> <\body> |>> <\table-of-contents|toc> |.>>>>|> Basic Facts from Stochastic Processes> |.>>>>|> Lebesgue Integral |.>>>>|> > Conditional Expectation |.>>>>|> > Stochastic Processes |.>>>>|> > Brownian Motion (Wiener Processes) |.>>>>|> > The Itô Integral and Formula> |.>>>>|> The Itô Construction |.>>>>|> > Itô's Formula |.>>>>|> > Deriving from the Chain Rule |.>>>>|> > SODEs |.>>>>|> > Some SPDEs> |.>>>>|> PDE/Sobolev Recap> |.>>>>|> Sobolev Spaces >> |.>>>>|> > SPDEs in Sobolev Spaces |.>>>>|> > Classical Theory |.>>>>|> > Stochastic Theory |.>>>>|> > Nonlinear Filtering (``Hidden Markov Models'')> |.>>>>|> Solutions of PDEs and SPDEs> |.>>>>|> Classical Solutions |.>>>>|> > Generalized Solutions |.>>>>|> > Mild Solutions |.>>>>|> > Generalization of the notion of a ``solution'' in SDE |.>>>>|> > Send corrections to . \; Example: Heat Equation. Suppose \\> is part of a probability space. Then chance can come in at any or all of these points: <\eqnarray*> u(t,x)|\t>>||>)|\x>u(t,x)+f(t,x,>)x\(a,b)>>|||(x,>)>>|||(t,>)>>|||(t,>)>>>> |||>|> -- elementary random event (outcomes)>|>|=\> -- probability space/space of outcomes>|> -- set>>|> subsets of \A>>|\\(\)> closed w.r.t. >/>/|\>>.>>|>, >, >=\\A>.>|>|\\\\>>|>| and are random events, then B>, B>, >> are r.e.>|>||>|[0,1]>, )=1>, additive for disjoint events.>|>>>> <\definition> A function (A)> on the sets of an algebra > is called a if <\enumerate-alpha> the values of > are non-negative and real, > is an additive function for any finite expression--explicitly, if A> and \A=\> iff j>, then <\equation*> \(A)=\(A). <\definition> A system \\(\)> is called a >-algebra> if it is an algebra and, in addition, if )>>, then also A\\>. It is an easy consequence that A\\>. <\definition> A measure is called >-additive> if <\equation*> \>A=>\(A) if the > are mutually disjoint. The above together form A tuple ,\,P)> is called a (> a set, > a >-algebra, a probability measure). <\lemma> Let > be a set of events. Then there is a smallest >-algebra > such that \\>. <\definition> A function \\> is called a if it is >-measurable, i.e. for arbitrary belonging to the Borel->-algebra (\)>, the set (A)\\>. <\definition> > with respect to >: For simplicity, =(0,1)>. is the , > the Borel->-algebra (0,1)> on =(0,1)>. > is called complete if it contains all subsets of > with the property: <\quote-env> There are subsets > and > from (0,1)> such that \B\B> and \B)=0>. This process maps ,\,P)> to ,|\>,P)>, where |\>> is the of > w.r.t. . Now suppose is a random variable in ,\,P)> in >. (\(\))\{X(A):A\\(\)}={\:X(\)\\(\)}>. > is called the >-algebra generated by . One reason to use this definition of a random variable is this: <\lemma> If > is generated by a random variable , then there exists a Borel function such that . <\definition> on ,\,P)> is called simple if it is >-measurable and takes a finite number of values: ,x,\,x>. ={\:X(\)=x}=X(x)>. Then the Lebesuge integral is <\equation*> >X\P=xP(\). <\definition> An arbitrary measurable function on ,\,P)> is called -integrable if there exists a sequence of such simple functions > so that \X> a.s. and <\equation*> lim\>>\|X-X\|\P=0. <\lemma> If is -integrable, then <\enumerate-numeric> There exists a limit <\equation*> >X\P=lim\>>X\P. This limit does not depend on the choice of the approximating system. If is a random variable \\>. Let > be Borel's >-algebra on >. Then <\equation*> \(>\>)=P(X(A))=P(\:X(\)\A) is called the of . <\theorem> <\equation*> >f(X)\P=>f(x)\(\x). Thus <\equation*> E[X]=>X\(\X). <\example> Let have values ,\,x>. =X(x).