MATH4210: Financial Mathematics IV. Stochastic Calculus .
The Brownian motionAn introduction to the stochastic calculusMATH4210: Financial MathematicsIV. Stochastic calculus: an elementary introductionMATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationProbability spaceA triple (Ω, F, P) is said to be a probability space, ifΩ is a (sample) sapce,F is a σ-field on Ω,Ω F,A F Ac F ,Ai F , i 1, 2, · · · i 1 Ai F .P is a probability measure on (Ω, F),P[A] [0, 1] for all A F .P[Ω] 1.Let2, · · · and such that Ai Aj if i 6 j, then Ai F , i 1,P P i 1 Ai F i 1 P[Ai ].MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationRandom variable, expectation A random variable is a map X : Ω R such that{X c} : {ω Ω : X(ω) c} F, c R. Law of X:Distribution functionF (x) : P[X x].Density function (if X follows a continuous law) ρ(x) F 0 (x),ρ(x)dx P X [x, x dx] .Characteristic function Φ(θ) : E eiθXMATH4210: Financial Mathematics
Probability theory reviewThe Brownian motionThe Brownian motion and heat equationThe Brownian motionAn introduction to the stochastic calculusRandom variable, expectation Expectation: E f (X) Zf (x)ρ(x)dx.R Variance: 2Var f (X) E f (X)2 E[f (X)] . Co-variance: Cov X, Y E XY E[X]E[Y ]. X1 , · · · , Xn are mutually independent ifnnhYi Y Efk (Xk ) E fk (Xk ) , bounded measurable functions fk .i 1k 1Remark: If X1 , · · · , Xn are mutually independent, thenf1 (X1 ), · · · , fn (Xn ) are also mutually independent.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationConditional expectation Let A, B F such that P[B] 0, we define the conditional probabilityof A knowing B by P[A B].P AB P[B] Let G be a sub-σ-field of F (i.e. G F and G is a σ-field), and X bea random variable such that E[ X ] , we define the conditionalexpectation of X knowing G as the random X such thatE[ Z ] ,Z is G-measurable,andE[Y X] E[Y Z], G-measurable bounded random variables Y.Denote the conditional expectation: E[X G] : Z. Denote the condition probability P[A G] : E[1A G].MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationConditional expectation A random variable Y is G-measurable if {Y c} G for all c R.Further, Y is G-measurable implies that f (Y ) is G-measurable for allbounded measurable functions f . Let Y be a random variable, we denote by σ(Y ) the smallest σ-field Gsuch that Y is G-measurable Denote thenE[X σ(Y )] E[X Y ].MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationConditional expectationRemember the following proposition !Proposition 1.1 (i). Tower property: E E[X G]] E[X].(ii). E[XY G] XE[Y G] if X is G-measurable.(iii). E[X Y] E[X] if X is independent of Y . (iv). E αX βY G αE[X G] βE[Y G].MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationBrownian MotionBrownian motion is a stochastic process B (Bt )t 0 , where each Bt isa random variable. Historically speaking, Brownian motion was observed by Robert Brown,an English botanist, in the summer of 1827, that “pollen grainssuspended in water performed a continual swarming motion.” Hence itwas named after Robert Brown, called Brownian motion. In 1905, Albert Einstein gave a satisfactory explanation and assertedthat the Brownian motion originates in the continued bombardment ofthe pollen grains by the molecules of the surrounding water, withsuccessive molecular impacts coming from different directions andcontributing different impulses to the particles. In 1923, Norbert Wiener (1894-1964) laid a rigorous mathematicalfoundation and gave a proof of its existence. Hence, it explains why it isnow also called a Wiener process. In the sequel, we will use bothBrownian motion and Wiener process interchangeably.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationBrownian MotionWe say a stochastic process (Bt )t 0 is a standard Brownian motion orWiener process on a given probability space (Ω, F, P) if it satisfies thefollowing conditions:(a) B0 0.(b) the map t 7 Bt is continuous for t 0.(c) stationary increment: the change Bt Bs is normally distributed:N (0, t s) for all t s 0.(d) independent increment: the changes Bt1 Bt0 , Bt2 Bt1 ,· · · ,Btn 1 Btn are mutually independent for all0 6 t0 t1 · · · tn tn 1 . We accept that the Brownian motion exists.