Diffusion equation for Brownian motion with a systematic drift component

2

$begingroup$

I know, that BM formula

$$X_{i+1} = X_{i}+sqrt{delta t}Z_i,$$ $$Z_isim N(0,1),$$

with $X_0equiv0$ leads to Diffusion equation $$u_t=-frac{1}{2}u_{xx}$$

for marginal distributions in $X_i=X(t),t=idelta t,$ in the limit case $delta tto 0$.

I want to derive Diffusion equation for Brownian motion with a systematic drift component, i.e.

$$X_{i+1} = X_{i}+delta tX_{i}+sqrt{delta t}Z_i,$$

using Chapmann-Kolmogorov equation and Taylor expression.

How can i derive this? What is the answer?

Any explanation is very appreciated.

$$$$

To my calculations:

Let density function for BM $pi_t (x)$ in time $t$ be defined as
$$pi_t (x)=frac{1}{sqrt{2pi t}} exp left (frac{-x^2}{2t}right).$$

According to properties of BW increments we consider Chapmann-Kolmogorov equation as follow:
$$pi_{t+delta t} (x')=int {frac{exp left (frac{-(x'-x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x) dx}.$$

$x = x'+ Delta x, Rightarrow dxrightarrow d Delta x,$
$$Rightarrow pi_{t+delta t} (x')= int {frac{exp left (frac{-(Delta x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x'+Delta x) dDelta x}.$$

Since $Delta x = delta tX_{i}+sqrt{delta t}Z_i$ we have Taylor expansion:
$$pi_t (x'+Delta x) = pi_t (x') + frac{dpi_t(x')}{dx} Delta x + ... + O(Delta x^4).$$

Then
$$pi_{t+delta t} (x')= int {frac{exp left (frac{-(Delta x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x'+Delta x) dDelta x}$$

$$=pi_t (x')+frac{dpi_t}{dx} delta t + frac{1}{2} frac{d^2pi_t}{dx^2}delta t+frac{1}{6} frac{d^3pi_t}{dx^3}3 delta t^2 + O(delta t^3);$$

$$frac{pi_{t+delta t} (x') - pi_t (x')}{delta t}=frac{dpi_t}{dx} + frac{1}{2} frac{d^2pi_t}{dx^2}+frac{1}{2}frac{d^3pi_t}{dx^3}delta t + O(delta t^2);$$

$delta t rightarrow 0:$

$$frac{d}{dt}pi_t (x')=frac{dpi_t}{dx} + frac{1}{2} frac{d^2pi_t}{dx^2}.$$

integration ordinary-differential-equations stochastic-calculus brownian-motion finance

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edited Jan 6 at 12:43

Thomas J

asked Dec 19 '18 at 16:26

Thomas JThomas J

114

$endgroup$

add a comment | 

2

$begingroup$

I know, that BM formula

$$X_{i+1} = X_{i}+sqrt{delta t}Z_i,$$ $$Z_isim N(0,1),$$

with $X_0equiv0$ leads to Diffusion equation $$u_t=-frac{1}{2}u_{xx}$$

for marginal distributions in $X_i=X(t),t=idelta t,$ in the limit case $delta tto 0$.

I want to derive Diffusion equation for Brownian motion with a systematic drift component, i.e.

$$X_{i+1} = X_{i}+delta tX_{i}+sqrt{delta t}Z_i,$$

using Chapmann-Kolmogorov equation and Taylor expression.

How can i derive this? What is the answer?

