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Stochastic Calculus and Applications
2nd Edition Samuel N. Cohen
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Stochastic Models for Fractional Calculus 2nd Edition
Chapter 16 provesthatSDEsarewellposedinageneralsetting,with Lipschitzcontinuouscoefficients.Thisisdonebyintroducingthespaces S p and Hp S (thesemimartingaleanalogueofthe Hp space).Variousotherbasic properties,includingstabilityandapproximationschemes,andaclosedform forgenerallinearequationsarepresented.Chapter 17 restrictsourattention toSDEsdrivenbyaBrownianmotionandaPoissonrandommeasure,and considerstheirbasicpropertiesasMarkovprocesses.Thekeyresultisthe generalFeynman–Kactheorem,whichconnectssolutionsofSDEswithsolutionsofcertainpartialintegro-differentialequations.Chapter 18 pushesthis connectionfurtherandoutlineshowmeasurechangetechniquesandsolutions ofmartingaleproblemscanbeusedtoconstructsolutionstoSDEsinanonLipschitzcontinuoussetting.
Part V considersapplicationsofthistheorytoproblemsincontroland filtering.Chapter 20 presentsthesimplecasewhereacontrollerdetermines theratesassociatedwithasinglejumpprocess.Chapter 21 givesthegeneral settingofacontrollerwhocandeterminethedriftandjumpratesofan SDE,byfirstconsideringtheconnectionbetweenBSDEsandthemartingale optimalityprinciple.Appendix A.10 supplementsthesechapters,providing theproofofBeneˇs’extensionofFilippov’simplicitfunctiontheorem,which allowsustoselectmeasurablecontrolsinageneralway.
Chapter 22 concludesbyconsideringaclassicalfilteringproblem,wherea Markovprocess X isobservedonlythroughthedriftofcontinuousprocess Y . ThefilteringequationandZakaiequationarederived,asistheKalmanfilter asaspecialcase.Wealsooutlinethecasewhen X isafinite-stateMarkov chain,andsothefinite-dimensionalWonhamfilterappearsforthestateprocess.Thecalculationofvariousassociatedquantities,whichareimportantfor statisticalcalibration,isalsopresented.
Remark1.0.1. Inthisbook,weadopttheconvention N = {1, 2,...},thatis, 0isnotconsideredanaturalnumber.Wewrite Z+ = N ∪{0} and ∅ forthe emptyset.
Wedenoteby R thesetofextendedrealnumbers,[−∞, ∞].Thishasa naturaltopology,whereintervalsoftheform[−∞, ∞],[−∞,a[, ]a, ∞]and ]a,b[,for a,b ∈ R,generatetheopensets(thatis,theopensetsarearbitrary unionsoftheseintervals).InkeepingwiththeFrenchstyleofnotation,we denoteby[a,b[theinterval {x : a ≤ x<b},andsimilarlyfor]a,b],]a,b[, etc...
1.1BooleanAlgebrasand σ -Algebras
Underlyingthemathematicaltheoryofprobabilityisthetheoryofsets.For muchofanalysisandprobability,akeystructureisgivenbycollectionsof sets,inparticular,bycollectionsofsubsetsofsomeset S
Definition1.1.1. Let S beaset.Acollectionofsubsets Σ of S iscalleda (Boolean)algebra of S (or field ofsubsetsof S )provided (i) ∅∈ Σ , (ii)if A ∈ Σ then Ac := S \ A ∈ Σ , (iii)if m ∈ N and An ∈ Σ for n =1, 2,...,m then m n=1 An ∈ Σ . If,furthermore,(iii)canbestrengthenedto (iii’)if An ∈ Σ for n =1, 2,...,then
=1 An ∈ Σ . then Σ iscalleda σ -algebra (or σ -field)on S .If Σ isa σ -algebraon S ,then thepair (S,Σ ) iscalleda measurablespace.
Remark1.1.2. Thedifferencebetweenanalgebraon S anda σ -algebraon S isthatanalgebraisassumedtobeclosedonlyunderfiniteunions(thatis, theunionofafinitenumberofelementsof Σ willalsobeanelementof Σ ), whereasa σ -algebraisassumedtobeclosedundercountableunions. Neitheranalgebranora σ -algebraisassumedtobeclosedunderuncountableunions.
Remark1.1.3. Clearly,(i)and(ii)implythat S ∈ Σ .Itiseasytoshowthat (ii)and(iii)imply:if An ∈ Σ for n =1, 2,... then ∞ n=1 An ∈ Σ .
