?

For Simplicity $ (\Omega ,\mathbb F,\mathbb P)$ is a probability space, the S.D.E is $dX_t=b(X_t)dt+dB_t$ and it's 1 dimensional, $b(X_t)$ is Lipschitz Continuous (and let's assume with constant 1),...
0
votes
0answers
38 views

mean evolution of 1D Fokker-Planck

Given the Fokker-Planck equation on 1D with drift term $D^{(1)}(x)$, and diffusion term $D^{(2)}(x)$, and the governing equation of probability density function $$\frac{\partial f(x,t)}{\partial t} = ...
0
votes
1answer
19 views

Simulating systems of SDEs in Mathematica and plotting solutions.

I would like to simulate the solution of the following system of SDE's: \begin{equation*}\begin{split} dX(t) &= - \frac{1}{2}X(t)\, dt - Y(t)\, dW(t)\\ dY(t) &...
3
votes
0answers
36 views

Computation of an Ito Integral [closed]

I have the following Ito Integral $$\begin{align}\int_{0}^{t}B_se^{-\sigma B_S}dB_s&& \end{align} $$ Here, $\sigma \gt 0$. Can someone please show me how to compute this Ito integral? ...
4
votes
1answer
27 views

Proving an SDE has a unique strong solution

I have the stochastic differential equation $$dX_t = \ln(1+ X_t^2) \, dt + X_t \, dB_t$$ In this equation, $X_0 = x$, and $x \in\mathbb R$. How can we show that this equation has a unique strong ...
0
votes
0answers
12 views

SDE with initial and terminal conditions

I have the simple SDE $$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$ with conditions $X_0$ and $X_T$ given. I would like to compute $E(e^{-\int_0^TX_sds})$ conditional on $X_0$ and $X_T$. What is the general ...
1
vote
1answer
25 views

Approximations in Stochastic Differential Equations

Given a general SDE: $dX_t=b(X_t,t)dt+\sigma (X_t,t)dB_t$ , $X_0 =x$ and a solution $X^x_t$ . Where $b|x-y|+\sigma |x-y|\leq |x-y|$ . Prove that: $E(X^x_t)^2\leq L exp(Lt)$ for some none ...
0
votes
0answers
45 views

Why are we allowed to multiply the differential form of an SDE by a function?

Suppose for example that we have the following SDE: $$dX_{t} = a(X_{t})\,dt + b(X_{t}) \,dB_{t}. $$ What rigorous justification is there for then saying, for example, that we can multiply both sides ...
0
votes
1answer
23 views

How to show that $I_tdX_t + I_tP_tX_t dt = d(I_tX_t)$ using Ito's Lemma? [closed]

$dX_t +P_tX_tdt = dB_t$ and the integrating factor is $I_t = e^{\int{P_t dt}}$
0
votes
1answer
55 views

Expected value of sde.

I have those sdes: $$dS(t) = mS(t)dt + e^{Y_t}S(t)dB(t)$$ and $$dY(t) = a(v − Y(t))dt + ab(pdB(t) +(1-p)dW(t))$$ where B,W independent Brownian motions and $a,v,p,b$ and $m$ known constants. How can I ...
0
votes
0answers
21 views

Uniform Boundary for S.D.E with Lipschitz Coefficients

Edit of progress: Since the SDE is linear, I got a solution in the form of $e^\int...$$\cdot e^\int$. By Jensen's inequality I can change the order of the left factor to have the integral on the ...
0
votes
0answers
27 views

How to solve this SDE

I have been learning basic stochastic analysis, and we have only been taught about Ito formula. The professor told us how can we solve this question below using it, but I miss it. Can anyone help me? ...
2
votes
0answers
38 views

Uniqueness of the trajectories of the solution of an SDE

Consider an SDE \begin{equation} dX_t=f(X_t,t)dt+b(X_t,t)dW_t \end{equation} Suppose firstly that the coefficient are Lipschitz continuous. So by the theorem of existence and unicity I have that exist ...
1
vote
0answers
30 views

Help with solving ODE

This is actually a part of an SDE problem, which I am having problem to solve. However i think it requires only ODE knowledge to solve. I have $$\mu(u) = x^3 + 3 \int_t^u a \mu(s) ds + 3 \sigma^2 \...
1
vote
1answer
41 views

Girsanov theorem calculations help

I need help to understand a couple of calculations in this Girsanov theorem related SDE problem. I have five questions as stated below. Let $X_t$ solve the Ornstein-Uhlenbeck equation $$dX_t = X_t\, ...
3
votes
1answer
45 views

Stochastic differential equation substitution reasoning?