Any explanation is very appreciated.

$$$$

To my calculations:

Let density function for BM $pi_t (x)$ in time $t$ be defined as
$$pi_t (x)=frac{1}{sqrt{2pi t}} exp left (frac{-x^2}{2t}right).$$

According to properties of BW increments we consider Chapmann-Kolmogorov equation as follow:
$$pi_{t+delta t} (x')=int {frac{exp left (frac{-(x'-x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x) dx}.$$

$x = x'+ Delta x, Rightarrow dxrightarrow d Delta x,$
$$Rightarrow pi_{t+delta t} (x')= int {frac{exp left (frac{-(Delta x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x'+Delta x) dDelta x}.$$

Since $Delta x = delta tX_{i}+sqrt{delta t}Z_i$ we have Taylor expansion:
$$pi_t (x'+Delta x) = pi_t (x') + frac{dpi_t(x')}{dx} Delta x + ... + O(Delta x^4).$$

Then
$$pi_{t+delta t} (x')= int {frac{exp left (frac{-(Delta x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x'+Delta x) dDelta x}$$

$$=pi_t (x')+frac{dpi_t}{dx} delta t + frac{1}{2} frac{d^2pi_t}{dx^2}delta t+frac{1}{6} frac{d^3pi_t}{dx^3}3 delta t^2 + O(delta t^3);$$

$$frac{pi_{t+delta t} (x') - pi_t (x')}{delta t}=frac{dpi_t}{dx} + frac{1}{2} frac{d^2pi_t}{dx^2}+frac{1}{2}frac{d^3pi_t}{dx^3}delta t + O(delta t^2);$$

$delta t rightarrow 0:$

$$frac{d}{dt}pi_t (x')=frac{dpi_t}{dx} + frac{1}{2} frac{d^2pi_t}{dx^2}.$$

integration ordinary-differential-equations stochastic-calculus brownian-motion finance

share|cite|improve this question

edited Jan 6 at 12:43

Thomas J

asked Dec 19 '18 at 16:26

Thomas JThomas J

114

$endgroup$

add a comment | 

2

2

2

1

$begingroup$

I know, that BM formula

$$X_{i+1} = X_{i}+sqrt{delta t}Z_i,$$ $$Z_isim N(0,1),$$

with $X_0equiv0$ leads to Diffusion equation $$u_t=-frac{1}{2}u_{xx}$$

for marginal distributions in $X_i=X(t),t=idelta t,$ in the limit case $delta tto 0$.

I want to derive Diffusion equation for Brownian motion with a systematic drift component, i.e.

$$X_{i+1} = X_{i}+delta tX_{i}+sqrt{delta t}Z_i,$$

using Chapmann-Kolmogorov equation and Taylor expression.

How can i derive this? What is the answer?

Any explanation is very appreciated.

$$$$

To my calculations:

Let density function for BM $pi_t (x)$ in time $t$ be defined as
$$pi_t (x)=frac{1}{sqrt{2pi t}} exp left (frac{-x^2}{2t}right).$$

According to properties of BW increments we consider Chapmann-Kolmogorov equation as follow:
$$pi_{t+delta t} (x')=int {frac{exp left (frac{-(x'-x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x) dx}.$$

$x = x'+ Delta x, Rightarrow dxrightarrow d Delta x,$
$$Rightarrow pi_{t+delta t} (x')= int {frac{exp left (frac{-(Delta x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x'+Delta x) dDelta x}.$$

Since $Delta x = delta tX_{i}+sqrt{delta t}Z_i$ we have Taylor expansion:
$$pi_t (x'+Delta x) = pi_t (x') + frac{dpi_t(x')}{dx} Delta x + ... + O(Delta x^4).$$

Then
$$pi_{t+delta t} (x')= int {frac{exp left (frac{-(Delta x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x'+Delta x) dDelta x}$$

$$=pi_t (x')+frac{dpi_t}{dx} delta t + frac{1}{2} frac{d^2pi_t}{dx^2}delta t+frac{1}{6} frac{d^3pi_t}{dx^3}3 delta t^2 + O(delta t^3);$$

$$frac{pi_{t+delta t} (x') - pi_t (x')}{delta t}=frac{dpi_t}{dx} + frac{1}{2} frac{d^2pi_t}{dx^2}+frac{1}{2}frac{d^3pi_t}{dx^3}delta t + O(delta t^2);$$