(i)Foranyset S ,thetrivial σ -algebra Σ = {∅,S } andthepowerset2S (that is,thesetofallsubsetsof S )areboth σ -algebrason S (ii)If A ⊆ S ,then Σ = {∅,A,Ac ,S } isa σ -algebraon S . (iii)Let I consistofallsetsoftheform {]a,b]: −∞≤ a ≤ b< ∞} or {]a, ∞[: −∞≤ a< ∞} andsuppose ΣI isthecollectionofallfinite (disjoint)unionsofsetsin I.Then ΣI isanalgebraofsubsetsof R (but nota σ -algebra).
Remark1.1.5. Inmanycircumstances,theset S mayonlybeimplicitlyconsidered,asourattentionwillbeonthealgebra Σ .Foranyalgebra Σ onaset S ,itistruethat S = A∈Σ A,sothisdoesnotleadtoconfusion.
Theorem1.1.6. Let G beacollectionofsubsetsofaset S .Thenthereexists asmallest σ -algebraon S whichcontains G .Thisisdenoted σ (G ) andiscalled the σ -algebrageneratedby G
Proof. Let {Σα }α∈A bethecollectionofall σ -algebrason S suchthat G⊂ Σα forevery α ∈A.Thisisnotempty,asitcontains2S .Set Σ = α∈A Σα .By Exercise 1.8.1, Σ isa σ -algebraandanyother σ -algebracontaining G also contains Σ .
Definition1.1.7. A topology onaset S isacollection T ofsubsetsof S satisfying (i) ∅∈T and S ∈T , (ii)if Oα ∈T ,forall α insome(possiblyuncountable)indexset A,then α∈A Oα ∈T , (iii)if m ∈ N and {On }m n=1 ⊂T ,then m n=1 On ∈T .
Thepair (S, T ) iscalleda topologicalspace andtheelementsof T arecalled theopensubsetsin (S, T ).
Example1.1.8. For S = R,theclassical‘open’sets,thatis,setsthatcanbe writtenasanarbitraryunionofopenintervals]a,b[,formatopologyof R
Definition1.1.9. Let S beatopologicalspacewithtopology T .The Borel σalgebra,denoted B (S ),isthe σ -algebrageneratedbytheopensetsin T ;that is, B (S ) isthesmallest σ -algebrathatcontains T .Theelementsof B (S ) are calledthe Borelsets of S
Weuse B (R) todenotetheBorel σ -algebrageneratedbythetopologyconsistingofallunionsofopenintervalsin R
Remark1.1.10. Wenotethat,followingthenotationofExample 1.1.4, σ (ΣI )= B (R).Itislefttothereadertofillinthedetails(Exercise 1.8.2).
Itisimportanttonotethatthereareusuallymanypossible σ -algebrason aset S .
Definition1.1.11. Let Σn beacollectionof σ -algebrasonaset S .Then wedefine n Σn := σ ( n Σn ),thatis, n Σn isthesmallest σ -algebrawith Σm ⊆ n Σn forall m.As n Σn issimplyacollectionofsubsetsof S , n Σn existsbyTheorem 1.1.6
Definition1.1.12. Afamilyofsets M issaidtobea monotoneclass if A ∈M whenever {An }n∈N isasequenceofsetsin M witheither
(i) An ⊆ An+1 and ∞ n=0 An = A,or (ii) An+1 ⊆ An and ∞ n=0 An = A.
Lemma1.1.13. Let K beafamilyofsets.Thenthereisasmallestmonotone classcontaining K.
Proof. Let M denotethecollectionofallmonotoneclassescontaining K.As M containsthepowersetof S = K,weknowthat K⊆ M,anditiseasy toverifythat M isamonotoneclass.Hence M isthesmallestmonotone classcontaining K
Theorem1.1.14(MonotoneClassTheorem). Let S beaset,and N analgebraofsubsetsof S (butnotnecessarilya σ -algebra).Suppose M is amonotoneclassofsubsetsof S whichcontains N .Then M containsthe σ -algebra σ (N ).Furthermore, σ (N ) isthesmallestmonotoneclasscontaining N
MA = B ∈ m(N ): A ∩ B,A ∪ B and A \ B ∈ m(N ) ⊆ m(N ).
