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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14 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) ...
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1answer
21 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 ...
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14 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, ...
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0answers
18 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- ...
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1answer
30 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 ...
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0answers
42 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 ...
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1answer
17 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 ...
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1answer
87 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 ...
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1answer
96 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 ...
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71 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 ...
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0answers
22 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, ...
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0answers
24 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 ...
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0answers
64 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 ...
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1answer
86 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 > ...
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0answers
24 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. ...
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1answer
68 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 ...
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1answer
60 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 ...
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0answers
76 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 ...
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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
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1answer
82 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 ...
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1answer
29 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 ...
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1answer
90 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
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1answer
77 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
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3answers
121 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
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2answers
103 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} ...
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1answer
59 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 ...
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1answer
48 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 ...
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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: ...
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0answers
37 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, ...
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1answer
102 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 $\delta$ ...
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1answer
61 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 ...
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1answer
77 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
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1answer
95 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
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0answers
42 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
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1answer
150 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 ...
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0answers
17 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 ???
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1answer
35 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 + ...
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0answers
41 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 ...
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0answers
21 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
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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 ...
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1answer
89 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. ...
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2answers
591 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 ...
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0answers
57 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. ...
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1answer
34 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 ...
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1answer
62 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?
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2answers
103 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
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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
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1answer
121 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
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0answers
27 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
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1answer
58 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 ...