I need help to explain reasoning behind why we choose certain substitutions to solve SDE. After choosing the correct substitution the solution of the SDE are often quite easy. However I have trouble ...
2
votes
0answers
46 views

Expected hitting time of a stochastic differential equation with jumps (neuroscience example)

The basic model I'm working with is a neuron that receives input from other neurons which cause instantaneous spikes in the voltage. In a nutshell, I have a differential equation that describes the ...
10
votes
1answer
117 views

Exploiting the Markov property

I've encountered the following problem when dealing with short-rate models in finance and applying the Feynman-Kac theorem to relate conditional expectations to PDEs. Let $(\Omega,\mathcal{F},\{\...
4
votes
2answers
31 views

Solving a nonlinear scalar Ito SDE

I need to solve the SDE: $$ dX_t = (X_t)^3 dt + (X_t)^2 dW_t ; X(0)=1 $$ Now what I found is this is an SDE of the form: $$dXt =a(X_t)dt+b(X_t)dW_t$$ where $a(x) = \frac{1}{2} b(x)b′(x)$ Using the ...
2
votes
1answer
81 views

Solution to General Linear SDE

In order to find a solution for the general linear SDE \begin{align} dX_t = \big( a(t) X_t + b(t) \big) dt + \big( g(t) X_t + h(t) \big) dB_t, \end{align} I assume that $a(t), b(t), g(t)$ and $h(t)$ ...
1
vote
0answers
18 views

Is the generated semigroup by an elliptic operator be the transition semigroup?

I am considering the time homogeneous Ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dW_t,$$ where $b,\sigma$ currently are only assumed to be global Lipschitz for the existence of solution $X_t$. The ...
0
votes
0answers
38 views

Stochastically perturbed fluid flow map ${\rm d}Φ_t(x_0)=u_t(Φ_t(x_0)){\rm d}t+ξ_t(Φ_t(x_0)){\rm d}W_t$

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q$ be a bounded, linear, nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\ge 0}...
0
votes
0answers
44 views

Feller property of Ito diffusion

Consider the following Ito diffusion $X_t$ satisfying $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad X_0=x,$$ with Lipschitz coefficients $b,\sigma$. It can be shown that if $g$ is bounded and continuous, ...
7
votes
0answers
89 views

Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Let$^1$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\...
2
votes
1answer
57 views

Simple question on interpreting Geometric Brownian Motion SDE

I'm writing an overview that is more economic than mathematical and I want to explain shortly the stochastic differential equation of Geometric Brownian Motion as simple and clear as possible $$dS_t ...
1
vote
0answers
48 views

Reformulate a SPDE parameterized by space and time as an SDE parameterized by time (as it is possible for PDEs)

Let $d\in\mathbb N$ $\mathcal V_t\subseteq\mathbb R^d$ be a bounded domain for $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ be bijective for $t\ge 0$ with $$\Phi(\;\cdot\;,x_0)\in C^1\left(\mathbb ...
0
votes
0answers
59 views

Why do we always consider real-valued $f$ in the Itō formula to find an expression for $f(t,X_t)$

The Itō formula (see Da Prato, Theorem 4.32) yields an expression for $f(t,X_t)$ where $${\rm d}X_t=\phi\;{\rm d}t+\Phi\;{\rm d}W_t\;,\;\;\;X_0=\xi\;.\tag 1$$ Even when $X$ takes values in a Hilbert ...
2
votes
1answer
31 views

Chain rule for derivatives in SDE

I'm having trouble understanding applying chain rule to SDEs or actually chain rules in general. It has been a while since I took rudimentary calculus classes, so I might be slipping on the basics. ...
1
vote
0answers
12 views

How to formulate and analyze systems of stochastic differential equations?

I'm having trouble finding reference material on how to deal with systems of stochastic differential equations. Specifically, I'm interested in ecological models. For example, consider the standard ...
1
vote
0answers
28 views

Mathematical tools for iterative / composed function analysis

Given the following: a. x0 = known scalar b. x1 = known scalar c. x2 = unknown scalar d. an unknown iteratively applied stochastic nonlinear function $\mathbf{g}$ where     &...
1
vote
0answers
25 views

For Ito diffusion, what is the difference between two measures $Q^x$ and $P$?

