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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3
votes
0answers
14 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
69 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
55 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
19 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
17 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
58 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
65 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 > ...
0
votes
0answers
15 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. ...
4
votes
1answer
49 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 ...
0
votes
1answer
43 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
48 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
45 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
72 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
27 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
74 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
67 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
114 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
80 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
55 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
44 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 ...
0
votes
0answers
34 views

basic Stochastic differential equation

I'm sorry but I'm having some troubles to find a solution of this simple stochastic differential equation, $dX_{t}=2\sqrt{X_{t}}dB_{t}+2dt$ where $B_{t}$ is a Brownian motion, please can you help ...
1
vote
0answers
23 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
30 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, ...
0
votes
1answer
94 views

How to solve a linear stochastic differential equation?

I don't know how to find a solution of this stochastic differential equation: $dX_{t}=(1+\delta \mu X_{t})dt+\delta X_{t}dB_{t}$ Where $B_{t}$ is a standard Brownian motion and $\mu$ and $\delta$ ...
1
vote
1answer
59 views

Rewriting Diffusion Processes: Combining Independent Wiener Processes

In stochastic calculus, a rule of thumb for computations is $(dW_t)^2 = dt$ for a Wiener process $W_t$. Say we have a diffusion process $dX_t = dW^1_t + X_t dW^2_t$, with $W_t^1, W_t^2$ independent ...
0
votes
1answer
60 views

Ito with the function containing stochastic integral

Statement of problem From Oksendal SDEs question 5.18: The geometric mean reversion process is a solution to: $$ dX_t = k (a - \log X_t) X_t dt + \sigma X_t dB_t $$ In showing that solution is ...
2
votes
1answer
88 views

Solve $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t) \, dt + \sqrt{1+X_t^2} \, dW_t$ explicitly

Solve explicitly the 1-dimensional equation: $dX_t = (\sqrt{1+X_t^2} + \frac{1}{2}X_t)dt + \sqrt{1+X_t^2}dW_t$ I have hopelessly been guessing solutions to this. Does anyone know how to solve this ...
2
votes
0answers
38 views

Explicit solution SDE?

I have the following SDE: $$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$ where ...
0
votes
1answer
74 views

Quadratic Variation of Diffusion Process and Geometric Brownian Motion

I'm looking to find out the stochastic differential equation satisfied by the quadratic variation of Geometric Brownian Motion, Diffusion Process. For example, for a diffusion process that ...
0
votes
0answers
15 views

Time homogeneous asset dynamics model

I'm studying asset process. As i know, Black scholes model and CEV model is time homogeneous diffusion model. Are there time homogeneous model ???
1
vote
1answer
33 views

Given a process what is the stochastic differential equation it fulfils?

Given the process $X_t = (2+t+\exp(W_t))_t$ where $W_t$ is Brownian motion. How can I find the SDE that it fulfils. I am actually looking for two functions $\sigma, \tau$ such that $X_t = X_0 + ...
0
votes
0answers
36 views

Matlab code for higher order scheme

Can somebody help me how to generate the code for the increment $\Delta$Z in the document I have attached? I know how to generate the rest of the increments but struggling in how to generate ...
0
votes
0answers
19 views

Milestein Scheme

Im struggling in the following schemes. I cant understand how the first scheme is equivalent to the second one. Can somebody help me? Thanks in advance. Moreover there is a typo error in the ...
3
votes
1answer
156 views

Stochastic inequality, true?

Consider two stochastic processes $X$ and $Y$ satisfying the following SDEs (with the same drift!): $$X_t = x + \int_0^t b(X_s)ds + B_t$$ $$Y_t = y + \int_0^t b(Y_s)ds + B_t.$$ If $0<x<y$, is ...
0
votes
1answer
79 views

Solution to the linear SDE $dX_t = \alpha X_t \, dt + \sqrt{2} dB_t$ using Itô calculus

So if I have the following generator and an initial condition: $$A(f)(x) = \alpha x f'(x) + f''(x) \\ X_0 = x \in \mathbb{R}^+$$ I've been asked to find $X_t$ and assume that $\alpha$ is a constant. ...
2
votes
2answers
430 views

Matlab Code to simulate trajectories of Ito process.

I need some help to generate a Matlab code in order to do the following question. Can somebody help me in this regard. Any sort of hint that could be helpful will surely be appreciated.. Q: "Simulate ...
0
votes
0answers
51 views

Stochastic Differential equations with $\sin(x^2)$ as drift.

Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance. ...
-1
votes
1answer
30 views

inverse function type SDE

SDE $dX_t=-a^2\sin X_t\cos^3X_tdt+a\cos^2X_tdW_t$ with $X_0=x_0$ I think this is inverse type of SDE, refer to Itô's formula and SDE. However, I can't find the inverse funcition. My try ...
0
votes
1answer
56 views

Strong solution of stochastic differential equation

Consider the stochastic differenctial equation: $dX_t=\frac34 X_t^2 dt-X_t^{3/2}dW_t$. How to find a strong solution?
-3
votes
2answers
92 views

How to solve the SDE $dX_t = \frac{b-X_t}{T-t} \,dt + dW_t$?

SDE: $$dX_t=\frac{b-X_t}{T-t}dt+dW_t,t<T, \qquad X_0 = a$$ Answer: Let $b(t)=\frac{-1}{T-t},c(t)=\frac{b}{T-t},\sigma(t)=1$, then $$\begin{align*} ...
0
votes
1answer
52 views

On the confinement of the solution of a SDE

One considers the stochastic differential equation $$dX_{t}=(1-X_{t})X_{t}dB_{t},$$ with $B$ Brownian motion, and one assumes that $0\leq X_{0}\leq1 $. One wants to show that ...
0
votes
1answer
113 views

Solve the SDE $dX_t = \frac{1}{2}\sigma(X_t)\sigma'(X_t)dt+\sigma(X_t)dW_t$

Solve this SDE: $dX_t=\frac{1}{2}\sigma(X_t)\sigma'(X_t)dt+\sigma(X_t)dW_t$ with $X_0=x_0$ My try is let $f(x)=\int_{x_0}^{x}\frac{dy}{\sigma(y)}$ and $(f^{-1})'=\sigma(x),(f^{-1})''=\sigma'(x)$ ...
0
votes
0answers
15 views

Are stochastic differential equation related to the uncertainty principle?

According to Newton it makes sense to assume that location and momentum are completely determined which motivates to model a molecule with a PDE on phase space. According to the uncertainty principle ...
0
votes
0answers
20 views

Stationary distribution for a square root process.

Consider the SDE $$ dX_t = \kappa (\omega - X_t) dt +\eta_1 \sqrt{X_t} dW_t $$ then I have been told that if $\kappa \cdot \omega >0 $ and I choose reflecting behaviour at 0 in case $\kappa ...
0
votes
1answer
51 views

Stochastic differential equation with trigonometric functions

I heard that the following SDE can be solved analitically by substitution: $dX(t) = - \left[ \sin (2 X(t) ) + \frac{1}{4} \sin (4 X(t) ) \right] dt + \sqrt{2} \cos^2 X(t) dB(t),$ $X(0) = 1, \; t \in ...
1
vote
1answer
81 views

Solve the linear SDE $dX_t = aX_t \, dt +(b+cX_t) \, dW_t$

I am trying to find the solution to the SDE: $$ dX_t=aX_tdt+(b+cX_t)dW_t $$ for $t\ge0$, $X_0>0$, constants $a,b,c$ Would appreciate any hints as to how to approach this using ito's formula, I'm ...
2
votes
1answer
125 views

Solving a Stochastic Differential Equation (SDE)

Question: Solve the stochastic differential equation: $$ dX_t=X^3_t\,dt-X^2_t\,dW_t $$ where: $$ X_0=1 $$ My Attempt: Using Ito's with: $$ f(x)=\log(x) $$ I get that: $$ ...
2
votes
0answers
116 views

How to solve this SDE ? stuck half way

Problem: $dX_t = \sigma X_tdB_t$, $X_0=x$. $dY_t=X_tdt-Z_tdt$ find $Y_t$, where $Z_t$ is a control and $B_t$ is standard Brownian motion. My attempt: From Ito's lemma, $\partial_BX_t=\sigma X_t$, ...
2
votes
0answers
34 views

Question about a Bessel process

Are there any explicit path solutions for a 3 dimensional Bessel process? E.g. the Ito SDE: $$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$ where $W_t$ is a standard Wiener process.
0
votes
1answer
48 views

Path solution for a SDE

I would like to get help in solving an Ito stochastic equation: $dY_t=-dW_t \, (Y_t^2+1)$ The process $W_t$ is the standard Brownian motion. Is it possible to get a path solution solution in terms ...