> (x)=P(\)>. Then <\equation*> E[X]=x\(x)=xP(\). > and > are are random variables with a joint density . <\equation*> E[\\|\=y]=x*p(x\|y)\x. <\equation*> P(A\|B)=B)|P(B)>. Now suppose is a -integrable random variable ,\,P)>. is a >-algebra on >, \\>. <\definition> Let > be >-measurable random variable. If there exists a -integrable >-measurable function > such that for any bounded >-measurable function > <\equation*> E(\\)=E(\\), the > will be called of > and denoted \|\]>. Properties of conditional expectation: <\enumerate-numeric> If > is >-measurable, then \|\]=\>. <\proof> (1) By assumption, > is >-measurable. (2) Let > be an arbitrary >-measurable function. Then <\equation*> E(\\)=E(E(\\|\)\)=E(\\). Prove that the conditional expectation is unique. If is bounded, >-measurable, then <\equation*> E[f(\)X\|\](\)=f*(\)E[X\|\]() Let ,X)> be an >-measurable function. Then <\equation*> E[g(\,X)\|\(X)]=E[g(\,c)\|\(X)]\|. Let \\> be >-algebras. Then <\equation*> E[E[X\|\]\|\]=E[X\|\]. This property can be memorized as ``Small eats big''. <\example> =\>, \\=\>. Let ={\,\,\}>. Then (\)={\>\\>\\}>. =\\>?>. Let > be a random variable <\equation> E[\\|\(\)]=\>]|P(\)>\>. Proof of (): <\enumerate-alpha> The right-hand side is a function of indicators of >>it is (\)>-measurable. E[\\|\(\)]g]=E\g> for all which are (\)>-measurable. Suppose >>. Then <\equation*> E[rhs*\>]=E\k>]|P(\)>\>=\>]|)>>)>=E(\\>). rhs: \>)>. What is a (\)>-measurable function? Answer: It is a function of the form <\equation*> \=y\>. Assume that for all , we are given a random variable =X(\)\\>. could be from }> or from , it does not matter. In the former case, > is called a sequence of r.v. or a discrete time stochastic process. In the latter, it is called a continuous time stochastic process. If \>, then > is a two-parameter random field. If is a random variable, (A)=P(\:X(\)\A>. <\definition> The (finite-dimensional) distribution of the stochastic process )T>> are the measures defined on =\\\\> given by <\equation*> \,\,t>(F\F\\\F)=P(\:X>\F,\,X>\F), where the \\(\)>. <\definition> A real-valued process > is called Gaussian if its finite dimensional distributions are Gaussian>>,\,X>)\\(k)>. Remember: A random variable > in > is called if there exists a vector \> and a symmetric non-negative k>-matrix )> such that <\equation*> \(\)\E[e,\)>]=e)-(R\,\)/2> for all \\>, where ,\)> represents an inner product, ]> and ,\)>. Fact: ,\,Y)> are normal vectors in > with ,R)>. Then elements of are independent iff <\equation*> \>(Y)=\>(Y), where =(\,\,\)>, where \\>. Fact 2: =(\,\,\)> is Gaussian iff for any \\>, <\equation*> (\,\)=\\ is Gaussian in 1D. <\definition> Brownian motion > is a one-dimensional continuous Gaussian process with <\equation*> E[W]=0,E[WW]=t\s\min(t,s). Alternative Definition: <\definition> Brownian motion > is a Brownian motion iff <\enumerate> =0> t,s:W-W\\(0,t-s)> >>, >-W>,\> are independent for all partitions \t\t\\>. Yet another: <\definition> The property (3) in Definition may be replaced by 3>. >-W>> is independent of >-W>>,> <\definition> <\equation*> \\\({W>,W>,\:s\t}). <\theorem> Brownian motion is a w.r.t. >> <\equation*> E[W\|\]=W for t>. (This is also the definition of a martingale.) <\remark> (W>,W>,\,W>)=\(W>,W>-W>,\,W>-W>)> (knowledge of one gives the other--add or subtract). This is important because RHS is independent, but LHS is not. <\corollary> \; <\enumerate> ]=t>. (So > grows roughly as >.) /t\0> almost surely. Proof: By Chebyshev's inequality, /t\|\c)\E[\|W/t\|]/c=t/tc\0> as \>. \ <\eqnarray*> =|>,>||>=|>,>>|0>\=1,>||\>\>=1,>>|0> \=-1,>||\> \>=-1.>>>> <\itemize> > is continuous. > is nowhere differentiable. <\theorem> \; <\eqnarray*> >||+>\sin(n*t)\t*\+\sin(n*t),>>|>|>|(0,2/\n) (n\1),>>|>|>|(0,1/\).>>>> <\proof> Consider [0,\]>. <\equation*> |~>\W->W>[0,\]>>. Then <\equation*> (t)=>\sin(n*t), where <\equation*> \=>>(t)sin(n*t)\t(n\0) and <\equation*> \=)|\>. First fact: > are Gaussian because linear combinations of normal r.v.s. are normal. <\equation*> E\\=>>>(t\s-t*s/\)sin(n*t)sin(k*s)=|n,>>|n>>|0.