Bydirectcalculation,wecanseethat MA isamonotoneclassforany A.As N isaBooleanalgebra, N⊆ MA forany A ∈N .As m(N )isthesmallest monotoneclasscontaining N ,itfollowsthat MA = m(N )forany A ∈N Therefore,weknowthatforany A ∈N andany B ∈ m(N ),thesets A ∩ B , A ∪ B and A \ B areallin m(N ).Thisimplies N⊆ MB ,andagainby minimalityof m(N )weknow m(N )= MB forall B ∈ m(N ).Itfollowsthat m(N )isaBooleanalgebra,andisclosedundercountableunions(asitisa monotoneclass),andishencea σ -algebra.Byminimalityof σ (N ),itfollows that σ (N ) ⊆ m(N ),asdesired.
Remark1.1.15. Atypicalapplicationofthisresultistoconsiderasimplealgebra Σ (forexample,theintervalsof R),andtodefine M tobethecollection ofsetsin σ (Σ )wheresomepropertyholds.Ifweshowthat (i)thealgebra Σ liesin M, (ii)limitsofmonotonesequencesin M liein M, thenthemonotoneclasstheoremallowsustoconcludethat M containsall of σ (Σ ).SeetheproofofTheorem 1.4.5 foranexampleofsuchanargument.
Definition1.1.16. Acollection N ofsetsiscalleda λ-system(or d-system) on S if (i) S ∈N , (ii)Forany A,B ∈N with A ⊆ B , B \ A ∈N ,
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money long since due.’ This was in 1658, but in March of the following year they wrote bitterly to the Council that, while such large debts were contracted and they were struggling with difficulties, it made them ‘exceeding unhappy’ to see that even their assignments on the customs were not handed over to them in full.[1482] In May 1659, among other items, £330,000 was owing for seamen’s and £43,000 for dockyard wages, and the £735 a week paid by the Navy Treasurer to the Savoy and Ely House hospitals was six months over-due.[1483] In September the army commissioners were directed to hand over £60,000 for naval purposes, although the soldiers’ pay was months in arrear When the Commonwealth accounts close on 7th July 1660 the debt was £1,056,000.[1484] For this large sum every year from 1640 furnished its quota, thus detailed:—1640-9, £10,200; 1650, £71,000; 1651, £25,000;1652, £16,000; 1653, £11,000; 1654, £5000; 1655, £50,000; 1656, £229,000; 1657, £218,000; 1660, £421,000. That the earlier amounts were not merely book debts carried forward for want of claimants is shown by the existence of a petition, of April 1658, begging for the settlement of a bill for freight incurred between 1643 and 1651.[1485] These liabilities, belonging to only one branch of the public service, help to explain why many classes of society, not actively royalist, may have welcomed a restorationwhichpromisedasettlementof debtsand a morestable financialsystem.
When the St George’s cross was made the national flag in February 1648-9, it was also ordered that an escutcheon should be carried on the stern of each man-of-war, containing a red cross in one compartment and a harp in another In 1653 the three Generals at sea used, besides their standards, a pendant of red, white, or blue, at the main, and their vice- and rear-admirals their respective colours at the fore and mizen. From 18th May 1658 the standard of the General of the fleet was to bear the arms of England, Scotland, and Ireland, ‘with his Highnes’ escutcheon of pretence according to the great seal of England.’ The jack flag for admirals was to consist of the arms of England and Scotland united, ‘according to the ancient form,’ with the harp added, ‘accordingtoamodelnowshown.’[1486] All saluting, whether from ships or forts, was strictly forbidden in 1652, except in honour of ambassadors; but the salute to the flag from foreigners was firmly upheld under all circumstances. By the treaty of 5th April 1654, the Dutch formally acknowledged the English right to the salute in the ‘British seas.’ In 1657 Opdam, with thirty Dutch sail, passing Dover struck his flag and saluted the castle; shortly afterwards he met the Dragon and the Colchester, whose captains ordered him again to strike. He refused, saying that he was not expected to pay this mark of respect to every ship he met, whereupon they replied that if he did not they would engage him till they sank alongside. Then ‘he struck in a great rage,’ and kept his flag down till out of sight of the Englishmen. Man-of-war captains sometimes displayed the same feeling of pride in their position at the expense of English ships. In 1654 a Virginiaman was run down and sunk in the Channel by the Ruby In the subsequent inquiry the master of the merchantman held that the Ruby should have gone astern of his vessel, to which her captain retorted by asking, ‘How many men-of-war have you known go under a merchantman’sstern?’
The prices of naval stores varied greatly, according to the confidence felt in the treasury and conditions of peace or war; the following are the rates for someoftheprincipalarticles:—
1655,£10, 12salast 1656,£12a last 1657,£12, 10salast 1658,£13a last
P
1654,£1,16sabarrel 1655,£15, 5salast
B
1654,£1,15satun 1659,£2,5s ”
S [1488] D 1656,12s6d each 1659,14s ”
O D 1657,£4,3sper 100of six score
W O 1659,£26, 15saton
E T 1658,£2,3sper cwt.
