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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33 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
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1answer
16 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. ...
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9 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 ...
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0answers
23 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 ...
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0answers
22 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 ...
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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 ...
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0answers
10 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 ...
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1answer
20 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, ...
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0answers
26 views

Itō formula for a scalar valued function of the solution of a scalar Itō SODE

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $B$ be a real-valued $\mathcal F$-Brownian motion on $(\Omega,\mathcal ...
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75 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 ...
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1answer
14 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 ...
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0answers
35 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 ...
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0answers
10 views

Find SDE coefficients such that solution follow given distribution

Consider a cdf F and suppsoe $F^{-1}$ is also known. Denote by W a standard Brownian Motion. 1) Find function G such that random variable $G(W_1)$ has cdf F. Answer: $G(x) = F^{-1}(Ф(x))$, where Ф ...
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1answer
13 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: ...
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0answers
18 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 ...
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1answer
57 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 ...
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0answers
25 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 ...
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0answers
25 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 ...
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1answer
21 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 ...
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1answer
44 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 ...
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0answers
19 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 ...
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0answers
15 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 = ...
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0answers
41 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 ...
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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 ...
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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 ...
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1answer
55 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 ...
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0answers
13 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 ...
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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\}] & = ...
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0answers
31 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
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0answers
48 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
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0answers
37 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): $$\partial X_t=\sigma(X_t,t)\partial B_t+b(t,X_t)\partial t$$. ...
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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
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1answer
46 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 ...
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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 ...
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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 ...
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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 ...
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0answers
25 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, ...
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1answer
59 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 ...
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1answer
42 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, ...
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0answers
25 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 = ...
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56 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 ...
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2answers
73 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 ...
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0answers
23 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 ...
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0answers
61 views

Geometric Brownian Motion Properties

I was reading Oksendal's book "Stochastic Differential Equations", fifth Ed., pp. 62-63 and came across some counter-intuitive properties of the Geometric Brownian motion (GBM). Let $\alpha,r>0$ ...
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1answer
40 views

Solving equation with Wiener process

I want to show that if $E(f(X_{t}))=E(f(W_{t})e^{\lambda W_{t}-0.5*\lambda^2*t})$, where $W_{t}$ is a Wiener Process, then $X_{t}\sim N(\lambda t,t)$. Does anyone have a clue how to solve this ...
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0answers
27 views

Simulating “Nested” Stochastic Differential Equations

I haven't had much luck over in Stats SE, so I'm going to try over here. I doubt many people here have experience with R, so I would like to know what are some methods to simulating a set of "nested" ...
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0answers
27 views

Noise with heavy tails

The main type of noise I know other than white noise is a colored noise (Ornstein-Uhlenbeck) of the form: $$d\eta = \lambda \eta dt + \alpha dW_t$$ with exponential correlation. I'm interested in ...
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0answers
48 views

Methods of SDE calibration

There is somewhere summary of methods that can be used to estimate parameters of SDE? I currently using MLE and regression due to linear dependence between samples. I searching for something ...
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1answer
41 views

Non-linear SDE: how to?

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1answer
46 views

Check solution to the SDE $dX_t = - \mu X_t \, dt+ \sigma \, dW_t$

I get stuck in this problem. I just can't get the hang of how we need to "guess" a function first and almost everything along the process of solving depends on it; It's not entirely logical to me when ...