>>>>> For , <\equation*> E[\]=E[\]|\>=>. \; Suppose we have some system described by > that has some additive noise >: =X+\>. (>) The ,\,\> are assumed to be <\enumerate> iid \N(\,\)> If the > satisfy the first property, they are called . If they satisfy both, it is . If we now consider > =W=0>, =W>-W>, =W>-W>>, ..., then <\enumerate> holds holds A popular model in dynamics is <\equation*> X>=A*X+B+\ for, say, the dynamics of an ``aircraft''. Another possibility is modeling the price of a risky asset <\equation*> X>=X+\X\+\X(W-W), where > is the individual trend of the stock, while > is market-introduced volatility. Equivalently, we might write <\equation*> >-X|\>=\X-\X-W|\> and then let t\0>, such that we obtain <\equation*> >=\X+\X>, which is all nice and well except that the derivative of white noise does not exist. But note that there is less of a problem defining the same equation in integral terms. Suppose we have a function , which might be random. Then define <\equation*> I(f)=f(s>)(W>-W>). But what happens if >. We get the term <\equation*> W>(W>-W>). Or is it <\equation*> W>(W>-W>)? Or even <\equation*> W+s|2>>(W>-W>)? In the Riemann integral, it does not matter where you evaluate the integrand--it all converges to the same value. But here, we run into trouble. Consider <\eqnarray*> >(W>-W>)\|>|>|>(W>-W>)\|>>|>||>>>|>\|E\|W>-W>\|>||>(W>-W>)\|>>|>||>>>|(s-s)>||+s|2>>(W>-W>)\|.>>>> Compute each of the above expectations, and show they are not equal. The idea here is to use simple functions: <\equation*> f(s)=e(\)\,t)>(s), where > is >>-measurable, where >=\(W>,\,W>:s\s)> <\eqnarray*> |>|>||=e(W, r\[0,t])>|>||>|>||> is ``adapted'' to >>>.>|>>> <\definition> <\equation*> I(f)=e(W>-W>). <\enumerate> \ <\eqnarray*> ||E*e(W>-W>)>>|||EEe(W>-W>)\>>>|||E(eE[(W>-W>)\|\>])>>|||E(eE[W>-W>])>>|||E(e*0)=0.>>>> <\eqnarray*> >||E\|e\|(t-t)=E\|f(s)\|\s>>|||e(W>-W>)>>|||Ee(W>-W>)-Eee(W>-W>)(W>-W>)>>|||E(e(W>-W>)\|\>))>>|||](t-t).>>>> is linear. Next: If is only >-measurable (but not a step function), and if E*f(s)\s\\>> could be approximated by a sequence of step functions (s)\f(s)>. [Insert here, courtesy of Mario.] Suppose we have a partition of a time interval as =0,t,t,\,t=T>. t=t-t>. We assume t\0>. Also, we assume we have a function <\equation*> f=f(t),\f=f(t)+f(t). <\enumerate-alpha> If , continuous, bounded variation. Then <\equation*> limt\0>\|\f\|=limt\0>maxf\||\>0>\|\f\||\>\>=0. If is Standard Brownian Motion, then <\equation*> limt>\|\W\|=T(> and in probability>). <\proof> We need \|\W\|\|-T\|\0>. So <\eqnarray*> ||(\W)-2(\W)T+T>>|||\|\W\|\|\W\|-2T+T>>|||\|\W\|+j>\|\W\|\|\W\|-T>>|||\|\t\|+j>\t\t-T>>|||\|\t\|+\|\t\||\>>-T>>|||\|\t\|\2max{\t}\T\0.>>>> So we essentially showed: <\eqnarray*> \|\W\|>|>|>|W)>|>|t,>>|W>|>|t>.()>>>> if C> and C>. Then <\equation*> |\t>F(x(t))=F(x(t))x(t). Alternatively, <\equation*> x(t)=x(0)+>(s)>\s. Then <\equation*> F(x(t))=F(x(0))+F(x(s))f(s)\s. First of all, there is no ``Stratonovich Formula''. Suppose \W> (double arrows: uniformly), then <\eqnarray*> (t)>||A(s)\s+B(s)|\>(s)\s|\>(X)(s)\s>,>>|||A(s)\s+B(s)\\W(s)|\>>.>>|(t))>||F(X(s))A(s)\s+F(X(s))B(s)|\>(s)\s>>|||F(X(s))A(s)\s+F(X(s))B(s)\(s)\W(s).>>>> In particular, <\equation*> X=W(t)=1\\W(s),F(y)=y,W(s)\\W(s)=W(t). <\remark> Itô integral is a martingale, Stratonovich is . Also: there is no connection between the two in the non-smooth case. Now, let's see what happens for Itô, again starting from a process given as <\eqnarray*> ||A(s)\s+B(s)\W(s).