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationBrownian MotionExample 1.1Given a set A R, compute the probability P(Bt h Bt A), h 0.Because Bt h Bt N (0, h), we have1P(Bt h Bt A) 2πhZx2e 2h dx.AMATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationBrownian MotionMarkov propertyWhich means that only the present value of a process is relevant forpredicting the future, while the past history of the process and the waythat the present has emerged from the past are irrelevant.P(Xu A Xs , 0 6 s 6 t) P(Xu A Xt ), u t 0.Markov processA stochastic process which satisfies the Markov property is called aMarkov process.Brownian motion is a Markov process because by property (d) Bu Bt isindependent of Bs B0 Bs for all 0 6 s 6 t 6 u.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationBrownian MotionExample 1.2Given a set A R and all the information up to time t, compute theconditional probability P(Bt h A Bs , 0 6 s 6 t), h 0.Brownian motion is a Markov process, so we haveP(Bt h A Bs , 0 6 s 6 t) P(Bt h A Bt ) g(Bt ),whereg(x) P[Bt h Bt Ax Bt ],Ax : {y x : y A}.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationBrownian MotionFiltrationA filtration (Ft )t 0 is a family of σ-field such that Fs Ft F, for alls t.MartingaleA process (Xt )t 0 is called a martingale with respect to a filtration(Ft )t 0 , if E[ Xt ] for all t 0, andE[Xt Fs ] Xs , 0 6 s 6 t.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationBrownian MotionLet us set Ft : σ{Bs , s t}.Proposition 1.2We haveCov(Bs , Bt ) min(s, t).{Bt , t 0} is a martingale.{Bt2 t, t 0} is a martingale.2{eaBt at/2, t 0} is a martingale.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationQuadratic variation of Brownian motion Let n (0 tn0 tn1 · · · tn2n t) a subdivision of interval [0, t],where tnk : k t with t t/2n . LetnZtn: 2Xk 1nBtnk Btnk 1 2andVtn: 2XBtnk Btnk 1 .k 1Proposition 1.3Let n , thenZtn t, and Vtn , a.s.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationThe heat equation Defineq(t, x, y) (y x)2 1exp .2t2πtProposition 1.4The function q(t, x, y) satisfies t q(t, x, y) 1 21 2 q(t, x, y) yyq(t, x, y).2 xx2MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationThe heat equationTheorem 1.1Let f : R R be such that, for some constant C 0, f (x) eC x forall x R. Define u(t, x) : E f (BT ) Bt x E f (BT Bt x) .Then u satisfies the heat equation1 2u 0, t u xx2with terminal condition u(T, x) f (x).MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationGeneralized Brownian MotionGive real scalars x0 , µ, and σ 0, we call processXt x0 µt σBtas generalized Brownian motion or Brownian motion with drift orgeneralized Wiener process starting at x0 , with a drift rate µ and avariance rate σ 2 .Let us denote(dXt µ dt σ dBt ,X0 x0 .By the property (c) of Brownian motion, we haveXt h Xt N (µh, σ 2 h).MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusProbability theory reviewThe Brownian motionThe Brownian motion and heat equationGeneralized Brownian motion and PDETheorem 1.2Let f : R R be such that, for some constant C 0, f (x) eC x forall x R. Define u(t, x) : E f (XT ) Xt x E f (XT Xt x) .Then u satisfies the heat equation12u 0, t u µ x u σ 2 xx2with terminal condition u(T, x) f (x).MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusStochastic Integral: motivationDynamic trading: let tk : k t, risky asset price (Stk )k 0 , interest rater 0.Discrete time dynamic trading between tk and tk 1 : Vtk 1 φtk Stk 1 Vtk φtk Stk er t Vtk er t φtk Stk 1 Stk er t Vtk Vtk φtk Stk r t φtk Stk 1 Stk .ThenVtn V0 n 1Xn 1X φtk Stk 1 Stk .Vtk φtk Stk r t k 0k 0Formal limit ?ZTZ(Vt φt St )rdt VT V0 0Tφt dSt .0MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusStochastic Integral: simple processLet θ : [0, T ] Ω R be a simple process, i.e.(n 1Xα0 ,θt α0 1{0} (t) αi 1(ti ,ti 1 ] (t) αk,i 0t 0;tk t 6 tk 1 ,for a discret time grid 0 t0 t1 · · · tn T and boundedFti -measurable random variables αi , we defineZTθt dBt 0n 1Xαi (Bti 1 Bti )i 0 α0 (Bt1 0) α1 (Bt2 Bt1 ) · · · αn 1 (Btn Btn 1 ).MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusStochastic Integral: simple processTheorem 2.1 (Itô Isometry)Let θ be a simple process, thenhZE0Tiθt dBt 0,Th Zand Eθt dBt 2 ihZ E0MATH4210: Financial Mathematics0Tiθt2 dt .
