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Continuous Time - Stochastic Calculus

# Continuous Time - Stochastic Calculus - 1 Temple University...

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1 Temple University Continuous Time Finance Stochastic Calculus

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2 Continuous Time Finance Brownian Motion
3 Scaled Random Walk = = = - = = = nt j j n j X n t W T H X 1 j j 3 2 1 1 ) ( if 1 if 1 .... ϖ ϖ ϖ ϖ ϖ ϖ We speed up time (number of steps in any length of time) and scale the step size: 0 0.5 1 1.5 2 2.5 3 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4

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4 Limiting Distribution – Brownian Motion t variance and zero mean with on distributi normal the to converges t at time evaluated ) t ( W walk random scaled the of on distributi the , n As 0. Fix t n s W(t)] - s) Var[W(t 0 W(t)] - s) E[W(t with increments d distribute normally y identicall and t independen with : W process continuous a is motion Brownian = + = + × +
5 Continuous Time Finance Stochastic Calculus

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6 Theory of Integration w.r.t. (Random) Stochastic Processes (t) be the position in an asset at time t W(t) be the asset price at time t dW(t) the change in price over t+dt (t)dW(t) the change in position value over interval of time dt
7 Theory of Integration w.r.t. (Random) Stochastic Processes (t)dW(t) the change in position value ) ( ) ( : of sense make T 0 t dW t Random variable versus Length (Lebesque measure) along x-axis

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8 Riemann-Stieltjes Integral Lebesgue integration is a mathematical construction that extends the integral to a larger class of functions (e.g., Dirichlet function 1 Q ) extends the domains on which these functions can be defined (e.g., arbitrary probability space Ω) The Riemann-Stieltjes integral of a real-valued function f of a real variable with respect to a real function g is denoted by: Georg Riemann 1826 –1866 Thomas Stieltjes 1856 –1894 ) ( ) ( b a x dg x f
9 Recall …. Riemann Integral Upper sum of f with respect to the partition Π is defined as: where M k is the supremum of f(x) in the interval [x k-1 , x k ] . Lower sum of f with respect to the partition Π is defined as where d k is the infimum of f(x) in the interval [x k-1 , x k ] . = - + Π - = n k k k k x x M f RS 1 1 ) ( ) ( = - - Π - = n k k k k x x m f RS 1 1 ) ( ) ( The upper and lower Riemann sums converge to the same limit which we call the Riemann integral: b a dx x f ) (

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10 Riemann-Stieltjes Integral Upper sum of f over g with respect to the partition Π is defined as: where M k is the supremum of f(x) in the interval [x k-1 , x k ] . Lower sum of f over g with respect to the partition Π is defined as where d k is the infimum of f(x) in the interval [x k-1 , x k ] . = - - = Π n k k k k x g x g M g f U 1 1 )) ( ) ( ( ) , , ( = - - = Π n k k k k x g x g m g f L 1 1 )) ( ) ( ( ) , , ( Sums converge to upper Riemann- Stieltjes integral : And lower Riemann- Stieltjes integral : ) ( ) ( b a x dg x f ) ( ) ( b a x dg x f
11 Theory of Integration w.r.t.

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