>>>> Now, what is ? Let's look at a Taylor expansion of <\equation*> F(X(t))-F(X(t))=F(X(t))\x+F(X(t))(\x)+(\)x)|\>(\t)> So, in continuous time <\eqnarray*> ||\F>>|||F(X(s))\X(s)+F(X(s))(\X(s))>>|||F(X(s))A(s)\s+F(X(s))B(s)\W(s)+F(X(s))B(s)\s>>>> <\theorem> If <\equation*> X(t)=X(0)+A(s)\s+B(s)\W(s) and C>, then <\equation*> F(X(t))=F(X(0))+F(X(s))A(s)\s+F(X(s))B(s)\W(s)+F(X(s))B(s)\s. Now if C(\,\)>,then <\equation*> X(t)=X(0)+A(s)\s+B(s)\W(s)\\, where we recall that \> with all components independent. Itô's Formula in multiple dimensions takes the form <\equation*> F(X(t))=F(X(0))+F|\x>A\s+F|\x>B\W+F|\x\x>BB\s. <\example> If > and <\equation*> X=\W(s), then <\equation*> W(s)\W(s)=(W(t)-t). <\example> If F=0> (i.e. is harmonic), then is a martingale. <\equation*> X(t)=X(0)+b(s,X(s))\s+\(s,X(s))\W(s) or, equivalently <\equation*> \X=b(t,X)\t+\(t,X)\W(t). <\example> . The equation <\equation*> \X(t)=a*X(t)\t+b*\W(t) has the solution <\equation*> X(t)=eX(0)+be\W(s). <\proof> Consider <\eqnarray*> >||X(0)+be\W>>|||X(0)+b*ee\W>>|||X(0)+b*eZ>>|||)\Z=e\W.>>>> Ito's Formula then gives <\eqnarray*> X>||g|\t>\t+g|\x>\Z+g|\x>(\Z)>>|||+b*e\Z+0>>|||+b*ee\W>>|||+b*\W.>>>> <\example> () <\equation*> \X(t)=a*X(t)\t+b*X(t)\W(t) is solved by <\equation*> X(t)=X(0)e/2)t+b*W(t)>. (Check this by Itô.) Solve <\equation*> \X(t)=(a+aX(t))\t+(b+bX)\W(t). <\theorem> If (s,x)-b(s,y)\|+\|\(s,x)-\(s,y)\C\|x-y\|> (a Lipschitz condition) and (s,x)\|+\|\(s,x)\|\C(1+\|X\|)> (a linear growth condition) and is independent of and \\>, then there exists a solution that is continuous in time. is measurable w.r.t (X(0),W(s),s\t)> and <\equation*> EsupT>\|X(t)\|\\. <\equation*> u=a*u,u(0,x)=u(x). (0>--ellipticity: if it holds, then the equation is called parabolic) General solution: <\equation*> u(t,x)=*a*t>>exp-|4*a*t>u(y)\y=E[u(x+>W(t)] ( formula--averaging over characteristics) Monte-Carlo simulation: <\equation*> area(A)=>|>. More general parabolic equation: <\equation*> u(x,t)=aDDu(x,t)+bDu(x,t)+c*u+f(t\0,x\\)u(0,x)=u(x) This equation is parabolic iff y y\a\|y\|> for all \> (the ellipticity property). If the highest order partial differential operator in the equation is elliptic, then the equation is parabolic. (The elliptic equation would be <\equation*> aDDu(x,t)+bDu(x,t)+c*u+f=0.) Now, onwards to PDEs. A model equation is <\equation*> \u(t,x)=a*u(t,x)\t+\u(t,x)\W. Recall from geometric Brownian motion: <\equation*> \u(t)=a*u(t)\t+\u(t)\W,u(0)=0. The solution is <\equation*> u(t)=uexpa-|2>+\W and <\equation*> E[u(t)]=uexp{u*t}E[exp2\W-\t]. Now consider <\equation*> Eb*W-bt|\>(t)>=1, which is an example of an , which satisfies the general property <\equation*> E[\(t)\|\]=\(s)s\t,\(0)=1. We find <\equation*> E(\(t)]=E[\(s)]=E[\(0)]=1. <\proof> By Ito's formula, <\equation*> \\(t)=b\(t)\W\\(t)=1+b\(s)\W. \; Here's a crude analogy: In stochastic analysis, (t)> plays the role of in ``regular'' real analysis. Going back to our above computation, we find <\equation*> E[u(t)]=uexp{u*t}E[exp2\W-\t]=uexp2a*t. So we find for geometric Brownian motion that it remains square-integrable for all time. (Consider that this is also the case for the regular heat equation.) Now, let's return to our SPDE, <\equation*> \u(t,x)=a*u(t,x)\t+\u(t,x)\W. We begin by applying the Fourier transform to , yielding >. <\equation*> \=-a*y+i\y(t,y)\W <\equation*> =(0,y)exp(-a-\/2)yt+i*y\W). Parseval's equality tells us <\equation*> \|u(t,x)\|=\|(t,y)\|\y\\ iff /2\0>. In SPDEs, first order derivatives in stochastic terms has the same strength as the second derivative in deterministic terms. The above condition is also called , and the whole evolution equation is then called . There's another example of SPDE in the lecture notes: <\equation*> \u(t,x)=a*u(t,x)\t+\u(t,x)\W. Here, the superellipticity equation is <\equation*> a-|2>\0\a\0. For the homework, see the notes as well. One of these problems is to consider the more general equation <\equation*> \u=aDDu+bDu+c*u\t+(\Du+\)\W(t)i,j=1,\,d,k=1,2,3,\ where we have <\equation*> \=>|>|>>|>||>|>|>|>>>>>. We have to assume <\equation*> \\>=>\\\\,\\\. <\equation*> \; A substitution that sometimes helps in the deterministic case is illustrated below: <\equation*> u|\t>=a(t,x)u+c*u Then we set u(t,x)> and obtain <\equation*> \v(t,x)=-c*eu(t,x)+a(t,x)*eu+c*eu=a(t,x)u*v. For the stochastic case, note: <\equation*> \\(t)=\(t)\\W. Then, let <\eqnarray*> (t)>|>|W(t)-(\/2)t>>>|\(t)>||(t)\\\>>|(t)>||(t)exp(\t)>>|\(t)>||(t)\\Wexp(\t)+\(t)exp(\t)\\t=-\\\W+\\(t)\t>>>> Applied to an SPDE, we get <\eqnarray*> u(t,x)>||t+\u(t,x)\W>>|||>>|||W(t)+(\/2)t>|\>(t)>u(t,x)>>|(u(t,x)\(t))>||\t+\v\W-v\\W+\v\t-\v\t>>|||\t.