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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4
votes
0answers
21 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 ...
4
votes
2answers
26 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 ...
1
vote
1answer
51 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
15 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
29 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
votes
0answers
39 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, ...
6
votes
0answers
76 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 ...
1
vote
1answer
54 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 ...
0
votes
0answers
39 views
+100

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
56 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
18 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
10 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
votes
0answers
13 views

Can someone help me solve this SDE?

I am having a problem about solving a SDE. The SDE is dx_t = dB_t + (v - x_t/(T-t)^2theta)dt, where T and v are fixed number, theta can vary. I have tried numerically solve this, but the ...
1
vote
1answer
38 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, ...
5
votes
0answers
83 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 ...
0
votes
1answer
16 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 ...
3
votes
0answers
36 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
21 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
19 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
62 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
27 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
28 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
22 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
46 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
20 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
17 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 = ...
1
vote
0answers
44 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
30 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 ...
0
votes
0answers
9 views

Decouple system of Ito SDEs

Consider $\begin{split} \newcommand{\d}{\mathrm d} \d x &= -yx \d t + x^2 \d B\\ \d y &= -2 y^2 \d t + 2xy \d B. \end{split}$ This is a system of two SDEs driven by the same standard ...
1
vote
1answer
57 views

Itos formula on a transformation of bessel Processes

Let $W$ be a Brownian motion and $z,\kappa>0$. Let $X_t(z)$ be a solution to the SDE $$dX_t(z)=dW_t+2/(\kappa X_t(z))dt.\quad X_0(z)=z.$$ The solution is well-defined on $t<\tau(z)$ where ...
0
votes
0answers
14 views

Stochastic Boundedness/Permanence

Suppose $X_t$ is a solution to an Ito stochastic differential equation $dX_t = f(X_t)dt + g(X_t)dW_t$ where $f,g$ are polynomials. According to some papers I've read, the solution is said to be ...
1
vote
1answer
19 views

Expectation equals to Black-Scholes Equation

Let $S_t$ be ageometric brownian motion with parameters $\sigma$ and $r$ and fix $T,K\in (0,\infty)$. How can I show that: \begin{align} \mathbb{E}[e^{-rT}max\{(S_T-K),0\}] & = ...
0
votes
0answers
32 views

Characterisations of Markov Processes: SDE's , Generators,…

There are different characterisations of a Markov Process: Probability Semigroups, Generators, even in some cases by Jumps Chains and Holding Times... And I know that, in "real life", the only thing ...
3
votes
0answers
54 views

Fokker Planck and SDE

I have the following Fokker-Planck equation in spherical coordinates $(\theta,\phi)$: $$ \partial f/ \partial t= D \cot\theta \quad \partial f/\partial \theta + \quad 1/\sin^2\theta \quad ...
3
votes
0answers
46 views

conversion from stratonovich SDE to Ito's form?

conversion of stratonovich SDE to Ito SDE (Where $\partial$ is differential in the stratonovich form and $d$ is in ito's form): $$\partial X_t=\sigma(X_t,t)\partial B_t+b(t,X_t)\partial t$$. ...
0
votes
1answer
20 views

Non-linear SDE: About the noise time-step

This is a follow-up on my previous post about stochastic differential equations. In the answer from @LuztL, and in the literature, I read commonly that the time-step of the noise should be somewhat ...
2
votes
1answer
48 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
25 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
10 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
29 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
28 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, ...
1
vote
1answer
60 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
47 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
votes
0answers
30 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
vote
0answers
58 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
74 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 ...
1
vote
0answers
24 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 ...