L V for blocks 1656,£6,15saton
Examples of that incongruity of expression usually associated with Puritan fervour are not frequent among the Navy papers, but they do occasionally occur On one occasion Lawson writes, ‘All that look towards Zion should hold Christian communion—we have all the guns aboard.’ Major Robert Sedgwick, starting for the West Indies, asks the Navy Commissioners, after official details, for ‘your prayers that we may be sent out with a blessing and be a blessing where we go.’ Major Sedgwick’s duties were to kill Spaniards, plunder their property, and annex their territory These men were too grimly earnest in the work they set their hands to do to trouble themselves about fine phrases. They lacked humour, and the court of Charles II was, we are taught, very witty; but when, in 1667, the roar of foreign guns was, for the only time in English history, heard in London, even that majority which always loves a royal jest must havebeguntoappreciatethedistinctionunderlyingStewartwit and Puritan dulness.
APPENDIX A
CHAPTER HOUSE BOOK V . XIII
Here ensuyth An Inventorie or boke of All such Stuff, tacle, apparell, Ordynaunce, Artillarie and habillamentes for the warre as Remayned in our soveraigne lord the kynges shippes the xxvij day of July the vjᵗʰ yere[1489] of his reign, By a vewe taken by Sir Henry Wyat, Sir Andrewe Wyndsore, knightes, George Dalyson, and Thomas Tamworth, commissioners in that behalf appoynted, Whuch Stuff, tacle, apparell, Ordynaunce, Artillaries, and habillamentes for the warre Was delyvered into the charge and kepyng of severall persons hereaftyr particlerly named to our seid soveraigne lord the kynges use by Indentures thereof made and also billes signed with the handes of the seid commissioners in the custodie of the seid persones remaynyng, That is to Sey
The kynges Shippe called the Henry Grace de Dewe:—Stuff, Tacle, and apparell of the seid shippe delyvered by the seid Commissioners into the charge of John Hopton by Indenture, that is to sey
ffyrst the foremast of the seid shippe j Shrowdes to the same xvj
Dedemens hyes[1490] to the same xvj
Tacles to the foremast iiij
Doble polles[1491] with Shyvers[1492] of Brasse iiij
Single polles with Shyvers of Brasse iij
Single polles with a colk[1493] of Brasse j
Swyfters to the foremast vj Doble polles with colkes of Brasse iij polles whuth Shyvers of wode iij polles with v colkes of Brasse and oone of wode vj
Garnettes to the foremast with iiij poles[1494] ij
Garnet with ij polles and shyvers of Brasse j
Garnet with a shever of Brasse and another of tymbre j
Trusses to the foremast ij
Drynges[1495] to the same j
Doble polles for the trusses with colkes of brasse ij
Single poles of tymbre ij
Drynges with a doble pole with a colk of brasse and oone single pole of wode j halyers to the foremast ij
Shyvers of Brasse to the brest[1496] of the forecastell iij
Ramehedes with ij shevers of Brasse j
Shetes to the foresayle ij
pollies with shevers of Brasse to the same ij lyftes to the foresayle ij
Doble polies with shyvers of Brasse to the same ij
Single polies with colkes of Brasse ij
Shetes to the toppe Sayle ij
Single polies with woden pynnes to the same ij
Tackes to the foresayle ij
Stodynges[1497] to the foreyerd ij pollies to the same with woden pynnes ij cranelynnes to the foremast j
Single poles with shyver of Brasse j
Bowelynnes to the foreyerd with the poleis and dedemanes hies and oone doble pole with a shever of brasse j
Stayes to the foreyerd with iiij dedemens heies ij
Sprete sayle yerdes ij halyers to the same ij
Single poleis with shyvers of Brasse to the same ij lyftes to the Sprete Sayle with iij single polies and woden pynnes j
Grapilles[1498] with the cheyne hangyng apon the bowspret with a pole havyng a colk of brasse j
knyghtes[1499] longyng to the lyftes of the foresayle with ij shevers of brasse ij
The fore topmast j
Shrowdes to the same xij
halyers with a doble polie and a colk of brasse ij single poleis with woden pynnes ij
Bowlynes to the foretop Sayle yerd with pawes[1500] and dedemens hyes to the same ij
Brasses[1501] for the foretop sayle yerd ij Single poles with pynnes of wode ij
lyftes to the foretopsayle yerd with iiij poleis with wooden pynnes ij
Shetes to the foretopsayle with ij woden poles ij