>>>> Let (t)> be a Wiener process independent of . <\eqnarray*> ||ut+.>>>> Then <\eqnarray*> ||ux+exp(\W-(\/2)t>>|||ux+exp(\W-(\/2)t\.>>>> <\example> Now consider <\equation*> \u(t,x)=a*u(t,x)+\u(t,x)\W\2a-\\0. (Remark: There is not a chance to reduce to =a>.) <\eqnarray*> v|\t>>||/2)v(t)>>|||W(t)) verifies equation>.>>>> <\eqnarray*> ||ux+>(t)>>||>|>|||ux+W+>\.>>>> (Note that, as above, the point of the conditional expectation is not measurability w.r.t. time (...), but with respect to and not w.r.t. >.) By a naive application of Ito's formula, we would get <\eqnarray*> ||W)>>|||W)>>|u(t,x-\W)>||/2u(t,x-\W)>>|u(t,x-\W)\W>|||2>v\t-\v\W.>>>> But this is wrong because Ito's formula only applies to functions of brownian motion. The function itself is random, though, so it does not work. To the rescue, the Ito-Wentzell formula. <\theorem> Suppose <\equation*> \F(t,x)=J(t,x)\t+H(t,x)\W and <\equation*> \Y(t)=b(t)\t+\(\)\W. Then <\equation*> \F(t,Y(t))=t+H(Y(t))\W|\>F>+F(Y(t))b\t+|2>F(Y(t))\t+\F(Y(t))\W +H(t,Y(t))\(t)\t For comparison, if we suppose G(t,x)=J(t,x)\t> and work out the regular Ito formula, we would find <\equation*> \G(t,Y(t))=t|\>G>+G(Y(t))b(t)\t+G\\t+G(Y)\W. <\itemize> Spaces: >=H>(\)> Heat equation: >>, (\)>, >. an SPDE: >>, (\)>, >. We will need: <\itemize> Gronwall Inequality: ... BDG Inequality () <\equation*> EsupT>g(s)\W\C*Eg(t)\t. >-inequality <\equation*> \|a*b\|\\a+>b. Itô-Wentzell formula. >>> <\definition> Suppose C>(\)>. Then <\equation*> (y)=)>>ef(x)\x. Then we have Parseval's Inequality <\equation*> >\|f\|\x=>\|\|\y and define <\equation*> |>\>(1+\|y\|)>\|(y)\|\y>, a norm. Then >> is the closure of >> in the norm |y|>>. (x)>, |^>(x)=const>, \H>> for what >? (\-d/2?>) =L>, >\H>> if \\>. Sobolev embeddings: +d/2>\C>> if \\1>. Alternative (but equivalent) definition: <\equation*> H={f:f,D f,\,Df\L} with <\equation*> \|>+f|L|>. >> is a Hilbert space with <\equation*> |>=>(1+\|y\|)>(y)(y)|\>\y. >> is dual to >> relative to >. (\0>) Because if H>> and H>>. Then <\equation*> =>(1+\|y\|)/2>(y)(y)|\>|(1+\|\\|)/2>>\y\|>|>. All this by S.L. Sobolev (1908-1989). Derived Sobolev spaces & generalized derivatives in the 1930s. Let's consider the heat equation in ,L,H)>, namely <\equation*> u=u+f,u\|=u. <\theorem> If is a classical solution and )> and > are in >(\)>, then <\equation*> supT>+\t\C(T)|0|2>+\t. (Note the slight abuse of notation with |>>.) <\proof> \; <\eqnarray*> uu|\t>\x>||u*u\x+u*f\x\|\u\x>>|>||>|v|\t>>|||0|2>+||>\2v(t)>>|||+(s)|0|2>\s+\s+2v(s)\s>>|\s>|>|\s+2v(s)\s+\s+C\s>>|\s>|>|v(s)\s>>||>|v(s)\s>>||>|>>> where > and all the constant-tweaking is done with the >-inequality. <\equation*> \u=(a(t)u+f)\t+(\(t)u+g)\W, where \\a(t)-\(t)/2\C>>. adapted to >, C>>, =u> independent of . Then <\equation*> Esup+E\t\E|0|2>+\t+\t. WLOG, =0> (check at home!). Use the substitution <\equation*> v(t,x)=ut,x-\(s)\W. : Ito formula for >. <\equation*> u=u+2a*u*u\s|\>>+f*u\s|\>+C>+g*u\W+g\s. Take expectation, which kills the W> term, giving a bound on <\equation*> E\sE. Take care of the sup, which is outside of the expectation, but needs to be inside. <\equation*> Esup>g*u\W\C*E\t\C*Esup\t\\Esup+C(\)\s. State/signal >: Markov process/chain. Observation =h(X)+g>(t)>. State is not observed directly. The inf about > comes ``only'' from >, t>. Find the best mean-squares estimate of )> given >, t>, where is a known function. This estimator is given by <\eqnarray*> >|>|f(X)\|\.