I am confused about the difference between $Q^{s,x}$ and $P$ for the following ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad t\ge s;\quad X_s=x.$$ Followings are from most books: given the ...
1
vote
0answers
19 views

Markov property of ito diffusion [duplicate]

Most books show Ito diffusions satisfy Markov property, that is, $E[f(X_{t+h})\mid F_t]=E^{X_t}[f(X_h)]$. But I was wondering whether it's true that $E[f(X_{t+h})\mid X_t]=E^{X_t}[f(X_h)]$. In this ...
1
vote
1answer
49 views

SDE Solution: Hull-White extension of Vasicek model

I am trying to figure out the particular ansatz (if that's all there is) for the solution to the SDE: $ dr_t = [v_t - ar_t]dt + \sigma dW_t, $ where $a$ is constant and $v,t$ are, potentially, time-...
5
votes
0answers
97 views

Brownian motion on sphere proof?

proving the brownian motion on the sphere equation the stratonovich form differential equation $$\partial X=n(X)\times \partial B$$ the equation in ito's form becomes $$dX=n(X)\times dB+H(X)n(X)...
0
votes
1answer
17 views

Solving a simple, linear type SDE

I am a bit confused by SDE's. I am trying to solve the SDE $dX=(c-\mu X )dt+\sigma dB$, with $\mu,\sigma,c$ constants and $X_0=x_0$ deterministic. I already know the solution of $dX=fdt+gdB$ with $X(...
3
votes
0answers
38 views

costruction of brownian motion on sphere?

i am trying to construct a brownian motion on the sphere using the method given in Price and williams paper.$\partial$ represents the SDE of stratonovich type which is converted to ito form in last ...
0
votes
1answer
23 views

Coefficient matching method for SDEs

I am using the book by M.Steele, Stochastic calculus and financial applications and came across the following solution method in chapter 9. The method is called coefficient matching and starts like: $...
0
votes
0answers
22 views

inverse Stochastic differential equation

SDE are really new for me, so I'm sorry if this is a silly question. Let $W_t$ be a Wiener process and let $x_0$ denote the initial value of the process. If I'm correct, for $\text{d}X_t = -(\beta X_t ...
4
votes
1answer
68 views

SDE Integration: Normal-Mean Reverting Process - Question

I am trying to figure out how a particular SDE can be integrated. The SDE is the normal mean-reverting model: $dX_t = \theta(\mu - X_t)dt + \sigma dW_t$ (1) Where $W_t - N(0,t)$. So far, I have ...
0
votes
0answers
29 views

Solve this simple Linear SDE?

How do I solve the following BSDE? $$ \left\{ \begin{aligned} dX_t &=(rX_t+\theta Z_t ) \, dt + Z_t \, dW_t \\ X_T &=\xi \end{aligned} \right. $$ There appears to be nothing online about ...
2
votes
0answers
31 views

Laplace transform of survival probability for stochastic diffusion

Let $Y_t$ be a killed process defined by \begin{eqnarray} Y_t = X_t \quad \mbox{if } t<\xi,\\ Y_t = 0 \quad \mbox{if } t\geq\xi. \end{eqnarray} where $\xi$ is a random time such that $$ \xi=\inf \...
0
votes
1answer
23 views

Applying ito integral to a specific example

Using the ito integral: $$x_t=\int_0^t b(s,x(s)) \, ds+\int_0^t \sigma (s,x(s)) \, dB(s)+x_0$$ I want to solve this: I have $$g(s,n(s))=\log(n(s))$$ Using ito's lemma taking the derivative I get ...
1
vote
1answer
53 views

Solve the stochastic differential equation $dX_t = \alpha X_t \, dW_t + \sigma X_t \, dt$

We want to solve: $dX_t = \alpha X_t dW_t + \sigma X_t dt$ where the initial condition $X_0$ is given and $\alpha$, $\sigma$ are constant. The solution goes as follows - We rewrite the ...
1
vote
0answers
21 views

calculation of mean curvature?

Converting a SDE from stratonovich to ito's form , Stratonovich form $$\partial X=P(X)\partial B$$ $$ P(X)=I-n(X)n(X)^T $$ Ito's conversion $$ dX=P(X)dB +\frac{1}{2}d(P(X))dB $$ $$ dX=P(X)dB +\...
2
votes
0answers
19 views

Solve the SDE $dX_t=\alpha \,dt + \sigma X_t \,dB_t$, $X_0=x_0$ [duplicate]

So I got the following SDE to solve: $$dX_t=\alpha\, dt + \sigma X_t \,dB_t, X_0=x_0$$ This is what I've tried: Using Ito's I should get the following relations: $$X_t=f(s,x)$$ $$\alpha = \frac{1}{...
1
vote
0answers
53 views

Is knowledge of PDE useful for SDE?

I am a stochastic analysis student and am particularly interested in stochastic differential equations. What always struck me as odd is how little PDE (or even ODE for that matter) seems to have ...
0
votes
0answers
33 views

Solved an SDE but the simulation does not match the solution

I solved the following: SDE $dS_t=k\frac{dt}{t}+\sigma dW_t$ with the solution: $S_t=k\ln(t)+\sigma W_t$ assuming that $S_0=0$ (First, is this correct?). The mean and the variance are $k \ln(t)$ and $\...