Steyes to the foretopmast j
Sayle yerdes to the foretop j
Toppe Galant apon the foretopmast j mastes to the same j
Shrowdes to the same viij
halyers with ij single poles with woden pynnes ij
Brasses to the same with ij single poleis and wodepynnes and dedemens hyes to the same ij
Bowlynes to the topgalant yerd the power and dedemens hies to the same ij lyftes to the foretopgalant yerd with iiij single polies with woden pynnes ij
Shetes with ij single poles with woden pynnes ij
Stayes to the foretopgalant mast j
Shevers of Brasse for the cattes in the forecastell iiij
Davettes[1502] with iiij shevers of Brasse ij
Smale davettes with oone shever of Brasse j
The mayne mast[1503] j
Shrowdes with cheynes of yron and dedemenes hies to the same xl
Bote tacles of sterebord syde with iiij doble poles and viii single poleis with xvj shyvers of Brasse[1504] iiij
Swifters on the same syde with vij doble poleis and vii single polees with colkes of Brasse and ij poles of tymbes[1505] pynnes viij
Garnettes with ij single poles with shivers of Brasse j
Garnettes with ij single polies with colkes of Brasse j
Garnettes with oone single pole with a shever of Brasse and an other pole with a colk of Brasse j
Stodynges with a single polie with a Shever of Brasse j
Bote tacles oon ladbord syde with iiij doble polies and viij single polies with xvj Shevers of Brasse iiij
Bretayn tacles[1506] with ij single polies and Shevers of Brasse to the same j
Swyfters with vij doble polies with colkes of Brasse and viij single poles with colkes of Brasse viij
Garnettes whereof oone with ij single polies and ij shevers of Brasse an other with ij single poleis with ij colkes of Brasse and an other with a shever of Brasse iij
Stodynges with a shever of Brasse j
tymber polies for the Shuts[1507] ij
The mayne yerde with the mayne parell j
Single poleis with a shever of Brasse to wynde up the mayne parell j
Trusses with iiij doble polleis and iiij single polies with xij shevers of Brasse iij
Drynges with ij doble polies and iiij shevers of Brasse ij
Single poleis of tymbre to the same ij
Tyes j payer
Whele Ropes[1508] j
Geers with vj single poleis whereof iiij with shevers of Brasse and ij of tymbre iij
knyghtes belonging to the same with iij Shevers of Brasse iij
Single poles for the topsayle iiij
Shutes with iiij shevers of Brasse ij
knyghtes with ij shevers of Brasse ij
The mayne yerd j
lyftes with ij doble poleis and ij single with vj
Shevers of Brasse to the same ij
Knyghtes with ij Shevers of Brasse ij
Shutes ij
Tackes ij
bowlynes with Brydelles and Dedemens hies ij
poleis to the mayne Bowlyne with ij Shevers of Brasse j
mayne Stayes with viij dedemens hies iiij
Brasses with ij single poles and colkes of Brasse ij
The mayne top j
The mayne top mast and a coler of yron j
Shrowdes to the same with dedemens hies xiiij
The mayne top Sayle yerd j
Tyes j
halyers with a doble and a single polie with ij shevers of Brasse j
Brases with iiij poles ij
lyftes with iiij polies and colkes of Brasse ij
Cranelynnes with a single pole and a colk of Brasse j
Steyes to the mayne top mast j
bowlynes with dedemens hies ij
The top Galant apon the mayne topmast j mastes for the same j
Rynges of yron for the same j
Shrowdes to the same with dedemens hies x
Sayle yerdes to the same j
Stayes to the same j
Bowlynes ij
Brases with ij poles to the same ij
Shutes ij
Grabulles with cheynes to the same ij
poleys apon the mayne yerd for the grabulles ij
Spare knyghtes standyng by the mast with ij shevers of Brasse ij
The mayne meson mast j
Shrowdes with xj doble poles and xj single poles, a doble and single polee with colkes of Brasse xij
Swyftyers with vj doble poles and vj single poles with colkes of Brasse vj
Tacles with ij doble poles of tymbre ij
Single poles oone of tymbre the other with a colk of Brasse ij
Steyes j
Shutes j
Single poles oon of tre[1509] the other with a colke of Brasse for the same Shutes ij
cranelynes with a single polie and a colk of Brasse j
Brases with ij single poles ij
Teyes[1510] ij
halyers ij
The Rame hede j
knyghtes with iij Shevers of Brasse j
The yerd to the meson Sayle j
lyftes with iij poles and dedemens hies j
Trusses with a double and a single polie with colkes of Brasse j
Toppe j
Topmast to the same j
Rynges of yron j
Shrowdes with dedemens hies x