>>>> <\proof> Let > be an >-measurable square-integable function>]\\>, =g(Y)>. <\eqnarray*> -g]>||)-+-g]>>|||)-(X)]+E[-g]>>||>|)-(X)]+2E[(f(Y)-)(-g)]>>|||)-)(-g)\|\]]=0.>>>> Geometric interpretation: conditional expectation, with respect ot the >-algebra > is an orthogonal projection on a space of >-measurable functions. <\eqnarray*> >|>|)\|\]>>|||f(x)P(X\\x\|\).>>>> State: <\eqnarray*> X>||)\t+\(X(t))\W>>|Y>||t+g(Y)\V,>>>> We assume > and > are independent Wiener processes. >, . Further , with E[f(X)]\\>. <\eqnarray*> >||)\|\].>>>> <\equation*> =f(x)u(t,x)\x|u(t,x)\x>, where is a solution of the <\equation*> \u(t,x)=\(x)u(t,x)-(b(x)u(t,x))\t+h(x)u(t,x)\Y, where A>. <\eqnarray*> (A)>||exp-h\s-h\V\P>>|Y>||V.>>>> If we add another term to the state process, <\equation*> \X=b(X)\t+\(X(t))\W+f(X(t))\V, then we get <\equation*> \u(t,x)=\(x)+\u(t,x)-(b(x)u(t,x))\t-(\u(t,x))\Y+h(x)u(t,x)\Y as the corresponding Zakai equation. () Here, we assume that is twice continuously differentiable in and once in . <\equation> >(t,x)=a(x)u,u(0,x)=u(x). \; First, let us talk about generalized functions. Suppose we wanted to find a derivative of . Classically, (0)> does not exist. Let be a differentiable function and > very smooth with compact support. Then <\equation*> f\(x)\x=-f(x)\(x)\x. If is not differentiable, <\equation*> f(x)\(x)\x=-\(x)\(x)\x for all \C>(\)>. Now reconsider the heat equation in a different form, namely <\equation> >(t,x)=(a(x)u),u(0,x)=u(x). A weak general solution of () is a function H(\)> such that for all 0> <\equation*> ||>=|\||>-|\||>\s for every function \C>(\)>. Going back to (), we find that a generalized solution is also a function from > so that <\equation*> ||>=|\||>-|(a\)||>\s for all \C>(\)>. This definition is equivalent to saying that <\equation*> u(t)=u+a*u\s as an equality in >. Let us now consider yet another different equation, namely <\equation> >(t,x)=u(t,x)+sin(u(t,x)),u(t,x)=u(x). Direct differentiation shows <\equation*> u(t,x)=>k(t,x-y)u(y)\y+>k(t-s,x-y)sin(u(s,y))\y\s, where is the heat kernel <\equation*> k(t,x-y)=t>>e|4t>>. Write this now in SPDE form <\equation*> \u(t,x)=a*u+f(u(t,x)). A is a solution that satisfies <\equation*> u(t,x)=>k(t,x-y)u(y)\y+>k(t-s,x-y)f(u(s,y))\y\s. OSDE <\equation*> \X=b(X(t))\t+\(X(t))\W,X=x. Given , >, >, ,P)>, . If and > are Lipschitz-continuous and <\equation*> \|b(x)\|\K(1+\|x\|),\|\(x)\|\K(1+\|x\|)\\!u. shows an OSDE that can't be solved in this way: <\equation*> \X=sign(X)\W. This equation has no solution for fixed ,P)>, . One could find |~>,)>, > such that X=sign(X)\>. The mechanism for this is Girsanov's theorem, by which you can kill the drift and obtain a different equation. If you specify the measure space and the Wiener process, you are looking for a . If you allow yourself the freedom of choosing these as part of your solution, your solution is . Simple Example: (0,b)>, \\>, \1-\>. L(0,b)>. For smooth functions , clearly <\equation*> f|f|H|>=)f|f|H|>=f(X)\x+f\x=:|2>. Let us consider the basis <\equation*> m(x)=sin(k-1)x|b>>, which is an ONS in . Observe <\equation*> \m=(1-\)m=m+(k-1)|b>m=1+(k-1)|b>m. Define <\equation*> \\1+(k-1)|b> as the eigenvalues of > w.r.t. the eigenbasis >. For (-\,\)>, we can construct an arbitrary power of the operator by defining its effect on the eigenbasis > by m\\m>. Further, we may observe <\equation*> f|f|H|>=\f=f|\f||>=|H|>, where <\equation*> f=|H|> are the Fourier coefficients. Then the <\equation*> H(0,b)\f\H:\f|H|2>\\. For 0>, define <\equation*> H(0,b)\\H. We may also define <\equation*> \1>f|\f||>>.1>f|\f||>> The spaces (0,b),s\\}> form the scale of spaces >\H>> if \s>. Properties: Let \s>. Then <\enumerate> >> is dense in >> in the norm |s|>>. > is a Hilbert space =f|\g|0|>>. For 0>, H(0,b)>, H(0,b)>, denote <\equation*> [u,v]\v|\>H>|u|\>H>||>. <\enumerate> If also belongs to , then >. Proof: > is self-adjoint in . <\remark> We will typically work with three elements of the Sobolev scale--the middle, e.g. >, then the space where the solution lives and finally the space that the solution gets mapped to by the operator. <\equation*> |\x>|\>>:H\H. <\definition> The triple of Hilbert spaces )> is called a normal triple if the following conditions hold: <\enumerate> H\V>. The imbeddings H\V> are dense and continuous. > is the space dual to with respect to the scalar product in . Note that we always assume that is identified with its dual. <\example> Any triple >,H,H>> for \0> is a normal triple. <\equation*> \u(t)=(A*u(t)=f(t))\t+>(Mu(t)+g(t))\W,u(0)=u\H. We will assume that V> and :V\H>, and further L(0,T;V)> and \L(0,T;H)>. We further assume and (t)> are >-measurable, and (\)>, (\)>, =H(\)>. <\equation*> A*u=(a(t,x)u>)>+b(t,x)u>+c. <\equation*> Mu=\(t,x)u>+h(t,x)u. We might also want to consider <\equation*> A u=\|\2n>a>\>u,Mu=\|\n>\>\>u. We assume we have a normal triple H\V>. Consider <\equation> \u(t)=(A*u(t)+f(t))\t+(\u(t)+g(t))\W(t), where we assume that > are infinitely many independent Brownian motions, >, V>, :\(t):V\H>, <\equation*> E\|H|2>\t\\, L(\\(0,T));V)>, i.e. <\equation*> E|2>\t\\, \L(\\(0,T);H)> and <\equation*> >E(t)|H|2>\t\\. If is )>, then > is >-adapted, and likewise for >.\ <\eqnarray*> ||,>>|u>||(t,x)u(t,x),>>|||(\),>>|||(\),>>|>||(\).>>>> Saying that \V> is >-adapted means that \\V>, ,\]> is an >-adapted random variable. Consider , which states that <\quote-env> Suppose we have a measure space ,\,P)>. Suppose and are Hilbert spaces. Then <\itemize> ):\\X> is >-measurable iff :f(\)\A\X}\\> is equivalent to <\itemize> )|X|>> is >-measurable for all > where > is a dense subset of . is a solution of () iff for all <\equation*> u(t)=u+(A u(s)+f(s))\s+(\u(s)+g(s))\W(s) with probability 1 in >, that is <\equation*> [u(t),\]=[u,\]+[A u(s)+f(s),\]\s+[\u+g,\]\W(s). If V>, we would have <\equation*> |H|>=|\|H|>+[A u(s)+f(s),\]\s+u+g|\||>\W(s). <\theorem> In addition to the assumptions we already made, assume <\description> (\\0> and \0>, so that <\equation*> \\\0,C\0:2[A\(t),\]+\|H|2>\-\|V|2>+C|H|2>. (``coercivity condition''>superellipticity) |V|>\C|V|>>. Then there is existence and uniqueness for the above equations. That means there is a L(\:C([0,T]);H)\L(\:C([0,T]);V)>, moreover <\equation*> E supT>+E\t\C*E|H|2>+|2>\t+|H|2>\t If >, > , is cont. in > and has one derivative in which is square-integrable. (We might have also used > and >, in which case is cont. in > and has two derivatives which are square-integrable.) Now consider the following fact leading up to the Suppose we have a function L(0,T)> and a generalized derivative (t)\L(0,T)>> is continuous on and <\eqnarray*> ||u(s)\s,>>|>||u(s)u(s)\s.>>>> Proof: Homework. In the infinite-dimensional setting, we have a very analogous statement: <\quote-env> Suppose L([0,T];V)> and (t)\L([0,T];V)>. Then C([0,T];H)> and <\equation*> =2[u(s),u(s)]\s. [Lectures 14-15 not typed, notes available from Prof. Rozovsky] [April 10, 2007, Lototsky, Lecture 16] <\equation*> \u=u\t+g(u)\W(t,x) on x\\> with <\eqnarray*> >||,>>|=u\|>>||>|\|=u\|>>||>>> Two different ways of writing this equation are <\equation*> u|\t>=u|\x>+g(u)W|\t\x> or <\equation*> \u=u\t+>g(u)h\W(t). <\theorem> If \C>>, then C>\C>>. Three kinds of space-time white noise: <\itemize> Brownian sheet ([0,t]\[0,x])> Cylindrical/Brownian motion family of Gaussian random variables =B(h)>, H> a Hilbert space, (h)]=0>, (h)B(g)]=> (s>) Space-time white noise W(t,x)=W|\t\x>=>h(x)\W(t)>, where }> is assumed a Basis of the Hilbert space we're in if ,k\1}> is a complete orthonormal system, then (h),k\1}>-independet standard Brownian motion. If (\)> or (0,\)>, then\ <\equation*> B(h)=W|\x>h(x)\x, and <\equation*> B(x)=B(\)=>h(y)\yW(t)=W(t,x) Consider 1>.