$delta t rightarrow 0:$

$$frac{d}{dt}pi_t (x')=frac{dpi_t}{dx} + frac{1}{2} frac{d^2pi_t}{dx^2}.$$

integration ordinary-differential-equations stochastic-calculus brownian-motion finance

share|cite|improve this question

edited Jan 6 at 12:43

Thomas J

asked Dec 19 '18 at 16:26

Thomas JThomas J

114

$endgroup$

I know, that BM formula

$$X_{i+1} = X_{i}+sqrt{delta t}Z_i,$$ $$Z_isim N(0,1),$$

with $X_0equiv0$ leads to Diffusion equation $$u_t=-frac{1}{2}u_{xx}$$

for marginal distributions in $X_i=X(t),t=idelta t,$ in the limit case $delta tto 0$.

I want to derive Diffusion equation for Brownian motion with a systematic drift component, i.e.

$$X_{i+1} = X_{i}+delta tX_{i}+sqrt{delta t}Z_i,$$

using Chapmann-Kolmogorov equation and Taylor expression.

How can i derive this? What is the answer?

Any explanation is very appreciated.

$$$$

To my calculations:

Let density function for BM $pi_t (x)$ in time $t$ be defined as
$$pi_t (x)=frac{1}{sqrt{2pi t}} exp left (frac{-x^2}{2t}right).$$

According to properties of BW increments we consider Chapmann-Kolmogorov equation as follow:
$$pi_{t+delta t} (x')=int {frac{exp left (frac{-(x'-x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x) dx}.$$

$x = x'+ Delta x, Rightarrow dxrightarrow d Delta x,$
$$Rightarrow pi_{t+delta t} (x')= int {frac{exp left (frac{-(Delta x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x'+Delta x) dDelta x}.$$

Since $Delta x = delta tX_{i}+sqrt{delta t}Z_i$ we have Taylor expansion:
$$pi_t (x'+Delta x) = pi_t (x') + frac{dpi_t(x')}{dx} Delta x + ... + O(Delta x^4).$$

Then
$$pi_{t+delta t} (x')= int {frac{exp left (frac{-(Delta x)^2}{2delta t}right)}{sqrt{2pi delta t}}pi_t (x'+Delta x) dDelta x}$$

$$=pi_t (x')+frac{dpi_t}{dx} delta t + frac{1}{2} frac{d^2pi_t}{dx^2}delta t+frac{1}{6} frac{d^3pi_t}{dx^3}3 delta t^2 + O(delta t^3);$$

$$frac{pi_{t+delta t} (x') - pi_t (x')}{delta t}=frac{dpi_t}{dx} + frac{1}{2} frac{d^2pi_t}{dx^2}+frac{1}{2}frac{d^3pi_t}{dx^3}delta t + O(delta t^2);$$

$delta t rightarrow 0:$

$$frac{d}{dt}pi_t (x')=frac{dpi_t}{dx} + frac{1}{2} frac{d^2pi_t}{dx^2}.$$

integration ordinary-differential-equations stochastic-calculus brownian-motion finance

integration ordinary-differential-equations stochastic-calculus brownian-motion finance

share|cite|improve this question

edited Jan 6 at 12:43

Thomas J

asked Dec 19 '18 at 16:26

Thomas JThomas J

114

share|cite|improve this question

edited Jan 6 at 12:43

Thomas J

asked Dec 19 '18 at 16:26

Thomas JThomas J

114

share|cite|improve this question

share|cite|improve this question

edited Jan 6 at 12:43

Thomas J

edited Jan 6 at 12:43

Thomas J

edited Jan 6 at 12:43

Thomas J

asked Dec 19 '18 at 16:26

Thomas JThomas J

114

asked Dec 19 '18 at 16:26

Thomas JThomas J

114

asked Dec 19 '18 at 16:26

Thomas JThomas J

114

114

add a comment | 

add a comment | 

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