The Brownian motionAn introduction to the stochastic calculusStochastic IntegralWe accept the following facts in mathematics:Let L2 (Ω) denote the space of all square integrable random variablesξ : Ω R, H2 [0, T ] be the set of (Ft )t 0 -adapted right-continuous andleft-limit processes θ, such thathZ TiEθt2 dt .022L (Ω) and H [0, T ] are both Hilbert space.In a Hilbert space E, let (en )n 1 be a Cauchy sequence, i.e. em en 0 as m, n , then there exists a unique e Esuch that en e as n .For each θ H2 ([0, T ]), there exists a sequence of simple processes(θn )n 1 such thathZ T 2 ilim Eθt θtn dt 0.n 0MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusStochastic IntegralLet θ H2 ([0, T ]), (θn )n 1 be a sequence of simple processes such thathZlim En Tθt θtn 2idt 0.0Proposition 2.1The sequence of stochastic integralsL2 (Ω) as n . Let us defineZ0θtn dBt has a unique limit inTZθt dBt : 0RTlimn Tθtn dBt .0MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusStochastic IntegralTheorem 2.2 (Itô Isometry)Let θ H2 [0, T ], thenhZE0Tiθt dBt 0,Th Zand Eθt dBt 2 ihZ E0MATH4210: Financial Mathematics0Tiθt2 dt .
The Brownian motionAn introduction to the stochastic calculusItô ProcessWe call processZXt X0 tZbs ds 0tσs dBs ,0as an Itô process, where the drift b and σ are (Ft )t 0 -adapted processessuch that the integrations are well defined.The process can be rewritten in differential form asdXt bt dt σt dBtAll of the stochastic processes mentioned above are Itô process.MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusItô Process: IntuitionConsider processdXt b(t, Xt ) dt σ(t, Xt ) dBt .In a short time interval [t, t t], the variable changes from x tox x, where x b(t, x) t σ(t, x)ε tand ε has a standard normal distribution N (0, 1).MATH4210: Financial Mathematics
The Brownian motionAn introduction to the stochastic calculusItô’s LemmaTheorem 2.3 (Itô’s Lemma)Let B be a Brownian motion, f (t, x) a smooth function. Then theprocess Yt f (t, Bt ) is also an Itô process andZYt Y0 0t 2f (s, Bs ) ds t f (s, Bs ) 12 xxZt x f (s, Bs ) dBs .0or equivalently,dYt 2f (t, Bt ) dt x f (t, Bt ) dBt . t f (t, Bt ) 21 xx Remark: More generally, one can show that when X is a Itô process,f (t, x) is a smooth function, then the process Yt f (t, Xt ) is also an Itôprocess.MATH4210: Financial Mathematics
An introduction to the stochastic calculus Probability theory review The Brownian motion The Brownian motion and heat equation Conditional expectation A random variable Y is G-measurable if fY cg2Gfor all c2R. Further, Y is G-measurable implies that f(Y) is G-measurable for all bounded measurable functions f.
Jul 09, 2010 · Stochastic Calculus of Heston’s Stochastic–Volatility Model Floyd B. Hanson Abstract—The Heston (1993) stochastic–volatility model is a square–root diffusion model for the stochastic–variance. It gives rise to a singular diffusion for the distribution according to Fell
are times when the fast stochastic lines either cross above 80 or below 20, while the slow stochastic lines do not. By slowing the lines, the slow stochastic generates fewer trading signals. INTERPRETATION You can see in the figures that the stochastic oscillator fluctuates between zero and 100. A stochastic value of 50 indicates that the closing
sion analysis on discrete-time stochastic processes. We now turn our focus to the study of continuous-time stochastic pro-cesses. In most cases, it is di cult to exactly describe the probability dis-tribution for continuous-time stochastic processes. This was also di cult for discrete time stochastic processes, but for them, we described the .
(e.g. bu er stocks, schedule slack). This approach has been criticized for its use of a deterministic approximation of a stochastic problem, which is the major motivation for stochastic programming. This dissertation recasts this debate by identifying both deterministic and stochastic approaches as policies for solving a stochastic base model,
Stochastic Programming Stochastic Dynamic Programming Conclusion : which approach should I use ? Objective and constraints Evaluating a solution Presentation Outline 1 Dealing with Uncertainty Objective and constraints Evaluating a solution 2 Stochastic Programming Stochastic Programming Approach Information Framework Toward multistage program
STOCHASTIC CALCULUS AND STOCHASTIC DIFFERENTIAL EQUATIONS 5 In discrete stochastic processes, there are many random times similar to (2.3). They are non-anticipating, i.e., at any time n, we can determine whether the cri-terion for such a random time is met or not solely by the “history” up to time n.
processes 1.Basics of stochastic processes 2.Markov processes and generators 3.Martingale problems 4.Exisence of solutions and forward equations 5.Stochastic integrals for Poisson random measures 6.Weak and strong solutions of stochastic equations 7.Stochastic equations for Markov processes in Rd 8.Convergenc
Access to Accounting Software – SAMS – Assessment book . 2 . Notes for students . This sample assessment is designed to demonstrate as many of the possible question types you may find in a live assessment. It is not designed to be used on its own to determine whether you are ready for a live assessment. In a live assessment, you will be required to upload documents as part of your evidence .