\ <\equation*> \u=u\t+>h(x)\W(t), where we assume that <\equation*> h(x)=>>sin(k*x). Observe that, strictly, the series on the RHS diverges in >. Now consider the setting of a Sobolev space <\equation*> H>=H>((0,\)), with <\equation*> |2>=>k>f,f=>f(x)h(x)\x for \\>. Now consider <\equation*> M(t,x)=>h(x)W(t)\H>, i.e. <\equation*> E|2>=t=1>>k>\\ if \-1/2>. <\equation*> u(t)=u+A*u\s+M(t), where <\equation*> A=|\x>:H+1>\H-1>. Then <\equation*> \!u\L(\;L(0,T);H+1>)\L(\;C(0,T);H>) for all \-1/2>, so is almost in > for almost all . We assume a Fourier point of view, so that <\equation*> u(t,x)=>u(t)h(x) and <\equation*> \u=-ku+\W(t). Then <\equation*> u(t)=e(t-s)>\W(s). Next, note\ <\quote-env> : If <\equation*> E\|X(x)-X(y)\|\C\|x-y\| for \>, then C>> for all \0>. Now, consider try to prove its assumption: <\eqnarray*> >||>u(t)(h(x)-h(y))>>|||BDG>>|>>(1-et>)\|h(x)-h(y)\|>>|||(\)>>|)p>.>>>> where we've used the BDG (Burkholder/Davis/Gundy) Inequality, i.e. <\equation*> E[M]\C*E\M\, where is assumed a martingale, which we can achieve by fixing time to \ in the expression for > above. Next, note <\eqnarray*> (t)]>||e(t-s)>\s=>(1-et>),>>>> also quadration variation if we fix time as hinted above. Once we get to )> above, realize that we <\equation*> k-2>\\, and usethe fact that <\equation*> \|h(x)-h(y)\|\\|sin(k*x)-sin(k*y)\|\C(K\|x-y\|)> for -2\-1>, i.e. \1/2>, i.e. =1/2-\>. So altogether, we obtain \C\|x-y\|)p>>. Thus <\equation*> u\C--\>=C>. Our above is ``a solution'' to our SPDE, but not in the variational sense defined so far. So we need a more general idea of what a solution is, to subsume both concepts. If you have a general PDE <\equation*> >=A(t)U, then u>. Then <\equation*> >=A(t)u+f(t) gives us <\equation*> u(t)=\u+\f(s)\s. For example, if we have <\equation*> u|\t>=u, then <\equation*> \:f\G(t,x,y)f(y)\y, where is given by <\equation*> G(t,x,y)=>et>h(x)h(y) if\ <\equation*> \u=u\t+h(x)\W,u=0. Then <\equation*> u(t,x)=>>G(t-s,x,y)h(y)\y\W(s). Now for <\equation*> \u=u\t+g(u)h\W, we write <\eqnarray*> ||>G(t,x,y)u(y)\y+>>G(t-s,x,y)g(u(y))h(y)\y\W(s).>>>> Then you define a to be a solution to the above integral equation. Now try <\eqnarray*> )-u(t,x)\|>|>|G(t-s,x,y)-G(t-s,x,y)h(y)g(u(s,y))\y\W(s)>>||>|>(G(t-s,x,y)-G(t-s,x,y))h(y)g\y\s>>|||>\|G(t-s,x,y)-G(t-s,x,y)\|g(u(x,y))\y\s.>>>> Then came Krylov (1996) and turned this ``hard analysis'' into clever ``soft analysis'' or so. <\initial> <\collection> <\references> <\collection> |1>> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|toc> |math-font-series||Table of contents> |.>>>>|> |math-font-series||1Basic Facts from Stochastic Processes> |.>>>>|> |1.1Lebesgue Integral |.>>>>|> > |1.2Conditional Expectation |.>>>>|> > |1.3Stochastic Processes |.>>>>|> > |1.4Brownian Motion (Wiener Processes) |.>>>>|> > |math-font-series||2The Itô Integral and Formula> |.>>>>|> |2.1The Itô Construction |.>>>>|> > |2.2Itô's Formula |.>>>>|> > |2.2.1Deriving from the Chain Rule |.>>>>|> > |2.2.2SODEs |.>>>>|> > |math-font-series||3Some SPDEs> |.>>>>|> |math-font-series||4PDE/Sobolev Recap> |.>>>>|> |4.1Sobolev Spaces |H>> |.>>>>|> > |4.2SPDEs in Sobolev Spaces |.>>>>|> > |4.2.1Classical Theory |.>>>>|> > |4.2.2Stochastic Theory |.>>>>|> > |math-font-series||5Nonlinear Filtering (``Hidden Markov Models'')> |.>>>>|> |math-font-series||6Solutions of PDEs and SPDEs> |.>>>>|> |6.1Classical Solutions |.>>>>|> > |6.2Generalized Solutions |.>>>>|> > |6.3Mild Solutions |.>>>>|> > |6.4Generalization of the notion of a ``solution'' in SDE |.>>>>|> > |math-font-series||7Existence and Uniqueness> |.>>>>|> |7.1Scales of Sobolev Spaces |.>>>>|> > |7.2Normal triples/Rigged Hilbert space/Gelfand's triple |.>>>>|> > |7.3Actual SPDEs |.>>>>|> > |math-font-series||8Existence and Uniqueness> |.>>>>|>