The Sayle yerd j
Tyes j poles to the same ij
lyftes with iij poles and dedemens hies j
The top Galant of the mayne meson j
The mast to the same j
Shrowdes to the same vj
lyftes with iij poleis and dedemens hies j
The Sayle yerd j
Tyes to the same j halyers j
The boneaventure mast j
Shrowdes with x Doble poles and x syngle poleis x
Sayle yerdes j
Tyes j halyers with a doble pole ij knyghtes with iiij Shevers of Brasse j
Shutes with ij poleis to the same j
The boneaventure top j mastes to the same j
Sayle yerdes j
Shrowdes viij
Steyes j
In the storehouse of the Shipp viij single pendaunt polies with shevers of Brasse viij
Smale single garnet poleis with shevers of Brasse j
Doble lyft poleis with shevers of Brasse iiij
Doble poleanker[1511] poleis with shevers of Brasse iiii
Snach polleis with gret Shevers of Brasse iiij
Single poleis with Shevers of Wode xiiij
Doble poleis with Shevers of Wode ij
Doble poleis with a colk of Brasse j
Single poleis with a colk of Brasse j pottes called piche pottes j ketilles to melt in pyche j boyes for ankers x boy Ropes x
Shevers of Brasse without poleis iij leddern[1512] bokettes xij dossen love[1513] hokes iiij lynch[1514] hokes iij
Copper ketill not sett in furnes weying by estimacon ccc[1515] j xiij ynch compas j
xvij ynch compas ij
xv ynch compas ij
ix ynch compas j
viij ynch compas j vij
iiij ynch compas iiij
vj ynch compas iij
vj ynch di[1516] compas j
v ynch compas j
viij ynch compas j
iiij ynch compas j
iij ynch compas j
v ynch compas j
iiij ynch compas vij
iij ynch compas j
iij ynch di compas j xxij
Smale lyne ij peces
Bygger lyne for lanyers[1517] ij peces
Brayle Ropes with iij poles to the same j
Grete doble Blockes ether of them ij Shyvers of Brasse ij
Single blokes with ij Shevers of Brasse ij
long Ores for the Grete bote lx
Tarre ij barelles
Ores for the Cocke bote xxiij
Standart Staves[1518] lix
Stremers viij
lytle flagges c
Top Armours vii
Targettes xx dossen large fflagges lx
To the mayne Sayle Acorse[1519] and ij bonettes doble j mayne sayle mayne topsayles j
Topgalant Sayle j
The meson Sayle j
The boneaventure Sayle j
The foresayle Acorse and a bonet doble and bonet single an other corse and iij bonettes single in all ij foresayles
The fore topsayle j
The foretopgalant Sayle j
The Bowspret Seyle j
The mayne Sayle for the gret Bote, a corse and ij bonettes single j sayll
The foreseyle acorse and ij bonettes single j
Top Seyle j
The meson Seyle j
The boneaventure Sayle j
An old corse of a hulk Sayle j
Sterbord bowers ij ladbord bowers ij
Destrelles[1520] of Sterbord ij
Destrelles on ladbord ij
Shot[1521] ankers j
Caggers[1522] j
Spare ankers ix xix
Trene[1523] platters iiij dossen
Trene cuppes v dossen
Tankerdes iij dossen
lantrons[1524] vj grete lantrons j middellantrons ij
Copper ketilles in furnes iij lede in oone pece by estimacon [1525]
Grete belles in the seid Ship of Brasse j
The grete botes mayne mast j
Shrowdes to the same xiiij
polles to the same xxviij
Tacles oone with a doble pole and colkes of Brasse the other with a single pole and a Shever of tymbre ij
Single poles with a shever of Brasse j mayne yerdis and the parell j
Trusses with ij poleis and Shevers of tymbre j
Tyes j
halyers with a doble pole and Shever of Brasse j
Single poleis on of them with a Shever of brasse and other of tymbre ij
Shutes ij
Tackes ij
bowlynes with a pole and Shever of tymbre ij
lyftes with ij Single poleis ij
Topsayle Shotes with ij single poleis ij yerde Ropes ij
The meyne Stey with ij doble poleis j
The toppe j
The topmast j
Shrowdes to the same vj
Sayle yerdes j
Tyes j
parell to the sayle yerd j
Bowlynes ij
lyftes ij
Cranelynes j
Brases ij
The foremast j
Shrowdes to the same vj
The Sayle yerd j
The parell j
Teyes j
Syngle halyers with a polie to the same j
Shetes vj tackes j lyftes with ij poleys ij
Steyes j
bowlynes with a polie j
Single Trusses with a polie j
Bowspretes j mayne meson mast j
Shrowdes to the same vj
The Sayle yerd j the parell to the same j
The Tye j
Single halyers with a pole j
Trusses with ij poles j lyftes with iij poles j
Brases with ij poles ij
Steys with ij Smale poles j
The boneaventure mast j
Shrowdes to the same iiij
Tyes j
Single halyers with oone pole j
The sayle yerd j
The parell to the same j
Ankers for the said bote iij
Cablettes of v ynch compas ij
Cocke bote j mastes to the same j
Sayle yerdes j
Shevers of Brasse ij
Ores to the same xij
bote hokes j
The skyff otherwise called Jolywatt j mastes to the same j
Sayles j
Ores to the same vj
Shevers of Brasse j
Shevers of Brasse called a Wyndyng Shever for the j Rame hede j hawsers of v ynch compas j hawsers of vj ynch di compas di hawser[1526]
hawsers of v ynch compas iij
Cables of ix ynch compas j hawsers of vj ynch compas di hawser
Soundyng ledes vj
Ordynaunce Artillarie and habillamentes for warr delyvered to the charge and custodie of Thomas Spert, master, and William Bonython, purser of the seid shipp by Indenture as aforeseid, that is to sey
Serpentynes of yron with miches[1527] boltes and forelockes cxxij
Chambers to the same ccxliiij
Stone gonnes of yron Apon trotill wheles and all other Apparell iiij
Chambers to the same iiij
Serpentynes of Brasse apon wheles shod with yron iij
Serpentynes of Brasse apon wheles unshodd j
Grete peces of yron of oon makyng and bygnes xij
Chambers to the same xxiiij
Grete yron gonnes of oone sort that come owt of fflaunders with myches bolts and forelockes iiij Chambers to the same viij
Grete Spanysh peces of yron of oone sorte ij
Chambers to the same iiij
Stone gonnes apon Trotill wheles with miches boltes and forelockes to the same xviij Chambers to the same xxxiiij
Smale vice peces of Brasse apon shodd wheles of Symondes makyng j long vice peces of Brasse of the same makyng iij ffawcons of Brasse apon Trotill wheles vj a fayre pece of Brasse of Arragows makyng j A Slyng of yron Apon Trotill wheles j Chambers to the same with other apparell j grete Stone gonnes of yron ij chambers to the same iiij
Grete culverynes of Brasse apon unshodd wheles of Symondes makyng ij
Grete bumberdes of Brasse apon iiij trotill wheles of herberd[1528] makyng j
Grete curtalles of Brasse apon iiij wheles and of the same makyng[1529] j hakebusshes of yron hole clxxxxiij hakbusshes of yron broken vjj
Shott of yron of Dyverse Sortes clx shott
Stone Shott of Dyverse Sortes in the balist of the ship A grete nomber not told
In the Grete Bote of the seid ship Remaynyng fyrst
Serpentynes of yron with myches boltes and forelockes viij
Chambers to the same xxv
Serpentynes of Brasse apon shodd wheles j ffawcons of Brasse apon Shodd wheles ij
In the Storehouse of the shipp
Bowes of Ewe cxxiiij chestes for the same ij hole chestes of Arrowes iij
Billys cxliiij moryspykes lxxx
Backes and Brestes of Almyne Ryvettes of ether cc
Splentes[1530] clxxxxviii payer
Salettes[1531] cc
Standardes of mayle cc
chargyng ladylles for Gonnes with staves vj staves withowt ladelles viij
Spare miches for Gonnes xiiij
Spare boltes ij
Javelyns ix dossen
Dartes lvij dossen
hamers for Gonnes xiiij
Crowes of yron iiij
Stokepykes of yron xiiij
APPENDIX B
THE MUTINY OF THE GOLDEN LION
On the 19th April 1587, Drake with the Bonaventure, Lion, Dreadnought, Rainbow, and Spy, of the Queen’s, and some twenty armed merchantmen attacked Cadiz, with results disastrous to Spain. Borough was vice-admiral and in command of the Lion. The fleet left Cadiz harbour on 21st April, and on the 30th Borough addressed a long and vigorously worded letter to Drake[1532] protesting that the councils of war called were only nominal consultations where the admiral declared his will, or else merely entertained his visitors who departed ‘without any consultacyon or counsell holden.’ Drake’s answer was to supersede him. All we know further is that on 27th May the Lion’s company put their new captain, Marchant, on the Spy, and sailed away for England with Borough who afterwards declared that he was in daily fear of his life, and therefore had no great reason to try and stop their action. If Borough did not incite them to mutiny the men of the Lion must have been for some time full of discontent and ready to desert. The chase of the Bark of Lyme, which took them from under the guns of the rest of the fleet, gave them their opportunity. On 30th May, Drake constituted a court-martial on the Bonaventure, of himself and the other superior officers, at which most of the mutineers were condemned to death in their absence. The account of this inquiry gives a vivid picture of the modes of thought among the men, and their ideas of their rights and duties.
Although time has settled the historical perspective in which we view Drake and Borough, it must be said for the latter that, in 1587, the admiral was only to him, one of half-a-dozen great seamen with whom Borough, and doubtless his contemporaries, thought he could claim equality. He was an experienced commander and one of the four Principal Officers of the Navy; he was, here, second in
command to Drake, and it was contrary to all the traditions of the service that the admiral should undertake any enterprise without the advice and consent of his captains. In this matter Drake was one of the first expedition leaders to strike out a line of his own, and Borough, tenacious of custom and what he considered his rights, at once came into collision with him. It was long before Drake’s principle of accepting sole responsibility was generally followed. In a private note of farewell to Burghley in 1596, and perhaps with this incident in his mind, Howard, when leaving for the Cadiz voyage wrote,
‘I have no meaning to ronne any rash or unadvysed course nor to settell any thyng for Her Maiesties servyce upon my own jugment but to yeald to those that shall show best reson.’[1533]
After their return an inquiry was held at which the vice-admiral was charged with neglect of duty at Cadiz.[1534] No actual result followed, but Borough came off with the honours of war since he was not disgraced, and remained one of the chief Officers of the Navy. Burghley appears to have been on his side, and Borough wrote subsequently an effusive letter thanking him ardently for his support. [1535] From one passage in this letter in which he says that he had hoped that after the inquiry his innocence would be proclaimed, but that ‘I have suppressed my greefe in respect of the comandment and charge given me,’ it may be inferred that the finding was actually favourable to Borough but not made public, perhaps from a desire not to offend Drake. One other point is worth noticing: if the crew of the Lion voiced the general feeling among English seamen, Drake was certainly not loved by them.
ADD. MSS., 12,505, f., 241.[1536]
A generall courte holden for the service of her Maᵗⁱᵉ abourde the Elizabeth Bonaventure the xxxth day of Maye before Sir Ffrauncis Drake, knighte, generall of Her Maᵗⁱᵉˢ
fleete; Thomas Fennard, Vice-Admirall; Anthony Plotte, Leivetenant-generall; John Marchant, serjant-major, and the reste of the captaines and masters of the fleete as followeth,
The generall, att this courte, called in question and judiciallye demaunded of Captayne Merchaunt howe he colde discharge himselfe to answere the departure of Her Maᵗⁱᵉˢ shippe the Golden Lyon which he latelye gave him in charge?
Captayne Marchaunt protestinge, with all earnest affeccon, his innocencye alledged and declared,—That there was a great Mutynie growen amonge the Company of the Lyon the 27 of this month; as sone as we had given over the chase undertaken, understandinge that she was the Barke of Lyme, [1537] when I requyred the Master that we mighte lye close by the wynde to recover our generall, the Master answered, ‘Well, Captaine, we will.’ But presentely one of the quartermasters came and delivered me a lettere in the behalfe of the whole company as followeth:—
‘Captayne Marchaunt, Captayne of the Golden Lyon appoynted by Sir Ffrauncis Drake, generall of this fleete,— Wee, the Quenes, and yours at this tyme desyre that, as you are a man and beare the name of a captayne over us, so to weighe of us like men, and lett us not be spoyled for wante of foode, for our allowaunce is so smale we are not able to lyve any longer of it; for when as three or foure were wonte to take a charge in hande, nowe tenne at the leaste, by reason of our weake victuallinge and filthie drinck, is scarce able to discharge it, and yet growe rather weaker and weaker; which suerly if it be not loked into, will growe to greate dishonour on your parte, and to a lastinge shame on our sydes, by reason of the moste worthie and the moste honorable challendge of our generall at Caste Calleys[1538] in daringe the kinges deputie, or the kinge himselfe if he were in place, or the proudest champyon he had to come fourthe and chaunge a Bullett with him; but none durste once adventure to come
